thermal transport from first principles Stefano Baroni Scuola - - PowerPoint PPT Presentation

thermal transport from first principles
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thermal transport from first principles Stefano Baroni Scuola - - PowerPoint PPT Presentation

thermal transport from first principles Stefano Baroni Scuola Internazionale Superiore di Studi AvanzaF, Trieste lectures given at the ICTP Caribbean School on Materials for Clean Energy , Cartagena de Indias, May 30 June 5, 2019 syllabus


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SLIDE 1

lectures given at the ICTP Caribbean School on Materials for Clean Energy, Cartagena de Indias, May 30 — June 5, 2019

thermal transport from first principles

Stefano Baroni

Scuola Internazionale Superiore di Studi AvanzaF, Trieste

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SLIDE 2

syllabus

  • What thermal transport is all about;
  • Conserved quanFFes: conFnuity and consFtuFve equaFons;
  • Onsager relaFons: thermal and electric conducFviFes, thermoelectric

coefficients;

  • The Green-Kubo & Einstein-Helfand formulas for transport coefficients;
  • The classical energy current;
  • Gauge invariance of transport coefficients;
  • Density-funcFonal theory of adiabaFc heat transport;
  • SeparaFng wheat from chaff: the Wiener-Khintchine theorem and

cepstral analysis

  • Heat conducFvity from laYce dynamics: crystals and glasses;
  • LaYce dynamics from density-funcFonal perturbaFon theory (if Fme

allows).

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SLIDE 3

what heat transport is all about

T2 T1

Q

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SLIDE 4

what heat transport is all about

T2 T1

Q

heat flows from the warm to the cool as Fme flows from the past to the future

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SLIDE 5

what heat transport is all about

T2 T1

Q

1 S dQ dt = −(T2 − T1)

  • S
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SLIDE 6

what heat transport is all about

T2 T1

Q

1 S dQ dt = −(T2 − T1)

  • S

jQ(r, t) = κT(r, t)

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SLIDE 7

what heat transport is all about

T2 T1

Q

∂T ∂t = κ ρcp ∆T 1 S dQ dt = −(T2 − T1)

  • S

jQ(r, t) = κT(r, t)

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SLIDE 8

why should we care?

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SLIDE 9

energy saving

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SLIDE 10

heat dissipaFon

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SLIDE 11

heat shielding

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SLIDE 12

heat shielding

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SLIDE 13

energy conversion

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SLIDE 14

planetary sciences

Uranus & Neptune

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SLIDE 15

why should we care?

… energy saving and heat dissipaFon heat shielding energy conversion earth and planetary sciences …

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SLIDE 16

why should we care?

… because it is important and sFll poorly understood

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SLIDE 17

E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2]

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extensive properFes

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SLIDE 18

E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2]

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extensive properFes

E[Ω] =

(r)dr

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SLIDE 19

dE(Ω, t) dt = −

  • ∂Ω

j(r, t)·ˆ n dS +

σ(r, t)dΩ

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conservaFon laws

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∂Ω

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ˆ n

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SLIDE 20

dE(Ω, t) dt = −

  • ∂Ω

j(r, t)·ˆ n dS +

σ(r, t)dΩ

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conservaFon laws

˙ (r, t)dΩ =

· j(r, t)dΩ

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∂Ω

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ˆ n

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slide-21
SLIDE 21

dE(Ω, t) dt = −

  • ∂Ω

j(r, t)·ˆ n dS +

σ(r, t)dΩ

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conservaFon laws

˙ (r, t)dΩ =

· j(r, t)dΩ

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∂Ω

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ˆ n

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continuity equation ˙ (r, t) + · j(r, t) = 0

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slide-22
SLIDE 22

adiabaFc decoupling of conserved densiFes

˙ (r, t) + · j(r, t) = 0

<latexit sha1_base64="wgHQ/yxv482WmNPQ/lehZPhCKc=">ACDnicbVC7SgNBFL0bXzG+Vi1tBoMQUcJuFLRAoJYRjAPcEOYmcwmY2YfzMwKIeQf/BAbGzuxtbYT/Rhn4zZJPNXhnHO591wSC6043xZuYXFpeWV/GphbX1jc8ve3moKJGU1WkItkiWDHBQ1bXAvWiXDARGsSQZXqd98ZFLxKLzTw5i1A9wLuc8p1kbq2NfI60YaeSxWXEQhKnkB1n3iI3msD9ER8kJMBEYenaQy72EmduF07KJTdiZA8TNSBEy1Dr2i9lLk4CFmgqs1AhLzalg4KXKBZjOsA9NqIBkbzX19MqDpQaBmSMDtIj1KyXiv9594n2z9sjHsaJZiE1EeP5iUA6QulrUJdLRrUYGoKp5OYeRPtYqrNAwsF09GdbTRPGpWye1Ku3J4Wq5dZ2zswT6UwIUzqMIN1KAOFJ7hE7hx3qyXq036/0vmrOymV2YgvXxC7dmV4=</latexit>

˙ ˜ (k, t) = k ·˜ (k, t)

<latexit sha1_base64="pHzR0Evo+6LDEThSgi+oOGqwbI=">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</latexit>
slide-23
SLIDE 23

adiabaFc decoupling of conserved densiFes

˙ (r, t) + · j(r, t) = 0

<latexit sha1_base64="wgHQ/yxv482WmNPQ/lehZPhCKc=">ACDnicbVC7SgNBFL0bXzG+Vi1tBoMQUcJuFLRAoJYRjAPcEOYmcwmY2YfzMwKIeQf/BAbGzuxtbYT/Rhn4zZJPNXhnHO591wSC6043xZuYXFpeWV/GphbX1jc8ve3moKJGU1WkItkiWDHBQ1bXAvWiXDARGsSQZXqd98ZFLxKLzTw5i1A9wLuc8p1kbq2NfI60YaeSxWXEQhKnkB1n3iI3msD9ER8kJMBEYenaQy72EmduF07KJTdiZA8TNSBEy1Dr2i9lLk4CFmgqs1AhLzalg4KXKBZjOsA9NqIBkbzX19MqDpQaBmSMDtIj1KyXiv9594n2z9sjHsaJZiE1EeP5iUA6QulrUJdLRrUYGoKp5OYeRPtYqrNAwsF09GdbTRPGpWye1Ku3J4Wq5dZ2zswT6UwIUzqMIN1KAOFJ7hE7hx3qyXq036/0vmrOymV2YgvXxC7dmV4=</latexit>

˙ ˜ (k, t) = k ·˜ (k, t)

<latexit sha1_base64="pHzR0Evo+6LDEThSgi+oOGqwbI=">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</latexit>

the smaller the wavevector, the slower the dynamics

slide-24
SLIDE 24

adiabaFc decoupling of conserved densiFes

˙ (r, t) + · j(r, t) = 0

<latexit sha1_base64="wgHQ/yxv482WmNPQ/lehZPhCKc=">ACDnicbVC7SgNBFL0bXzG+Vi1tBoMQUcJuFLRAoJYRjAPcEOYmcwmY2YfzMwKIeQf/BAbGzuxtbYT/Rhn4zZJPNXhnHO591wSC6043xZuYXFpeWV/GphbX1jc8ve3moKJGU1WkItkiWDHBQ1bXAvWiXDARGsSQZXqd98ZFLxKLzTw5i1A9wLuc8p1kbq2NfI60YaeSxWXEQhKnkB1n3iI3msD9ER8kJMBEYenaQy72EmduF07KJTdiZA8TNSBEy1Dr2i9lLk4CFmgqs1AhLzalg4KXKBZjOsA9NqIBkbzX19MqDpQaBmSMDtIj1KyXiv9594n2z9sjHsaJZiE1EeP5iUA6QulrUJdLRrUYGoKp5OYeRPtYqrNAwsF09GdbTRPGpWye1Ku3J4Wq5dZ2zswT6UwIUzqMIN1KAOFJ7hE7hx3qyXq036/0vmrOymV2YgvXxC7dmV4=</latexit>

(x)

<latexit sha1_base64="Hd3XMRibWVq42uYngx5Yd2lnZRU=">ACB3icbVDLSgMxFM3UV62vqks3g0WomzJTBbsuHFZwT5gOpRMmlD8xiSO2IZ+gF+gFv9BHfi1s/wC/wN08fCth4IHM7JSe49UcKZAc/7dnIbm1vbO/ndwt7+weFR8fikZVSqCW0SxZXuRNhQziRtAgNO4mWESctqPR7dRvP1JtmJIPME5oKPBAspgRDFYKujQxjCtZfrsFUtexZvBXSf+gpTQAo1e8afbVyQVALh2JjA9xIM6yBEU4nhW5qaILJCA9oYKnEgpowm408cS+s0ndjpe2R4M7Uv4mMiEizwRCW3smwMGYsIpsXGIZm1ZuK/3lBCnEtzJhMUqCSzL+PU+6CcqetuH2mKQE+tgQTzewGLhlijQnY7mwz/moP6RVrfhXler9daleW3SUR2foHJWRj25QHd2hBmoighR6Qa/ozXl23p0P53N+NecsMqdoCc7XL274mjw=</latexit>

x

<latexit sha1_base64="wjzaU1m9VEYejmaHtOYVQTzLsA=">AB/XicbVDLTgIxFL3jE/GFunTSExckRk0kSWJG5eQyCOBCemUCzS0M5O2YyQT4ge41U9wZ9z6LX6Bv2GBWQh4kiYn5/S0954gFlwb1/12Nja3tnd2c3v5/YPDo+PCyWlTR4li2GCRiFQ7oBoFD7FhuBHYjhVSGQhsBeO7md96RKV5FD6YSYy+pMOQDzijxkr1p16h6JbcOcg68TJShAy1XuGn249YIjE0TFCtO54bGz+lynAmcJrvJhpjysZ0iB1LQypR+l80Cm5tEqfDCJlT2jIXP2bSJkMFB+OzNI7KZVaT2Rg85KakV71ZuJ/Xicxg4qf8jBODIZs8f0gEcREZNYF6XOFzIiJZQpbjcgbEQVZcY2ZpvxVntYJ81ybsules3xWol6ygH53ABV+DBLVThHmrQAYIL/AKb86z8+58OJ+LqxtOljmDJThfvxCljE=</latexit>

+

<latexit sha1_base64="LGIDd6u3i7l6Xp9Ya7W2yrPHmWs=">AB/XicbVDLSgMxFL3js9ZX1aWbYBEocxUwS4Lbly2YB/QDiWT3rahmcyQZIQyFD/ArX6CO3Hrt/gF/oZpOwvbeiBwOCcnufcEseDauO63s7G5tb2zm9vL7x8cHh0XTk6bOkoUwaLRKTaAdUouMSG4UZgO1ZIw0BgKxjfz/zWEyrNI/loJjH6IR1KPuCMGivVr3uFolty5yDrxMtIETLUeoWfbj9iSYjSMEG17nhubPyUKsOZwGm+m2iMKRvTIXYslTRE7afzQafk0ip9MoiUPdKQufo3kbIwUHw4MkvpDTUehIGNh9SM9Kr3kz8z+skZlDxUy7jxKBki+8HiSAmIrMuSJ8rZEZMLKFMcbsBYSOqKDO2MduMt9rDOmWS95NqVy/LVYrWUc5OIcLuAIP7qAKD1CDBjBAeIFXeHOenXfnw/lcXN1wswZLMH5+gWVbpXk</latexit>

<latexit sha1_base64="BRkoKSZHplkDWfDyp4Cw51sAmf8=">AB/XicbVDLSgMxFL3js9ZX1aWbYBHcWGaqYJcFNy5bsA9oh5Jb9vQTGZIMkIZih/gVj/Bnbj1W/wCf8O0nYVtPRA4nJOT3HuCWHBtXPfb2djc2t7Zze3l9w8Oj4LJ6dNHSWKYNFIlLtgGoUXGLDcCOwHSukYSCwFYzvZ37rCZXmkXw0kxj9kA4lH3BGjZXq171C0S25c5B14mWkCBlqvcJPtx+xJERpmKBadzw3Nn5KleFM4DTfTGlI3pEDuWShqi9tP5oFNyaZU+GUTKHmnIXP2bSFkYKD4cmaV3UhpqPQkDmw+pGelVbyb+53USM6j4KZdxYlCyxfeDRBATkVkXpM8VMiMmlCmuN2AsBFVlBnbmG3GW+1hnTLJe+mVK7fFquVrKMcnMFXIEHd1CFB6hBAxgvMArvDnPzrvz4Xwurm4WeYMluB8/QKYoJXm</latexit>

λ 2 = π k

<latexit sha1_base64="r0V4/7umNTd9toN1siEupn1rvQ=">ACHXicbZDLSgMxGIUz9VbrbdSlC4NFcFVmqmA3QsGNywr2Ap2hZDKZNjSZGZKMUMIsfQ4fwK0+gjtxKz6Br2HazsK2/hA4nJOT8H9ByqhUjvNtldbWNza3ytuVnd29/QP78Kgjk0xg0sYJS0QvQJIwGpO2oqRXioI4gEj3WB8O827j0RImsQPapISn6NhTCOKkTLWwD71IoGw9piphCjX9fymcFKa63E+sKtOzZkNXBVuIaqgmNbA/vHCBGecxAozJGXfdVLlayQUxYzkFS+TJEV4jIakb2SMOJG+ni2Sw3PjhDBKhDmxgjP3b0NjHg6HKmFdzTiUk54YPocqZFczqbmf1k/U1HD1zROM0ViP8+yhUCZygiEVBCs2MQJhQc0GEI+QoaMUPGXeawKjr1mntZq9fVZuNglEZnIAzcAFcA2a4A60QBtg8ARewCt4s56td+vD+pxfLVlF5xgsjPX1CwWOo7I=</latexit>

˙ ˜ (k, t) = k ·˜ (k, t)

<latexit sha1_base64="pHzR0Evo+6LDEThSgi+oOGqwbI=">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</latexit>

the smaller the wavevector, the slower the dynamics

slide-25
SLIDE 25

consFtuFve relaFons: the heat equaFon

  • t = · j
<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit>
slide-26
SLIDE 26

consFtuFve relaFons: the heat equaFon

  • t = · j
<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit>

j = κT

<latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit>
slide-27
SLIDE 27

consFtuFve relaFons: the heat equaFon

  • t = · j
<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit>

j = κT

<latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit>

= cpT

<latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit>
slide-28
SLIDE 28

consFtuFve relaFons: the heat equaFon

  • t = · j
<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit>

j = κT

<latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit>

= cpT

<latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit>

α = κ cpρ

<latexit sha1_base64="EPm/9eh/7c4A4uXHrDIw5pOo3CY=">ACHicbZDLSgMxGIUz9VbrepSkGARXJWZKuhGKbhxWcFeoDM/6SZTmhmJiQZoQzd+Rw+gFt9BHfiVvAJfA3Ty8K2/hA4nJOT8H+B4Exp2/62Ciura+sbxc3S1vbO7l5/6Cl0kwS2iQpT2UnAEU5S2hTM81pR0gKcBpOxjcjvP2I5WKpcmDHgrqxdBPWMgIaGP5WMXuIgAX7uhBJK7AxACRjnxhSujdOSXK3bVngxeFs5MVNBsGn75x+2lJItpogkHpbqOLbSXg9SMcDoquZmiAsgA+rRrZAIxV4+2WOET43Tw2EqzUk0nrh/GzmJA8n6kZ57J4dYqWEcmH4MOlKL2dj8L+tmOrzycpaITNOETL8PM451iseocI9JSjQfGgFEMrMBJhEYTNoANWScRQ7LolWrOufV2v1FpX4zY1RER+gEnSEHXaI6ukMN1EQEPaEX9IrerGfr3fqwPqdXC9asc4jmxvr6Ba0Rovk=</latexit>

∂T ∂t = α∆T

<latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit>
slide-29
SLIDE 29

consFtuFve relaFons: the heat equaFon

  • t = · j
<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit>

j = κT

<latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit>

= cpT

<latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit>

α = κ cpρ

<latexit sha1_base64="EPm/9eh/7c4A4uXHrDIw5pOo3CY=">ACHicbZDLSgMxGIUz9VbrepSkGARXJWZKuhGKbhxWcFeoDM/6SZTmhmJiQZoQzd+Rw+gFt9BHfiVvAJfA3Ty8K2/hA4nJOT8H+B4Exp2/62Ciura+sbxc3S1vbO7l5/6Cl0kwS2iQpT2UnAEU5S2hTM81pR0gKcBpOxjcjvP2I5WKpcmDHgrqxdBPWMgIaGP5WMXuIgAX7uhBJK7AxACRjnxhSujdOSXK3bVngxeFs5MVNBsGn75x+2lJItpogkHpbqOLbSXg9SMcDoquZmiAsgA+rRrZAIxV4+2WOET43Tw2EqzUk0nrh/GzmJA8n6kZ57J4dYqWEcmH4MOlKL2dj8L+tmOrzycpaITNOETL8PM451iseocI9JSjQfGgFEMrMBJhEYTNoANWScRQ7LolWrOufV2v1FpX4zY1RER+gEnSEHXaI6ukMN1EQEPaEX9IrerGfr3fqwPqdXC9asc4jmxvr6Ba0Rovk=</latexit>

∂T ∂t = α∆T

<latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit>

∂ ˜ T(k, t) ∂t = −αk2 ˜ T(k, t)

<latexit sha1_base64="d+UF3wdRyVt3QEV1BGYB59NRIgs=">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</latexit>
slide-30
SLIDE 30

consFtuFve relaFons: the heat equaFon

  • t = · j
<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit>

j = κT

<latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit>

= cpT

<latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit>

α = κ cpρ

<latexit sha1_base64="EPm/9eh/7c4A4uXHrDIw5pOo3CY=">ACHicbZDLSgMxGIUz9VbrepSkGARXJWZKuhGKbhxWcFeoDM/6SZTmhmJiQZoQzd+Rw+gFt9BHfiVvAJfA3Ty8K2/hA4nJOT8H+B4Exp2/62Ciura+sbxc3S1vbO7l5/6Cl0kwS2iQpT2UnAEU5S2hTM81pR0gKcBpOxjcjvP2I5WKpcmDHgrqxdBPWMgIaGP5WMXuIgAX7uhBJK7AxACRjnxhSujdOSXK3bVngxeFs5MVNBsGn75x+2lJItpogkHpbqOLbSXg9SMcDoquZmiAsgA+rRrZAIxV4+2WOET43Tw2EqzUk0nrh/GzmJA8n6kZ57J4dYqWEcmH4MOlKL2dj8L+tmOrzycpaITNOETL8PM451iseocI9JSjQfGgFEMrMBJhEYTNoANWScRQ7LolWrOufV2v1FpX4zY1RER+gEnSEHXaI6ukMN1EQEPaEX9IrerGfr3fqwPqdXC9asc4jmxvr6Ba0Rovk=</latexit>

∂T ∂t = α∆T

<latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit>

∂ ˜ T(k, t) ∂t = −αk2 ˜ T(k, t)

<latexit sha1_base64="d+UF3wdRyVt3QEV1BGYB59NRIgs=">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</latexit>
  • T(k, t) = e−αk2t

T(k, 0)

<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit>
slide-31
SLIDE 31

consFtuFve relaFons: the heat equaFon

∂T ∂t = α∆T

<latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit>
  • T(k, t) = e−αk2t

T(k, 0)

<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">ACT3icdVHLSgMxFM3UV31XboJFkFBy0wVdKMIblxaFXo1JLJ3GlDk5khuaOUod/ld7h04VY/wZ2Y1ln4vBA4OSfnJvckSKUw6LpPTmlqemZ2rjy/sLi0vLJaWVu/MkmObR4IhN9EzADUsTQoESblINTAUSroPB+Vi/vgNtRBI3cZhCR7FeLCLBGVqW2n49yIEFDIE2tzxFcN+ENHBHu7SEzrZapXD6Dbf95lM+4wObusUR5T+43N3u5WqW3MnRX8DrwBVUtRlt/LqhwnPFMTIJTOm7bkpdnKmUXAJowU/M5AyPmA9aFsYMwWmk09GH9Fty4Q0SrRdMdIJ+9WRcxVo0evjtz45U8YMVWD945ebn9qY/EtrZxgd3IRpxlCzD+vjzJMaHjdGkoNHCUQwsY18JOQHmfacbR/oFNxvuZw29wVa95B7V647B6dlpkVCabZIvsEI8ckTNyQS5Ji3DyQJ7JC3l1Hp03571UHC05Bdg36o0/wGI3LML</latexit>

T(r, t) = 1 (4παt)

3 2

  • e |rr|2

4αt T(r, 0)dr

<latexit sha1_base64="QezugKTriq0vDXB6FuodfXuCJ4E=">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</latexit>
slide-32
SLIDE 32

consFtuFve relaFons: the heat equaFon

∂T ∂t = α∆T

<latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit>
  • T(k, t) = e−αk2t

T(k, 0)

<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit>

T(r, t) = 1 (4παt)

3 2

  • e |rr|2

4αt T(r, 0)dr

<latexit sha1_base64="QezugKTriq0vDXB6FuodfXuCJ4E=">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</latexit>

T

<latexit sha1_base64="to9WkwJZ3IcO7M/jmX6wxaXS36M=">AB/XicbVDLSgMxFL3js9ZX1aWbwSK4KjNV0JU3LhsoS9oh5J0zY0yQzJHaEMxQ9wq5/gTtz6LX6Bv2HazsK2HgczslJ7j1hLhBz/t2Nja3tnd2c3v5/YPDo+PCyWnTRImrEjEel2SAwTXLEGchSsHWtGZChYKxw/zPzWE9OGR6qOk5gFkgwVH3BK0Eq1eq9Q9EreHO468TNShAzVXuGn249oIplCKogxHd+LMUiJRk4Fm+a7iWExoWMyZB1LFZHMBOl80Kl7aZW+O4i0PQrdufo3kVIZaj4c4dI7KZHGTGRo85LgyKx6M/E/r5Pg4C5IuYoTZIouvh8kwsXInXh9rlmFMXEkI1txu4dEQ0oWgbs834qz2sk2a5F+XyrWbYuU+6ygH53ABV+DLVTgEarQAoMXuAV3pxn5935cD4XVzecLHMGS3C+fgHYvZYT</latexit>

x

<latexit sha1_base64="RvBtAn0pWJHXswtsHX8MrBDp/TM=">AB/XicbVDLSgMxFL3js9ZX1aWbwSK4KjNV0JU3LhswT6gHUomTdvQJDMkd8QyFD/ArX6CO3Hrt/gF/oZpOwvbeiBwOCcnufeEseAGPe/bWVvf2Nzazu3kd/f2Dw4LR8cNEyWasjqNRKRbITFMcMXqyFGwVqwZkaFgzXB0N/Wbj0wbHqkHMcskGSgeJ9TglaqPXULRa/kzeCuEj8jRchQ7RZ+Or2IJpIpIY0/a9GIOUaORUsEm+kxgWEzoiA9a2VBHJTJDOBp2451bpuf1I26PQnal/EymVoeaDIS68kxJpzFiGNi8JDs2yNxX/89oJ9m+ClKs4Qabo/Pt+IlyM3GkXbo9rRlGMLSFUc7uBS4dE4q2MduMv9zDKmUS/5lqVy7KlZus45ycApncAE+XEMF7qEKdaDA4AVe4c15dt6dD+dzfnXNyTInsADn6xcSUJY3</latexit>
slide-33
SLIDE 33

consFtuFve relaFons: the heat equaFon

∂T ∂t = α∆T

<latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit>
  • T(k, t) = e−αk2t

T(k, 0)

<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit>

T(r, t) = 1 (4παt)

3 2

  • e |rr|2

4αt T(r, 0)dr

<latexit sha1_base64="QezugKTriq0vDXB6FuodfXuCJ4E=">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</latexit>

T

<latexit sha1_base64="to9WkwJZ3IcO7M/jmX6wxaXS36M=">AB/XicbVDLSgMxFL3js9ZX1aWbwSK4KjNV0JU3LhsoS9oh5J0zY0yQzJHaEMxQ9wq5/gTtz6LX6Bv2HazsK2HgczslJ7j1hLhBz/t2Nja3tnd2c3v5/YPDo+PCyWnTRImrEjEel2SAwTXLEGchSsHWtGZChYKxw/zPzWE9OGR6qOk5gFkgwVH3BK0Eq1eq9Q9EreHO468TNShAzVXuGn249oIplCKogxHd+LMUiJRk4Fm+a7iWExoWMyZB1LFZHMBOl80Kl7aZW+O4i0PQrdufo3kVIZaj4c4dI7KZHGTGRo85LgyKx6M/E/r5Pg4C5IuYoTZIouvh8kwsXInXh9rlmFMXEkI1txu4dEQ0oWgbs834qz2sk2a5F+XyrWbYuU+6ygH53ABV+DLVTgEarQAoMXuAV3pxn5935cD4XVzecLHMGS3C+fgHYvZYT</latexit>

x

<latexit sha1_base64="RvBtAn0pWJHXswtsHX8MrBDp/TM=">AB/XicbVDLSgMxFL3js9ZX1aWbwSK4KjNV0JU3LhswT6gHUomTdvQJDMkd8QyFD/ArX6CO3Hrt/gF/oZpOwvbeiBwOCcnufeEseAGPe/bWVvf2Nzazu3kd/f2Dw4LR8cNEyWasjqNRKRbITFMcMXqyFGwVqwZkaFgzXB0N/Wbj0wbHqkHMcskGSgeJ9TglaqPXULRa/kzeCuEj8jRchQ7RZ+Or2IJpIpIY0/a9GIOUaORUsEm+kxgWEzoiA9a2VBHJTJDOBp2451bpuf1I26PQnal/EymVoeaDIS68kxJpzFiGNi8JDs2yNxX/89oJ9m+ClKs4Qabo/Pt+IlyM3GkXbo9rRlGMLSFUc7uBS4dE4q2MduMv9zDKmUS/5lqVy7KlZus45ycApncAE+XEMF7qEKdaDA4AVe4c15dt6dD+dzfnXNyTInsADn6xcSUJY3</latexit>
slide-34
SLIDE 34

consFtuFve relaFons: Onsager’s equaFons

P conserved (current) densities:

  • a1(r), · · · aP (r)
  • ,
  • j1(r), · · · jP (r)
  • <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit>

P conserved quantities:

  • A1, A2, · · · AP
  • <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit>
slide-35
SLIDE 35

consFtuFve relaFons: Onsager’s equaFons

P conserved (current) densities:

  • a1(r), · · · aP (r)
  • ,
  • j1(r), · · · jP (r)
  • <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit>

P conserved quantities:

  • A1, A2, · · · AP
  • <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit>

S = S[a1, a2, · · · aP ]

<latexit sha1_base64="XCViKXKeS7eUFse3lctwUm45QyU=">ACFXicbVC7SgNBFJ2NrxhfUQsLm8EgWISwGwVtlICNZSQmBpJluTuZJENmH8zcFcKS7/ADbPUT7MTW2i/wN5w8CpN4MLhnHtmuMePpdBo29WZmV1bX0ju5nb2t7Z3cvHzR0lCjG6ySkWr6oLkUIa+jQMmbseIQ+JI/+oPbsf/4xJUWUfiAw5i7AfRC0RUM0Ehe/qh2XWuB5xTBKxfbrBOhpuBVXS9fsEv2BHSZODNSIDNUvfxPuxOxJOAhMglatxw7RjcFhYJPsq1E81jYAPo8ZahIQRcu+nkgBE9NUqHdiNlJkQ6Uf8mUhb4SvT6OPdOCoHWw8A3+QCwrxe9sfif10qwe+WmIowT5CGbft9NJMWIjuiHaE4Qzk0BJgS5gLK+qCAoWnSNOMs9rBMGuWSc14q318UKjezjrLkmJyQM+KQS1Ihd6RK6oSREXkhr+TNerberQ/rc7qasWaZQzIH6+sXRzKeSw=</latexit>
slide-36
SLIDE 36

consFtuFve relaFons: Onsager’s equaFons

P conserved (current) densities:

  • a1(r), · · · aP (r)
  • ,
  • j1(r), · · · jP (r)
  • <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit>

P conserved quantities:

  • A1, A2, · · · AP
  • <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">ACPXicbVBNbxMxFPS20KYBSgpHLhYJUpCiaDcqHoQVw4Bok0kZIo8r59Sax47cV+Gyla5ef0d/QHcIUDPwBxQb1yxfk4kJaRbI1mPO/JE2dKOgrDH8HB4YOHR8elk/Kjx09On1bOnl05k1vALhlbD8WDpXU2CVJCvuZRZHGCnvx/MPa7y3QOmn0Z1pmOErFVMuJBEFeGlfe1To1DkY7tAtM+JdcaD9ForvgtWEsp4rX+ftx1PBXqzGExJDztMPXnuWva+NKNWyG/D7JNqRKtuhM678GiYG8hQ1gRLODaIwo1EhLElQuCoPc4eZgLmY4sBTLVJ0o2Lz0RV/5ZWET4z1RxPfqP8mCkhjK6cz2ptTiNS5ZRr7fCpo5u56a/F/3iCnyfmokDrLCTVs109yxcnwdZc8kRaB1NITAdbXBhxmwgog37hvJrbw31y1WpGb5qtT61q+3LXUYm9YC9ZnUXsLWuzj6zDugzYNfvKvrHvwU3wM/gd3G6fHgS7zHO2h+DPX0THrFM=</latexit>

S = S[a1, a2, · · · aP ]

<latexit sha1_base64="XCViKXKeS7eUFse3lctwUm45QyU=">ACFXicbVC7SgNBFJ2NrxhfUQsLm8EgWISwGwVtlICNZSQmBpJluTuZJENmH8zcFcKS7/ADbPUT7MTW2i/wN5w8CpN4MLhnHtmuMePpdBo29WZmV1bX0ju5nb2t7Z3cvHzR0lCjG6ySkWr6oLkUIa+jQMmbseIQ+JI/+oPbsf/4xJUWUfiAw5i7AfRC0RUM0Ehe/qh2XWuB5xTBKxfbrBOhpuBVXS9fsEv2BHSZODNSIDNUvfxPuxOxJOAhMglatxw7RjcFhYJPsq1E81jYAPo8ZahIQRcu+nkgBE9NUqHdiNlJkQ6Uf8mUhb4SvT6OPdOCoHWw8A3+QCwrxe9sfif10qwe+WmIowT5CGbft9NJMWIjuiHaE4Qzk0BJgS5gLK+qCAoWnSNOMs9rBMGuWSc14q318UKjezjrLkmJyQM+KQS1Ihd6RK6oSREXkhr+TNerberQ/rc7qasWaZQzIH6+sXRzKeSw=</latexit>

At thermodynamic equilibrium: S = max

<latexit sha1_base64="1VSIyahDpqvwiWTLkDnFul+d4=">ACK3icbVDLSgMxFM34tr6qLt0Eq+DGMlMXiqAoblxWtFZoh5Jb9vQZGZM7kiHoV/hd/gBbvUTXCluxd8wrV1Y9UDgcE5Ocu8JYikMu6rMzE5NT0zOzefW1hcWl7Jr65dmyjRHCo8kpG+CZgBKUKoEAJN7EGpgIJ1aB7NvCrd6CNiMIrTGPwFWuHoiU4Qys18runSLEDWkXNGRKcAq3iZAi0CJRh3Tr8ojWEXqYKdbrbzXyBbfoDkH/Em9ECmSEciP/W9GPFEQIpfMmJrnxuhnTKPgEvq5emIgZrzL2lCz1E4Axs+Ga/XptlWatBVpe0KkQ/VnIuPKjtnu4Ng7GVPGpCqwecWwY357A/E/r5Zg68DPRBgnCH/r6VSIoRHTRHm0IDR5lawrgWdgPKO0wzjrZf24z3u4e/5LpU9PaKpYtS4eR41NEc2SCbZId4ZJ+ckHNSJhXCyT15JE/k2XlwXpw35/376oQzyqyTMTgfX25kqGg=</latexit>
slide-37
SLIDE 37

consFtuFve relaFons: Onsager’s equaFons

P conserved (current) densities:

  • a1(r), · · · aP (r)
  • ,
  • j1(r), · · · jP (r)
  • <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit>

P conserved quantities:

  • A1, A2, · · · AP
  • <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit>

S = S[a1, a2, · · · aP ]

<latexit sha1_base64="XCViKXKeS7eUFse3lctwUm45QyU=">ACFXicbVC7SgNBFJ2NrxhfUQsLm8EgWISwGwVtlICNZSQmBpJluTuZJENmH8zcFcKS7/ADbPUT7MTW2i/wN5w8CpN4MLhnHtmuMePpdBo29WZmV1bX0ju5nb2t7Z3cvHzR0lCjG6ySkWr6oLkUIa+jQMmbseIQ+JI/+oPbsf/4xJUWUfiAw5i7AfRC0RUM0Ehe/qh2XWuB5xTBKxfbrBOhpuBVXS9fsEv2BHSZODNSIDNUvfxPuxOxJOAhMglatxw7RjcFhYJPsq1E81jYAPo8ZahIQRcu+nkgBE9NUqHdiNlJkQ6Uf8mUhb4SvT6OPdOCoHWw8A3+QCwrxe9sfif10qwe+WmIowT5CGbft9NJMWIjuiHaE4Qzk0BJgS5gLK+qCAoWnSNOMs9rBMGuWSc14q318UKjezjrLkmJyQM+KQS1Ihd6RK6oSREXkhr+TNerberQ/rc7qasWaZQzIH6+sXRzKeSw=</latexit>

At thermodynamic equilibrium: S = max

<latexit sha1_base64="1VSIyahDpqvwiWTLkDnFul+d4=">ACK3icbVDLSgMxFM34tr6qLt0Eq+DGMlMXiqAoblxWtFZoh5Jb9vQZGZM7kiHoV/hd/gBbvUTXCluxd8wrV1Y9UDgcE5Ocu8JYikMu6rMzE5NT0zOzefW1hcWl7Jr65dmyjRHCo8kpG+CZgBKUKoEAJN7EGpgIJ1aB7NvCrd6CNiMIrTGPwFWuHoiU4Qys18runSLEDWkXNGRKcAq3iZAi0CJRh3Tr8ojWEXqYKdbrbzXyBbfoDkH/Em9ECmSEciP/W9GPFEQIpfMmJrnxuhnTKPgEvq5emIgZrzL2lCz1E4Axs+Ga/XptlWatBVpe0KkQ/VnIuPKjtnu4Ng7GVPGpCqwecWwY357A/E/r5Zg68DPRBgnCH/r6VSIoRHTRHm0IDR5lawrgWdgPKO0wzjrZf24z3u4e/5LpU9PaKpYtS4eR41NEc2SCbZId4ZJ+ckHNSJhXCyT15JE/k2XlwXpw35/376oQzyqyTMTgfX25kqGg=</latexit>

δ δai

  • S −
  • j

λjAj

  • = 0

δS δai = αi(r) = λi

<latexit sha1_base64="3sMxC2Co5d01PHJMOmG3M6dPTaI=">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</latexit>
slide-38
SLIDE 38

consFtuFve relaFons: Onsager’s equaFons

P conserved (current) densities:

  • a1(r), · · · aP (r)
  • ,
  • j1(r), · · · jP (r)
  • <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit>

P conserved quantities:

  • A1, A2, · · · AP
  • <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit>

S = S[a1, a2, · · · aP ]

<latexit sha1_base64="XCViKXKeS7eUFse3lctwUm45QyU=">ACFXicbVC7SgNBFJ2NrxhfUQsLm8EgWISwGwVtlICNZSQmBpJluTuZJENmH8zcFcKS7/ADbPUT7MTW2i/wN5w8CpN4MLhnHtmuMePpdBo29WZmV1bX0ju5nb2t7Z3cvHzR0lCjG6ySkWr6oLkUIa+jQMmbseIQ+JI/+oPbsf/4xJUWUfiAw5i7AfRC0RUM0Ehe/qh2XWuB5xTBKxfbrBOhpuBVXS9fsEv2BHSZODNSIDNUvfxPuxOxJOAhMglatxw7RjcFhYJPsq1E81jYAPo8ZahIQRcu+nkgBE9NUqHdiNlJkQ6Uf8mUhb4SvT6OPdOCoHWw8A3+QCwrxe9sfif10qwe+WmIowT5CGbft9NJMWIjuiHaE4Qzk0BJgS5gLK+qCAoWnSNOMs9rBMGuWSc14q318UKjezjrLkmJyQM+KQS1Ihd6RK6oSREXkhr+TNerberQ/rc7qasWaZQzIH6+sXRzKeSw=</latexit>

At thermodynamic equilibrium: S = max

<latexit sha1_base64="1VSIyahDpqvwiWTLkDnFul+d4=">ACK3icbVDLSgMxFM34tr6qLt0Eq+DGMlMXiqAoblxWtFZoh5Jb9vQZGZM7kiHoV/hd/gBbvUTXCluxd8wrV1Y9UDgcE5Ocu8JYikMu6rMzE5NT0zOzefW1hcWl7Jr65dmyjRHCo8kpG+CZgBKUKoEAJN7EGpgIJ1aB7NvCrd6CNiMIrTGPwFWuHoiU4Qys18runSLEDWkXNGRKcAq3iZAi0CJRh3Tr8ojWEXqYKdbrbzXyBbfoDkH/Em9ECmSEciP/W9GPFEQIpfMmJrnxuhnTKPgEvq5emIgZrzL2lCz1E4Axs+Ga/XptlWatBVpe0KkQ/VnIuPKjtnu4Ng7GVPGpCqwecWwY357A/E/r5Zg68DPRBgnCH/r6VSIoRHTRHm0IDR5lawrgWdgPKO0wzjrZf24z3u4e/5LpU9PaKpYtS4eR41NEc2SCbZId4ZJ+ckHNSJhXCyT15JE/k2XlwXpw35/376oQzyqyTMTgfX25kqGg=</latexit>

δ δai

  • S −
  • j

λjAj

  • = 0

δS δai = αi(r) = λi

<latexit sha1_base64="3sMxC2Co5d01PHJMOmG3M6dPTaI=">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</latexit>

ji =

  • j

Λijfj f = α

<latexit sha1_base64="M6y7rALkPqCoKP31PCfXrdEF4Hg=">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</latexit>
slide-39
SLIDE 39

Onsager’s linear- response equations Λij = Λji

<latexit sha1_base64="Wr/tm4lDRlhP4qFs0ULU4AxpquI=">ACGnicdVDLSgMxFM34rPU16lIXwSK4GmZadaYLoeDGhYsK9gHtUDJp2qbNPEgyQhlm43f4AW71E9yJWzd+gb9hpq1oRQ8ETs69J+EcL2JUSN81xYWl5ZXVnNr+fWNza1tfWe3LsKY1LDIQt50OCMBqQmqSkWbECfI9Rhre6CKbN24JFzQMbuQ4Iq6P+gHtUYykjr6AWxfqe0u6iR0mMLz7+uQph29YBp20bZPTWga5lm5XLYyUnKlgMtw5ygAGaodvSPdjfEsU8CiRkSomWZkXQTxCXFjKT5dixIhPAI9UlL0QD5RLjJEUKj5TShb2QqxNIOF/OhLse5z2B3LunQT5Qox9T/l9JAfi9ywT/5q1Ytlz3IQGUSxJgKf92IGZQizomCXcoIlGyuCMKcqAcQDxBGWqk7VzFd8+D+pFw2rZBSvTwoVZ9ZRDuyDQ3AMLGCDCrgEVADGNyB/AInrR7Vl70V6nqwvazLMH5qC9fQL5JqF3</latexit>

consFtuFve relaFons: Onsager’s equaFons

P conserved (current) densities:

  • a1(r), · · · aP (r)
  • ,
  • j1(r), · · · jP (r)
  • <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit>

P conserved quantities:

  • A1, A2, · · · AP
  • <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit>

S = S[a1, a2, · · · aP ]

<latexit sha1_base64="XCViKXKeS7eUFse3lctwUm45QyU=">ACFXicbVC7SgNBFJ2NrxhfUQsLm8EgWISwGwVtlICNZSQmBpJluTuZJENmH8zcFcKS7/ADbPUT7MTW2i/wN5w8CpN4MLhnHtmuMePpdBo29WZmV1bX0ju5nb2t7Z3cvHzR0lCjG6ySkWr6oLkUIa+jQMmbseIQ+JI/+oPbsf/4xJUWUfiAw5i7AfRC0RUM0Ehe/qh2XWuB5xTBKxfbrBOhpuBVXS9fsEv2BHSZODNSIDNUvfxPuxOxJOAhMglatxw7RjcFhYJPsq1E81jYAPo8ZahIQRcu+nkgBE9NUqHdiNlJkQ6Uf8mUhb4SvT6OPdOCoHWw8A3+QCwrxe9sfif10qwe+WmIowT5CGbft9NJMWIjuiHaE4Qzk0BJgS5gLK+qCAoWnSNOMs9rBMGuWSc14q318UKjezjrLkmJyQM+KQS1Ihd6RK6oSREXkhr+TNerberQ/rc7qasWaZQzIH6+sXRzKeSw=</latexit>

At thermodynamic equilibrium: S = max

<latexit sha1_base64="1VSIyahDpqvwiWTLkDnFul+d4=">ACK3icbVDLSgMxFM34tr6qLt0Eq+DGMlMXiqAoblxWtFZoh5Jb9vQZGZM7kiHoV/hd/gBbvUTXCluxd8wrV1Y9UDgcE5Ocu8JYikMu6rMzE5NT0zOzefW1hcWl7Jr65dmyjRHCo8kpG+CZgBKUKoEAJN7EGpgIJ1aB7NvCrd6CNiMIrTGPwFWuHoiU4Qys18runSLEDWkXNGRKcAq3iZAi0CJRh3Tr8ojWEXqYKdbrbzXyBbfoDkH/Em9ECmSEciP/W9GPFEQIpfMmJrnxuhnTKPgEvq5emIgZrzL2lCz1E4Axs+Ga/XptlWatBVpe0KkQ/VnIuPKjtnu4Ng7GVPGpCqwecWwY357A/E/r5Zg68DPRBgnCH/r6VSIoRHTRHm0IDR5lawrgWdgPKO0wzjrZf24z3u4e/5LpU9PaKpYtS4eR41NEc2SCbZId4ZJ+ckHNSJhXCyT15JE/k2XlwXpw35/376oQzyqyTMTgfX25kqGg=</latexit>

δ δai

  • S −
  • j

λjAj

  • = 0

δS δai = αi(r) = λi

<latexit sha1_base64="3sMxC2Co5d01PHJMOmG3M6dPTaI=">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</latexit>

ji =

  • j

Λijfj f = α

<latexit sha1_base64="M6y7rALkPqCoKP31PCfXrdEF4Hg=">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</latexit>
slide-40
SLIDE 40

consFtuFve relaFons: Onsager’s equaFons

A E V Ni

α = ∂S ∂A 1 T p T − μi T

Onsager’s linear- response equations Λij = Λji

<latexit sha1_base64="Wr/tm4lDRlhP4qFs0ULU4AxpquI=">ACGnicdVDLSgMxFM34rPU16lIXwSK4GmZadaYLoeDGhYsK9gHtUDJp2qbNPEgyQhlm43f4AW71E9yJWzd+gb9hpq1oRQ8ETs69J+EcL2JUSN81xYWl5ZXVnNr+fWNza1tfWe3LsKY1LDIQt50OCMBqQmqSkWbECfI9Rhre6CKbN24JFzQMbuQ4Iq6P+gHtUYykjr6AWxfqe0u6iR0mMLz7+uQph29YBp20bZPTWga5lm5XLYyUnKlgMtw5ygAGaodvSPdjfEsU8CiRkSomWZkXQTxCXFjKT5dixIhPAI9UlL0QD5RLjJEUKj5TShb2QqxNIOF/OhLse5z2B3LunQT5Qox9T/l9JAfi9ywT/5q1Ytlz3IQGUSxJgKf92IGZQizomCXcoIlGyuCMKcqAcQDxBGWqk7VzFd8+D+pFw2rZBSvTwoVZ9ZRDuyDQ3AMLGCDCrgEVADGNyB/AInrR7Vl70V6nqwvazLMH5qC9fQL5JqF3</latexit>

ji =

  • j

Λijfj f = α

<latexit sha1_base64="M6y7rALkPqCoKP31PCfXrdEF4Hg=">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</latexit>
slide-41
SLIDE 41

Green-Kubo linear-response theory

Λij = Ω kB ∞ Ji(t)Jj(0)dt

<latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit>
slide-42
SLIDE 42

Green-Kubo linear-response theory

Λij = Ω kB ∞ Ji(t)Jj(0)dt

<latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit>

J = 1 Ω

j(r)dr

<latexit sha1_base64="j1CMhF2qvqFg7ZQoOPL9+IsYeiI=">ACNXicbVDNSgMxGMz6W+vfqkcvwSLopeyqoBeh4EW8WMH+QLeUbJpto0l2Sb4VytJH8Tl8AK/6AB68iR59BbNtEbUOBCYzmSTfhIngBjzvxZmZnZtfWCwsFZdXVtfW3Y3NuolTVmNxiLWzZAYJrhiNeAgWDPRjMhQsEZ4e5b7jTumDY/VNQwS1pakp3jEKQErdzjC3yKg0gTmvnDLiUrEeGOAKOuMNvtkLJIF+G93/2mHbfklb0R8DTxJ6SEJqh23I+gG9NUMgVUEGNavpdAOyMaOBVsWAxSwxJCb0mPtSxVRDLTzkYDvGuVbo4irVdCvBI/ZnIqAw17/Xh1z0ZkcYMZGjz+bfNXy8X/NaKUQn7YyrJAWm6Pj5KBUYpx3iLtcMwpiYAmhmtsJMO0T2yHYpm0z/t8epkn9oOwflg+ujkqVk0lHBbSNdtAe8tExqBzVEU1RNE9ekRP6Nl5cF6dN+d9fHTGmWS20C84n1/x+auV</latexit>
slide-43
SLIDE 43

Green-Kubo linear-response theory

Λij = Ω kB ∞ Ji(t)Jj(0)dt

<latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit>

J = 1 Ω

j(r)dr

<latexit sha1_base64="j1CMhF2qvqFg7ZQoOPL9+IsYeiI=">ACNXicbVDNSgMxGMz6W+vfqkcvwSLopeyqoBeh4EW8WMH+QLeUbJpto0l2Sb4VytJH8Tl8AK/6AB68iR59BbNtEbUOBCYzmSTfhIngBjzvxZmZnZtfWCwsFZdXVtfW3Y3NuolTVmNxiLWzZAYJrhiNeAgWDPRjMhQsEZ4e5b7jTumDY/VNQwS1pakp3jEKQErdzjC3yKg0gTmvnDLiUrEeGOAKOuMNvtkLJIF+G93/2mHbfklb0R8DTxJ6SEJqh23I+gG9NUMgVUEGNavpdAOyMaOBVsWAxSwxJCb0mPtSxVRDLTzkYDvGuVbo4irVdCvBI/ZnIqAw17/Xh1z0ZkcYMZGjz+bfNXy8X/NaKUQn7YyrJAWm6Pj5KBUYpx3iLtcMwpiYAmhmtsJMO0T2yHYpm0z/t8epkn9oOwflg+ujkqVk0lHBbSNdtAe8tExqBzVEU1RNE9ekRP6Nl5cF6dN+d9fHTGmWS20C84n1/x+auV</latexit>

J(t) = J(Γt) = J(t, Γ0)

<latexit sha1_base64="Y/ipwzwJRIv28OvaTUiqAkTysk=">ACJHicbVDNSgMxGMz6W+vfqkcvwWKpIGW3CvZSKHhQeqpgW6FbSjZN29Bkd0m+FcrSN/A5fACv+gjexIMXr76GabsHrQ4EJjOZL3zjR4JrcJwPa2l5ZXVtPbOR3dza3tm19/abOowVZQ0ailDd+UQzwQPWA6C3UWKEekL1vJHl1O/dc+U5mFwC+OIdSQZBLzPKQEjde18rQAnOF/BtYJ3RaQkXP1vGy+YozTVHJOunbOKToz4L/ETUkOpah37S+vF9JYsgCoIFq3XSeCTkIUcCrYJOvFmkWEjsiAtQ0NiGS6k8z2meBjo/RwP1TmBIBn6s9EQqWv+GAIv+YkRGo9lr7JSwJDvehNxf+8dgz9cifhQRQDC+j8+34sMIR4WhnucUoiLEhCpuNsB0SBShYIo1zbiLPfwlzVLRPSuWbs5z1XLaUQYdoiNUQC6QFV0jeqogSh6QE/oGb1Yj9ar9Wa9z58uWnmAP2C9fkN9qiCQ=</latexit>
slide-44
SLIDE 44

Green-Kubo linear-response theory

Λij = Ω kB ∞ Ji(t)Jj(0)dt

<latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit>

J = 1 Ω

j(r)dr

<latexit sha1_base64="j1CMhF2qvqFg7ZQoOPL9+IsYeiI=">ACNXicbVDNSgMxGMz6W+vfqkcvwSLopeyqoBeh4EW8WMH+QLeUbJpto0l2Sb4VytJH8Tl8AK/6AB68iR59BbNtEbUOBCYzmSTfhIngBjzvxZmZnZtfWCwsFZdXVtfW3Y3NuolTVmNxiLWzZAYJrhiNeAgWDPRjMhQsEZ4e5b7jTumDY/VNQwS1pakp3jEKQErdzjC3yKg0gTmvnDLiUrEeGOAKOuMNvtkLJIF+G93/2mHbfklb0R8DTxJ6SEJqh23I+gG9NUMgVUEGNavpdAOyMaOBVsWAxSwxJCb0mPtSxVRDLTzkYDvGuVbo4irVdCvBI/ZnIqAw17/Xh1z0ZkcYMZGjz+bfNXy8X/NaKUQn7YyrJAWm6Pj5KBUYpx3iLtcMwpiYAmhmtsJMO0T2yHYpm0z/t8epkn9oOwflg+ujkqVk0lHBbSNdtAe8tExqBzVEU1RNE9ekRP6Nl5cF6dN+d9fHTGmWS20C84n1/x+auV</latexit>

J(t) = J(Γt) = J(t, Γ0)

<latexit sha1_base64="Y/ipwzwJRIv28OvaTUiqAkTysk=">ACJHicbVDNSgMxGMz6W+vfqkcvwWKpIGW3CvZSKHhQeqpgW6FbSjZN29Bkd0m+FcrSN/A5fACv+gjexIMXr76GabsHrQ4EJjOZL3zjR4JrcJwPa2l5ZXVtPbOR3dza3tm19/abOowVZQ0ailDd+UQzwQPWA6C3UWKEekL1vJHl1O/dc+U5mFwC+OIdSQZBLzPKQEjde18rQAnOF/BtYJ3RaQkXP1vGy+YozTVHJOunbOKToz4L/ETUkOpah37S+vF9JYsgCoIFq3XSeCTkIUcCrYJOvFmkWEjsiAtQ0NiGS6k8z2meBjo/RwP1TmBIBn6s9EQqWv+GAIv+YkRGo9lr7JSwJDvehNxf+8dgz9cifhQRQDC+j8+34sMIR4WhnucUoiLEhCpuNsB0SBShYIo1zbiLPfwlzVLRPSuWbs5z1XLaUQYdoiNUQC6QFV0jeqogSh6QE/oGb1Yj9ar9Wa9z58uWnmAP2C9fkN9qiCQ=</latexit>

J(t)J(0) =

  • J(t, Γ0)J(0, Γ0)P (Γ0)dΓ0
  • <latexit sha1_base64="oN8l1/xSxXZw/0tCosyTRx3u4VU=">ACpnicbVFdb9MwFHXC1ygwynjkxaJi6gRUyZi0vQxN4gHEAyrSuk6qu3DjOK0124nsG0QV9f/wl/gF/A2ctIi140q2zj3nfsjHamkwyj6FYR37t67/2DnYefR4ye7T7vP9i5cUVkuRrxQhb1MwQkljRihRCUuSytAp0qM0+sPjT7+LqyThTnHRSmGmZG5pIDeirp/mQKzEwJ+rmPB/6KDihldkXtn1ImDTbSG/YRtIYkamtuZMrxqXl/X9M9hdSxjr7DMrSFj9YboHX8bI+f4vLZmgSXbXYz37NEKqNBVtE1uRJtxcNojbobRCvQY+sY5h0f7Os4JUWBrkC5yZxVOK0BouSK7HsMqJEvg1zMTEQwNauGndOrqkrzyT0byw/ngHWvZmR81auVsjhtzatDOLXTq+zXg3G1rDfk/bVJhfjKtpSkrFIav1ueVoljQ5tNoJq3gqBYeALfSv4DyOXhL0X+tdybe9uE2uDgcxO8Gh1+Pemcna492yAvykvRJTI7JGflEhmREeLAbHAWnwfuwH34JR+F4VRoG657nZCPCb38AdZPJ9w=</latexit>
slide-45
SLIDE 45

Green-Kubo linear-response theory

Λij = Ω kB ∞ Ji(t)Jj(0)dt

<latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit>

J = 1 Ω

j(r)dr

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J(t) = J(Γt) = J(t, Γ0)

<latexit sha1_base64="Y/ipwzwJRIv28OvaTUiqAkTysk=">ACJHicbVDNSgMxGMz6W+vfqkcvwWKpIGW3CvZSKHhQeqpgW6FbSjZN29Bkd0m+FcrSN/A5fACv+gjexIMXr76GabsHrQ4EJjOZL3zjR4JrcJwPa2l5ZXVtPbOR3dza3tm19/abOowVZQ0ailDd+UQzwQPWA6C3UWKEekL1vJHl1O/dc+U5mFwC+OIdSQZBLzPKQEjde18rQAnOF/BtYJ3RaQkXP1vGy+YozTVHJOunbOKToz4L/ETUkOpah37S+vF9JYsgCoIFq3XSeCTkIUcCrYJOvFmkWEjsiAtQ0NiGS6k8z2meBjo/RwP1TmBIBn6s9EQqWv+GAIv+YkRGo9lr7JSwJDvehNxf+8dgz9cifhQRQDC+j8+34sMIR4WhnucUoiLEhCpuNsB0SBShYIo1zbiLPfwlzVLRPSuWbs5z1XLaUQYdoiNUQC6QFV0jeqogSh6QE/oGb1Yj9ar9Wa9z58uWnmAP2C9fkN9qiCQ=</latexit>
  • 1

T t T t J(t + τ, Γ0)J(τ, Γ0)dτ

<latexit sha1_base64="oN8l1/xSxXZw/0tCosyTRx3u4VU=">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</latexit>

J(t)J(0) =

  • J(t, Γ0)J(0, Γ0)P (Γ0)dΓ0
  • <latexit sha1_base64="oN8l1/xSxXZw/0tCosyTRx3u4VU=">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</latexit>
slide-46
SLIDE 46

Einstein-Helfand relaFons

Einstein (1905)

  • |x(t) x(0)|2

=

  • t

v(t)dt

  • 2

2Dt D = v(t)v(0)dt

<latexit sha1_base64="gxqmbM4YXxOWgvLt4/qMElHRs=">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</latexit>
slide-47
SLIDE 47

Einstein-Helfand relaFons

Einstein (1905)

  • |x(t) x(0)|2

=

  • t

v(t)dt

  • 2

2Dt D = v(t)v(0)dt

<latexit sha1_base64="gxqmbM4YXxOWgvLt4/qMElHRs=">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</latexit>

Helfand (1960)

  • t

J(t)dt

  • 2

2Λt Λ = J(t)J(0)dt

<latexit sha1_base64="QaXBtrlPDlz6zV/YrGvBPDfA9c=">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</latexit>
slide-48
SLIDE 48

the classical energy current

J = 1 Ω

j(r)dr

<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>
slide-49
SLIDE 49

the classical energy current

J = 1 Ω

j(r)dr

<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>

· j(r) = ˙ (r)

  • <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>
slide-50
SLIDE 50

the classical energy current

J = 1 Ω

j(r)dr

<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>

·

r

  • · j(r)
  • dr =

r˙ (r)dr

  • <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>

· j(r) = ˙ (r)

  • <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>
slide-51
SLIDE 51

the classical energy current

J = 1 Ω

j(r)dr

<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>

j(r)dr =

r˙ (r)dr + surface terms

<latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>

·

r

  • · j(r)
  • dr =

r˙ (r)dr

  • <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>

· j(r) = ˙ (r)

  • <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>
slide-52
SLIDE 52

the classical energy current

J = 1 Ω

j(r)dr

<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>

J = 1 Ω

r˙ (r)dr

<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit>

j(r)dr =

r˙ (r)dr + surface terms

<latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>

·

r

  • · j(r)
  • dr =

r˙ (r)dr

  • <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>

· j(r) = ˙ (r)

  • <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>
slide-53
SLIDE 53

J = 1 Ω

r˙ (r)dr

<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">ACVXicbVHLSgMxFE3H+n606tJNsAi6KTMq6EYQ3YgbK1grdErJpHfaYJIZkjtCGfpfoe4duFGP0Ewfj2QuBwzrn3JidRKoVF38seFPF6ZnZufmFxaXlVJ5de3aJpnhUOeJTMxNxCxIoaGOAiXcpAaYiQ0otvTod64A2NFoq+wn0JLsa4WseAMHdUuN0LFsBfF9Jwe0TA2jOfBIA8vFHTZIBQa2NM6YfR0LCTYAipFTLRdPuT36GdL0+7XPGr/qjoXxBMQIVMqtYuP7uxPFOgkUtmbTPwU2zlzKDgEgYLYWYhZfyWdaHpoGYKbCsfBTCgW47p0Dgx7mikI/Z7R85VZES3hz/m5ExZ21eR6x/e2/7WhuR/WjPD+LCVC51mCJqP18eZpJjQYca0IwxwlH0HGDfCvYDyHnPZovsJl0zwO4e/4Hq3GuxVdy/3K8cnk4zmyAbZJNskIAfkmJyRGqkTu7JE3khr4WHwptX9GbGVq8w6VknP8orvQPECrU/</latexit>

the classical energy current

(r, t) =

  • I
  • r − RI(t)
  • eI
  • R(t), V(t)
  • <latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">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</latexit>
slide-54
SLIDE 54

J = 1 Ω

r˙ (r)dr

<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit>

the classical energy current

(r, t) =

  • I
  • r − RI(t)
  • eI
  • R(t), V(t)
  • <latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">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</latexit>

eI(R, V) = 1 2MIV2

I + 1

2

  • J=I

v(|RI − RJ|)

<latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit>
slide-55
SLIDE 55

J =

  • I

˙ RIeI + RI ˙ eI

  • <latexit sha1_base64="hTV/28IlByLW+14TZOSP3HgwK5k=">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</latexit>

J = 1 Ω

r˙ (r)dr

<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit>

the classical energy current

(r, t) =

  • I
  • r − RI(t)
  • eI
  • R(t), V(t)
  • <latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">ACnHicbVHtShtBFJ1dP6q2amp/FspgKETRsKuC/hGk9UdDKag0UciGZXZyNxmcmV1m7gph2Vfpe/UJfA1nY4QYvTBwOPecO9xzk1wKi0Hw3/OXldWP6ytb3z8tLm13fi807NZYTh0eSYzc5cwC1Jo6KJACXe5AaYSCbfJ/c+6f/sAxopM/8VJDgPFRlqkgjN0VNz4F0Fuhcx0K1IMx0lamupgCo0qsdo7j2yh4g6NhiCRYkYSTonPXyBN1Xcac35aC01dA/izoLprXHbzQveo9e9xoBu1gWvQtCGegSWZ1FTceo2HGCwUauWTW9sMgx0HJDAouodqICgs54/dsBH0HNVNgB+U0x4p+d8yQplxTyOdsvOkqvEiNEYX80pmbJ2ohLnrxewi72afK/XLzA9G5RC5wWC5s/fp4WkmNH6VHQoDHCUEwcYN8JtQPmYGcbRHdQlEy7m8Bb0jtrhcfvo+qR58WOW0Rr5SnZJi4TklFyQX+SKdAn3lr1979g78b/5l/5v/8+z1Pdmni/kVfm9J9MCzmU=</latexit>

eI(R, V) = 1 2MIV2

I + 1

2

  • J=I

v(|RI − RJ|)

<latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit>
slide-56
SLIDE 56

J =

  • I

˙ RIeI + RI ˙ eI

  • <latexit sha1_base64="hTV/28IlByLW+14TZOSP3HgwK5k=">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</latexit>

J = 1 Ω

r˙ (r)dr

<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">ACVXicbVHLSgMxFE3H+n606tJNsAi6KTMq6EYQ3YgbK1grdErJpHfaYJIZkjtCGfpfoe4duFGP0Ewfj2QuBwzrn3JidRKoVF38seFPF6ZnZufmFxaXlVJ5de3aJpnhUOeJTMxNxCxIoaGOAiXcpAaYiQ0otvTod64A2NFoq+wn0JLsa4WseAMHdUuN0LFsBfF9Jwe0TA2jOfBIA8vFHTZIBQa2NM6YfR0LCTYAipFTLRdPuT36GdL0+7XPGr/qjoXxBMQIVMqtYuP7uxPFOgkUtmbTPwU2zlzKDgEgYLYWYhZfyWdaHpoGYKbCsfBTCgW47p0Dgx7mikI/Z7R85VZES3hz/m5ExZ21eR6x/e2/7WhuR/WjPD+LCVC51mCJqP18eZpJjQYca0IwxwlH0HGDfCvYDyHnPZovsJl0zwO4e/4Hq3GuxVdy/3K8cnk4zmyAbZJNskIAfkmJyRGqkTu7JE3khr4WHwptX9GbGVq8w6VknP8orvQPECrU/</latexit>

the classical energy current

(r, t) =

  • I
  • r − RI(t)
  • eI
  • R(t), V(t)
  • <latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">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</latexit>

eI(R, V) = 1 2MIV2

I + 1

2

  • J=I

v(|RI − RJ|)

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J =

  • I

eIVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ)

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slide-57
SLIDE 57

hurdles towards an ab iniFo Green-Kubo theory

Thermal Conductivity of Periclase (MgO) from First Principles

Stephen Stackhouse*

Department of Geological Sciences, University of Michigan, Ann Arbor, Michigan, 48109-1005, USA

Lars Stixrude†

Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, United Kingdom

Bijaya B. Karki‡

Department of Computer Science, Louisiana State University, Baton Rouge, Louisiana 70803, USA and Department of Geology and Geophysics, Louisiana State University, Baton Rouge, Louisiana 70803, USA

PRL 104, 208501 (2010) P H Y S I C A L R E V I E W L E T T E R S

week ending 21 MAY 2010

sensitive to the form of the potential. The widely used Green-Kubo relation [14] does not serve our purposes, because in first-principles calculations it is impossible to uniquely decompose the total energy into individual con- tributions from each atom.

slide-58
SLIDE 58

insights from classical mechanics

Je =

  • I

IVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ) I(R, V) = 1 2MIV2

I + 1

2

  • J=I

v(|RI − RJ|) E =

  • I

I(R, V) =

slide-59
SLIDE 59

insights from classical mechanics

  • I

I(R, V) = I(R, V) = 1 2MIV2

I + 1

2

  • J=I

v(|RI − RJ|)(1 + ΓIJ) Je =

  • I

IVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ) + 1 2

  • I=J

ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]

slide-60
SLIDE 60

insights from classical mechanics

Je =

  • I

IVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ) + 1 2

  • I=J

ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]

slide-61
SLIDE 61

insights from classical mechanics

Je =

  • I

IVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ) + 1 2

  • I=J

ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]

0.1 0.2

t

J(t)J(0)

ΓIJ =         

1 2 (AIJ − AJI) ,

AIJ ∼ U[0, γ] sgn(I − J) × γ

slide-62
SLIDE 62

insights from classical mechanics

Je =

  • I

IVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ) + 1 2

  • I=J

ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]

0.1 0.2

t

J(t)J(0)

ΓIJ =         

1 2 (AIJ − AJI) ,

AIJ ∼ U[0, γ] sgn(I − J) × γ

t

0.4 0.2

κ

ΓIJ =         

1 2 (AIJ − AJI) ,

AIJ ∼ U[0, γ] sgn(I − J) × γ

slide-63
SLIDE 63

insights from classical mechanics

Je =

  • I

IVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ) + 1 2

  • I=J

ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]

slide-64
SLIDE 64

˙ P = d dt 1 4

  • I=J

ΓIJ v(|RI − RJ|)(RI − RI)

insights from classical mechanics

Je =

  • I

IVI + 1 2

  • I=J

(VI · FIJ)(RI − RJ) + 1 2

  • I=J

ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]

slide-65
SLIDE 65

J′ = J + ˙ P

insights from classical mechanics

slide-66
SLIDE 66

J′ = J + ˙ P

insights from classical mechanics

var

  • D
  • D(t) =

t J(t)dt

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κ ∼ 1 2tvar

  • D(t)
  • D
  • J
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slide-67
SLIDE 67

D(t) = D(t)+P(t) − P(0)

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J′ = J + ˙ P

insights from classical mechanics

var

  • D
  • D(t) =

t J(t)dt

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κ ∼ 1 2tvar

  • D(t)
  • D
  • J
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slide-68
SLIDE 68

D(t) = D(t)+P(t) − P(0)

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var

  • D(t)
  • = var
  • D(t)
  • + var
  • ∆P(t)
  • + 2cov
  • D(t) · ∆P(t)
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J′ = J + ˙ P

insights from classical mechanics

var

  • D
  • D(t) =

t J(t)dt

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κ ∼ 1 2tvar

  • D(t)
  • D
  • J
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slide-69
SLIDE 69

D(t) = D(t)+P(t) − P(0)

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var

  • D(t)
  • = var
  • D(t)
  • + var
  • ∆P(t)
  • + 2cov
  • D(t) · ∆P(t)
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J′ = J + ˙ P

insights from classical mechanics

var

  • D
  • D(t) =

t J(t)dt

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κ ∼ 1 2tvar

  • D(t)
  • D
  • J
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var

  • D
  • var
  • D
  • O(t)

var

  • P
  • cov
  • D

· P

  • O(1)
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slide-70
SLIDE 70

insights from classical mechanics

κ′ = κ J′ = J + ˙ P

slide-71
SLIDE 71

E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

slide-72
SLIDE 72

E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

∂𝝯

slide-73
SLIDE 73

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

E[Ω] =

e(r)dr

∂𝝯

slide-74
SLIDE 74

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

E[Ω] =

e(r)dr

E′[Ω] = E[Ω] + O[∂Ω]

∂𝝯

slide-75
SLIDE 75

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

E[Ω] =

e(r)dr

E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r)

∂𝝯

slide-76
SLIDE 76

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

˙ e′(r, t) = −∇ ·

  • j(r, t) + ˙

p(r, t)

  • E[Ω] =

e(r)dr

E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r)

∂𝝯

slide-77
SLIDE 77

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

˙ e′(r, t) = −∇ ·

  • j(r, t) + ˙

p(r, t)

  • E[Ω] =

e(r)dr

E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r) j′(r, t) = j(r, t) + ˙ p(r, t)

∂𝝯

slide-78
SLIDE 78

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

˙ e′(r, t) = −∇ ·

  • j(r, t) + ˙

p(r, t)

  • J′(t) = J(t) + ˙

P(t)

E[Ω] =

e(r)dr

E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r) j′(r, t) = j(r, t) + ˙ p(r, t)

∂𝝯

slide-79
SLIDE 79

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

˙ e′(r, t) = −∇ ·

  • j(r, t) + ˙

p(r, t)

  • J′(t) = J(t) + ˙

P(t)

E[Ω] =

e(r)dr

E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r) j′(r, t) = j(r, t) + ˙ p(r, t)

∂𝝯

any two energy densiFes that differ by the divergence of a (bounded) vector field are physically equivalent

slide-80
SLIDE 80

E ∪ E E

?

= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]

gauge invariance

˙ e′(r, t) = −∇ ·

  • j(r, t) + ˙

p(r, t)

  • J′(t) = J(t) + ˙

P(t)

E[Ω] =

e(r)dr

E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r) j′(r, t) = j(r, t) + ˙ p(r, t)

∂𝝯

any two energy densiFes that differ by the divergence of a (bounded) vector field are physically equivalent the corresponding energy fluxes differ by a total Fme derivaFve, and the heat transport coefficients coincide

slide-81
SLIDE 81

EDF T = 1 2

  • I

MIV2

I + e2

2

  • I=J

ZIZJ RIJ +

  • v

v−1 2EH + XC(r) − µXC(r)

  • (r)dr

density-funcFonal theory

slide-82
SLIDE 82

EDF T = 1 2

  • I

MIV2

I + e2

2

  • I=J

ZIZJ RIJ +

  • v

v−1 2EH + XC(r) − µXC(r)

  • (r)dr

the DFT energy density

eDF T (r) = e0(r) + eKS(r) + eH(r) + eXC(r) r

  • r

R

  • r

Re r r r r r r r r r

slide-83
SLIDE 83

EDF T = 1 2

  • I

MIV2

I + e2

2

  • I=J

ZIZJ RIJ +

  • v

v−1 2EH + XC(r) − µXC(r)

  • (r)dr

r r r r r e0(r) =

  • I

(r − RI) 1 2MIV 2

I + wI

  • eKS(r) = Re
  • v

v(r)

ˆ HKSv(r)

  • eH(r) = −1

2(r)vH(r) eXC(r) = (XC(r) − vXC(r)) (r)

the DFT energy density

eDF T (r) = e0(r) + eKS(r) + eH(r) + eXC(r) r

  • r

R

  • r

Re r r r r r r r r r

slide-84
SLIDE 84

the DFT energy current

JDF T =

eDF T (r, t)dr = JKS + JH + J

0 + J0 + JXC

slide-85
SLIDE 85

the DFT energy current

JKS =

  • v
  • v|r ˆ

HKS| ˙ v + v ˙ v|r|v

  • JH = 1

4

  • ˙

vH(r)vH(r)dr J

0 =

  • v,I

v |(r RI) (VI · Iˆ v0)| v J0 =

  • I
  • VIe0

I +

  • L=I

(RI RL) (VL · LwI)

  • JXC =
  • (LDA)
  • (r) ˙

(r)GGA(r)dr (GGA) JDF T =

eDF T (r, t)dr = JKS + JH + J

0 + J0 + JXC

slide-86
SLIDE 86

JKS =

  • v
  • v|r ˆ

HKS| ˙ v + v ˙ v|r|v

  • J
  • r

r r J r R V J V R R V J (LDA) r r r r (GGA) J r r

  • JH = 1

4

  • ˙

vH(r)vH(r)dr J

0 =

  • v,I

v |(r RI) (VI · Iˆ v0)| v J0 =

  • I
  • VIe0

I +

  • L=I

(RI RL) (VL · LwI)

  • JXC =
  • (LDA)
  • (r) ˙

(r)GGA(r)dr (GGA)

  • | ˙

ϕv and ˆ HKS| ˙ ϕv orthogonal to the

  • ccupied-state manifold
  • ˆ

Pcr|ϕv computed from standard DFPT

the DFT energy current

JDF T =

eDF T (r, t)dr = JKS + JH + J

0 + J0 + JXC

slide-87
SLIDE 87

a benchmark

108 “LDA Ar” atoms @bp density, T= 250 K 100 ps CP trajectory 100 ps classical FF trajectory 1 ns classical FF trajectory J(t) · J(0)

a

1 κ C✏

slide-88
SLIDE 88

a benchmark

108 “LDA Ar” atoms @bp density, T= 250 K 100 ps CP trajectory 100 ps classical FF trajectory 1 ns classical FF trajectory 1 3V kBT 2 t J(t) · J(0)dt

t"[ps]

b

1" 100 κ

J(t) · J(0)

a

1 κ C✏

slide-89
SLIDE 89

a benchmark

108 “LDA Ar” atoms @bp density, T= 250 K 100 ps CP trajectory 100 ps classical FF trajectory 1 ns classical FF trajectory 1 3V kBT 2 t J(t) · J(0)dt

t"[ps]

b

1" 100 κ

same behavior at T=400 K J(t) · J(0)

a

1 κ C✏

slide-90
SLIDE 90

liquid (heavy) water

1 3V kBT 2 t J(t) · J(0)dt

1 2 t[ps]

64 molecules, T=385 K expt density @ac

slide-91
SLIDE 91

liquid (heavy) water

1 3V kBT 2 t J(t) · J(0)dt

1 2 t[ps]

64 molecules, T=385 K expt density @ac ω S(ω) = ∞

−∞

J(t) · J(0) eiωtdt

slide-92
SLIDE 92

1 3V kBT 2 t J(t) · J(0)dt

1 2 t[ps]

liquid (heavy) water

64 molecules, T=385 K expt density @ac t 3V kBT 2 t J(t) · J(0)dt 1 6V kBT 2

  • t

J(t)dt

  • 2

≈ Einstein’s relaFon

slide-93
SLIDE 93

liquid (heavy) water

1 6V kBT 2

  • t

J(t)dt

  • 2

1 2 t[ps]

64 molecules, T=385 K expt density @ac 1 3V kBT 2 t J(t) · J(0)dt

1 2 t[ps]

slide-94
SLIDE 94

liquid (heavy) water

1 6V kBT 2

  • t

J(t)dt

  • 2

1 2 t[ps]

64 molecules, T=385 K expt density @ac 1 3V kBT 2 t J(t) · J(0)dt

1 2 t[ps]

κDFT = 0.74 ± 0.12 W/(mK) κexpt = 0.61 (light@AC) κDFT = 0.60 (heavy@AC)

slide-95
SLIDE 95

sensitive to the form of the potential. The widely used Green-Kubo relation [14] does not serve our purposes, because in first-principles calculations it is impossible to uniquely decompose the total energy into individual con- tributions from each atom.

PRL 104, 208501 (2010) P H Y S I C A L R E V I E W L E T T E R S

week ending 21 MAY 2010

Stephen Stackhouse, Lars Stixrude, and Bijaya B. Karki

Thermal Conductivity of Periclase (MgO) from First Principles

hurdles towards an ab iniFo Green-Kubo theory

slide-96
SLIDE 96

Ab Initio Green-Kubo Approach for the Thermal Conductivity of Solids

PRL 118, 175901 (2017) P H Y S I C A L R E V I E W L E T T E R S

week ending 28 APRIL 2017

Christian Carbogno, Rampi Ramprasad, and Matthias Scheffler

ulations: Because of the limited time scales accessible in aiMD runs, thermodynamic fluctuations dominate the HFACF, which in turn prevents a reliable and numerically stable assessment of the thermal conductivity via Eq. (2). Furthermore, achieving convergence with respect to system sensitive to the form of the potential. The widely used Green-Kubo relation [14] does not serve our purposes, because in first-principles calculations it is impossible to uniquely decompose the total energy into individual con- tributions from each atom.

PRL 104, 208501 (2010) P H Y S I C A L R E V I E W L E T T E R S

week ending 21 MAY 2010

Stephen Stackhouse, Lars Stixrude, and Bijaya B. Karki

Thermal Conductivity of Periclase (MgO) from First Principles

hurdles towards an ab iniFo Green-Kubo theory

slide-97
SLIDE 97

Green-Kubo vs. Einstein-Helfand

  • J(t)
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κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt

slide-98
SLIDE 98

Green-Kubo vs. Einstein-Helfand

  • J(t)J(0)
<latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit>
  • J(t)
<latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit>

κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt

slide-99
SLIDE 99

Green-Kubo vs. Einstein-Helfand

  • J(t)J(0)
<latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit><latexit sha1_base64="kR+GEuspQyDw8D5dX7n4U81hHKw=">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</latexit>
  • J(t)
<latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit><latexit sha1_base64="lSlNKg3IiMkLZAlHsEBRxTftqO8=">ACOXicbVDLSgNBEJz1GeMrUTx5GQxCvIRdEcxR8CKeIhgVkizk95kyMzsMtMrhCX/4V/wy/x6E28+gPOJnswakNDUdVNV1eYSGHR9+8hcWl5ZXV0lp5fWNza7tS3bmxcWo4tHksY3MXMgtSaGijQAl3iQGmQgm34eg8128fwVgR62scJ9BTbKBFJDhDR913FcNhGWXkzoe0YdKzW/406J/QVCAGimq9VD19r9mKcKNHLJrO0EfoK9jBkUXMKk3E0tJIyP2A6DmqmwPayqe0JPXRMn0axca2RTtmfGxlT1o5V6CZzm/a3lpP/aZ0Uo2YvEzpJETSfHYpSTGmeQa0LwxwlGMHGDfCeaV8yAzj6JKau5Ibswlw94kFVEzonMnOmRShEXP/ZaBTJRDUpFx2Qa/Y/sLbo4bgd8Irk5qZ80i0hLZJwekTgJySs7IBWmRNuHEkCfyTF68V+/d+/A+Z6MLXrGzS+bK+/oGLZGtBA=</latexit>
  • t

J(t)J(0)dt

<latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">ACaHicbVDBSiNBEO3M6qpxd42KiHhpDKJ7CTOysIXwYt4UjAqZLKhplMTG7t7hu6ahTDMx+zX7FWP/oJfYU+Sg1EfFLx6VUVvSRX0lEYPjeCLwuLX5eWV5qr37/WGutb9y4rLACuyJTmb1LwKGSBrskSeFdbhF0ovA2eTir67d/0TqZmWsa59jXMDIylQLIS4PWSwNDcI/JVU8VmBGCnmsge6TtLyoDung59s09JmdNg3pYNBqh51wAv6RDPSZjNcDtYbW/EwE4VGQ0KBc70ozKlfgiUpFbNuHCYg3iAEfY8NaDR9cvJlxXf98qQp5n1YhP1LcTJWjnxjrxnfXF7n2tFj+r9QpKj/ulNHlBaMR0UVoThmvLeNDaVGQGnsCwkp/Kxf3YEGQN3ZuS32Yy1H4TxySBmlqpTwDJRMr5/4r0RaEuq2fRGRu9t+0hujpR2ImufrVPj2eWLrNdtscOWcR+s1N2zi5Zlwn2j/1nj+yp8RK0gu1gZ9oaNGYzm2wOwd4rap61g=</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit>

κ

<latexit sha1_base64="HQZyO9P6Lsay47RpTE2XjdMsf4=">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</latexit><latexit sha1_base64="HQZyO9P6Lsay47RpTE2XjdMsf4=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit>

κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt

slide-100
SLIDE 100

Green-Kubo vs. Einstein-Helfand

  • J(t)J(0)
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  • J(t)
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  • t

J(t)J(0)dt

<latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit><latexit sha1_base64="HsK6pvg4OKlBdE5OFin+gYIDZ0=">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</latexit>

κ

<latexit sha1_base64="HQZyO9P6Lsay47RpTE2XjdMsf4=">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</latexit><latexit sha1_base64="HQZyO9P6Lsay47RpTE2XjdMsf4=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit>

κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt

  • t

J(t)dt

  • 2

2t J(t)J(0)dt

<latexit sha1_base64="wLZhjaWaiCPS2hmyYBewowj2U=">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</latexit>
slide-101
SLIDE 101

Green-Kubo vs. Einstein-Helfand

  • J(t)J(0)
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  • J(t)
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  • t

J(t)J(0)dt

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κ

<latexit sha1_base64="HQZyO9P6Lsay47RpTE2XjdMsf4=">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</latexit><latexit sha1_base64="HQZyO9P6Lsay47RpTE2XjdMsf4=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit>

κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt

  • κ
<latexit sha1_base64="t+sn9D5jokghWQCsUi1PhQAahw=">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</latexit><latexit sha1_base64="t+sn9D5jokghWQCsUi1PhQAahw=">ACMXicdVDBSiNBEO3RXTfG1TUunrw0hoU9hZmY1eQmeNmjgolCEqSmU9E23T1Nd81CGPIPXvU3/Bpvstf9CXtiBLPsFhQ8XlVR73UKukpjp+ildUPH9c+VdarG583t75s13Z6PsudwK7IVOYuU/CopMEuSVJ4aR2CThVepJOTcn7xC52XmTmnqcWhmsjx1IABao3mIC1cLVdjxudw8PWQYfHjXheJWg3O80fPFkwdbao06tatDsYZSLXaEgo8L6fxJaGBTiSQuGsOsg9WhATuMZ+gAY0+mExlzvj3wIz4uPMhTbE5+z7iwK091Odhk0NdOP/npXkv2b9nMbtYSGNzQmNeH0zhWnjJfe+Ug6FKSmAYBwMmjl4gYcCAoJLX0phXmLIjxSBqkKZniBJRMnVzyV6DJtSTUs2o1BPmWFv8/6DUbSdxIzlr14/Yi0grbY/vsO0vYETtmP9kp6zLBbtkdu2cP0WP0FD1Hv19XV6LFzVe2VNGfF8h3qmc=</latexit><latexit sha1_base64="t+sn9D5jokghWQCsUi1PhQAahw=">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</latexit><latexit sha1_base64="t+sn9D5jokghWQCsUi1PhQAahw=">ACMXicdVDBSiNBEO3RXTfG1TUunrw0hoU9hZmY1eQmeNmjgolCEqSmU9E23T1Nd81CGPIPXvU3/Bpvstf9CXtiBLPsFhQ8XlVR73UKukpjp+ildUPH9c+VdarG583t75s13Z6PsudwK7IVOYuU/CopMEuSVJ4aR2CThVepJOTcn7xC52XmTmnqcWhmsjx1IABao3mIC1cLVdjxudw8PWQYfHjXheJWg3O80fPFkwdbao06tatDsYZSLXaEgo8L6fxJaGBTiSQuGsOsg9WhATuMZ+gAY0+mExlzvj3wIz4uPMhTbE5+z7iwK091Odhk0NdOP/npXkv2b9nMbtYSGNzQmNeH0zhWnjJfe+Ug6FKSmAYBwMmjl4gYcCAoJLX0phXmLIjxSBqkKZniBJRMnVzyV6DJtSTUs2o1BPmWFv8/6DUbSdxIzlr14/Yi0grbY/vsO0vYETtmP9kp6zLBbtkdu2cP0WP0FD1Hv19XV6LFzVe2VNGfF8h3qmc=</latexit><latexit sha1_base64="t+sn9D5jokghWQCsUi1PhQAahw=">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</latexit>

1 2

  • t

J(t)dt

  • 2
<latexit sha1_base64="9aLIg7wOHFJajNecvx/tbd/KL78=">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</latexit><latexit sha1_base64="9aLIg7wOHFJajNecvx/tbd/KL78=">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</latexit><latexit sha1_base64="9aLIg7wOHFJajNecvx/tbd/KL78=">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</latexit><latexit sha1_base64="9aLIg7wOHFJajNecvx/tbd/KL78=">ACh3icbZDBThsxEIadpS2Q0jZQ9dSLRYSgl3Q3qoAjFZeqJyo1gBqHyOvMJha2d2XPIkVm343X6Av02j5CvckiEWAkS7/mfF4vrRQ0mEc/25Fay9evlrf2Gy/3nrz9l1ne+fc5aUVMBC5yu1lyh0oaWCAEhVcFha4ThVcpNendf7iBqyTufmJ8wJGmk+NzKTgGKx5xfLBc+qXy/YgoypExM1VAl7dbyqTBcXzlsaJMc5ylmf9eHeD+J0onuE+ZldNZqLvq30tmlw+MO924Fy+CPhVJI7qkibPxdusDm+Si1GBQKO7cMIkLHluUQoFVZuVDgourvkUhkEarsGN/AJCRfeCM6FZbsMxSBfuw7PtXNznYbKeg3OFebz+WGJWbHIy9NUSIYsRyUlYpiTmuidCItCFTzILiwMvyVihkPVDFwX5lSf8wVIMImDlBzaWrHn3IlUytX9vNgSi0RdNVuB5DJY2xPxXm/l8S95MeX7slxg3SDfCS75IAk5IickG/kjAyIHfkD/lL/kWb0efoMGpqo1bT856sRPT1P/OoxjE=</latexit><latexit sha1_base64="9aLIg7wOHFJajNecvx/tbd/KL78=">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</latexit>
slide-102
SLIDE 102

Einstein vs. Green-Kubo

κGK(T) = T v(t)v(0)dt ∞ v t v 0

T t dt

Cv

T

d

slide-103
SLIDE 103

Einstein vs. Green-Kubo

κGK(T) = T v(t)v(0)dt ∞ v t v 0

T t dt

Cv

T

d

ΘT(t)

T 1

GK T

T v t v 0 dt = ∞ v(t)v(0)ΘT(t)dt ∞ Cv

T

d

slide-104
SLIDE 104

Einstein vs. Green-Kubo

κGK(T) = T v(t)v(0)dt = ∞ v(t)v(0)ΘT(t)dt ∞ Cv

T

d κE(T) = 1 2T

  • T

v(t)dt

  • 2

∞ v t v 0

T t dt

Cv

T

d

ΘT(t)

T 1

slide-105
SLIDE 105

ΛT(t) = 1 − t T

T 1

ΘT(t)

Einstein vs. Green-Kubo

κGK(T) = T v(t)v(0)dt = ∞ v(t)v(0)ΘT(t)dt ∞ Cv

T

d κE(T) = 1 2T

  • T

v(t)dt

  • 2

∞ v t v 0

T t dt

Cv

T

d

E T

1 2T

  • T

v t dt

  • 2

= ∞ v(t)v(0)ΛT(t)dt ∞ Cv

T

d

slide-106
SLIDE 106

Einstein vs. Green-Kubo

κGK(T) = T v(t)v(0)dt = ∞ v(t)v(0)ΘT(t)dt = ∞

−∞

˜ Cv(ω)˜ ΘT(ω)dω 2π κE(T) = 1 2T

  • T

v(t)dt

  • 2

= ∞ v(t)v(0)ΛT(t)dt = ∞

−∞

˜ Cv(ω)˜ ΛT(ω)dω 2π

T 1

ΛT(t) = 1 − t T ΘT(t)

2 Π 4 Π 6 Π 1

˜ ΘT(ω) ˜ ΛT(ω)

slide-107
SLIDE 107

separaFng wheat from chaff

κ ∞ C(t)dt C(t) = J(t)J(0) κ S(ω = 0) S(ω) = ∞

−∞

C(t)e−iωtdt

<latexit sha1_base64="dlr6XNFARaPUay7AnG/iQTlyg9w=">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</latexit>
slide-108
SLIDE 108

separaFng wheat from chaff

κ ∞ C(t)dt C(t) = J(t)J(0) κ S(ω = 0) S(ω) = ∞

−∞

C(t)e−iωtdt

<latexit sha1_base64="dlr6XNFARaPUay7AnG/iQTlyg9w=">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</latexit>

the Wiener-Khintchine theorem S(ω) = lim

T →∞

  • 1

T

  • T

2

− T

2

J(t)eiωt

  • 2
<latexit sha1_base64="GYd+a4OkLTcKI+yGh5R3gNrDZ98=">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</latexit>
slide-109
SLIDE 109

separaFng wheat from chaff

κ ∞ C(t)dt C(t) = J(t)J(0) κ S(ω = 0) S(ω) = ∞

−∞

C(t)e−iωtdt

<latexit sha1_base64="dlr6XNFARaPUay7AnG/iQTlyg9w=">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</latexit>

the Wiener-Khintchine theorem S(ω) = lim

T →∞

  • 1

T

  • T

2

− T

2

J(t)eiωt

  • 2
<latexit sha1_base64="GYd+a4OkLTcKI+yGh5R3gNrDZ98=">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</latexit>

in pracFce: S (k) = N

  • N−1
  • n=0

Jne−i 2πnk

N

  • 2
<latexit sha1_base64="QT2WmUBZ5tb85ArVUCszmJGDNzU=">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</latexit>
slide-110
SLIDE 110

separaFng wheat from chaff

  • ˆ

S(k) = N

  • ˜

J(k)

  • 2
  • = 1

2S(k) × 2

2

H2O

slide-111
SLIDE 111

separaFng wheat from chaff

  • ˆ

S(k) = N

  • ˜

J(k)

  • 2
  • = 1

2S(k) × 2

2

H2O

slide-112
SLIDE 112

separaFng wheat from chaff

  • ˆ

S(k) = N

  • ˜

J(k)

  • 2
  • = 1

2S(k) × 2

2

H2O

slide-113
SLIDE 113

separaFng wheat from chaff

S(ω) log

  • S(ω)
  • ˆ

S(k) = S(ωk)ˆ ξk

  • log

ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

slide-114
SLIDE 114

separaFng wheat from chaff

S(ω) log

  • S(ω)
  • ˆ

S(k) = S(ωk)ˆ ξk

  • log

ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

slide-115
SLIDE 115

separaFng wheat from chaff

log ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

  • log
  • S(ω)
  • n

Cn 1 N

N−1

  • k=0

log ˆ S(k)

  • e−i 2πkn

N

= Cn + white noise

slide-116
SLIDE 116

separaFng wheat from chaff

log ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

  • log
  • S(ω)
  • n

Cn 1 N

N−1

  • k=0

log ˆ S(k)

  • e−i 2πkn

N

= Cn + white noise

slide-117
SLIDE 117

log

  • S(ωk)
  • =

P ∗−1

  • n=0

Cnei 2πkn

N

+ less noise

separaFng wheat from chaff

n

Cn log

  • S(ω)
  • 1

N

N−1

  • k=0

log ˆ S(k)

  • e−i 2πkn

N

= Cn + white noise log ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

slide-118
SLIDE 118

log

  • S(ωk)
  • =

P ∗−1

  • n=0

Cnei 2πkn

N

+ less noise

separaFng wheat from chaff

n

Cn log

  • S(ω)
  • 1

N

N−1

  • k=0

log ˆ S(k)

  • e−i 2πkn

N

= Cn + white noise log ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

slide-119
SLIDE 119

log

  • S(ωk)
  • =

P ∗−1

  • n=0

Cnei 2πkn

N

+ less noise

separaFng wheat from chaff

n

Cn log

  • S(ω)
  • 1

N

N−1

  • k=0

log ˆ S(k)

  • e−i 2πkn

N

= Cn + white noise log ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

slide-120
SLIDE 120

log

  • S(ωk)
  • =

P ∗−1

  • n=0

Cnei 2πkn

N

+ less noise

separaFng wheat from chaff

n

Cn log

  • S(ω)
  • 1

N

N−1

  • k=0

log ˆ S(k)

  • e−i 2πkn

N

= Cn + white noise log ˆ S(k)

  • = log
  • S(ωk)
  • + log

ˆ ξk

slide-121
SLIDE 121

log

  • S(ωk)
  • =

P ∗−1

  • n=0

Cnei 2πkn

N

+ less noise

separaFng wheat from chaff

log(κ) = λ + C0 + 2

P ∗−1

  • n=1

Cn ± σ

  • 4P ∗ − 2

N∗

constants independent of the Fme series being sampled

  • pFmal number of coefficients, to be determined
slide-122
SLIDE 122

log

  • S(ωk)
  • =

P ∗−1

  • n=0

Cnei 2πkn

N

+ less noise

separaFng wheat from chaff

log(κ) = λ + C0 + 2

P ∗−1

  • n=1

Cn ± σ

  • 4P ∗ − 2

N∗ ∆κ κ =        Ar (100 ps) 10 % H2O (100 ps) 5 % a-SiO2 (100 ps) 12 % c-MgO (500 ps) 15 %

constants independent of the Fme series being sampled

  • pFmal number of coefficients, to be determined
slide-123
SLIDE 123

sensitive to the form of the potential. The widely used Green-Kubo relation [14] does not serve our purposes, because in first-principles calculations it is impossible to uniquely decompose the total energy into individual con- tributions from each atom.

PRL 104, 208501 (2010) P H Y S I C A L R E V I E W L E T T E R S

week ending 21 MAY 2010

Stephen Stackhouse, Lars Stixrude, and Bijaya B. Karki

Thermal Conductivity of Periclase (MgO) from First Principles Ab Initio Green-Kubo Approach for the Thermal Conductivity of Solids

PRL 118, 175901 (2017) P H Y S I C A L R E V I E W L E T T E R S

week ending 28 APRIL 2017

Christian Carbogno, Rampi Ramprasad, and Matthias Scheffler

ulations: Because of the limited time scales accessible in aiMD runs, thermodynamic fluctuations dominate the HFACF, which in turn prevents a reliable and numerically stable assessment of the thermal conductivity via Eq. (2). Furthermore, achieving convergence with respect to system

hurdles towards an ab iniFo Green-Kubo theory

slide-124
SLIDE 124

molecular dynamics is less and less ergodic as temperature decreases

slide-125
SLIDE 125

molecular dynamics is less and less ergodic as temperature decreases molecular dynamics cannot account for quantum effects, which are more and more important as temperature decreases

slide-126
SLIDE 126

molecular dynamics is less and less ergodic as temperature decreases molecular dynamics cannot account for quantum effects, which are more and more important as temperature decreases at lower temperatures the harmonic approximaFon becomes more and more accurate

slide-127
SLIDE 127

molecular dynamics is less and less ergodic as temperature decreases molecular dynamics cannot account for quantum effects, which are more and more important as temperature decreases at lower temperatures the harmonic approximaFon becomes more and more accurate

do BTE!

slide-128
SLIDE 128

molecular dynamics is less and less ergodic as temperature decreases molecular dynamics cannot account for quantum effects, which are more and more important as temperature decreases at lower temperatures the harmonic approximaFon becomes more and more accurate

do BTE! what about glasses and alloys?

slide-129
SLIDE 129

LaYce dynamics in the harmonic approximaFon

M¨ x = −∂V ∂x

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V (x)

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x

<latexit sha1_base64="KCBxiMsuGj6/9+qinApIUdrJUdk=">ABx3icbY87TwJBFIXv4AvXF2ps5GYWJFdNHKkNhoB4k8EiFkdrjAhJmdzcwsgRAKa1v9If4c/42zug3gqU7OTe5X5QIbmwQfJPC1vbO7l5x3zs4PDo+KZ2etYxKNcMmU0LpTkQNCh5j03IrsJNopDIS2I4mj1nfnqI2XMUvdp5gT9JRzIecUeuixqxfKgeV4Ff+pglzU4Zc9X7pqztQLJUYWyaoMQuqLWcCl143NZhQNqEjXDAZaT4a29WUSmPmMlr6V5LasVnvsvC/7jW1w/vegsdJajFmbuK6YSp8q/yMyR9wjcyKuTOUae7+8dmYasqsI/c8xiuE2aVrUS3lSqjdty7SGnLcIFXMI1hHAHNXiCOjSBAcI7fMAneSaKTMnsb1og+c05rIi8/QDHd3/o</latexit>
slide-130
SLIDE 130

1 2Φ

  • x − x0

2

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LaYce dynamics in the harmonic approximaFon

M¨ x = −∂V ∂x

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V (x0 + u) ≈ V (x0) + 1 2Φu2 + O

  • u3
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V (x)

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x

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slide-131
SLIDE 131

1 2Φ

  • x − x0

2

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LaYce dynamics in the harmonic approximaFon

M¨ x = −∂V ∂x

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M ¨ u = −Φu

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V (x0 + u) ≈ V (x0) + 1 2Φu2 + O

  • u3
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V (x)

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x

<latexit sha1_base64="KCBxiMsuGj6/9+qinApIUdrJUdk=">ABx3icbY87TwJBFIXv4AvXF2ps5GYWJFdNHKkNhoB4k8EiFkdrjAhJmdzcwsgRAKa1v9If4c/42zug3gqU7OTe5X5QIbmwQfJPC1vbO7l5x3zs4PDo+KZ2etYxKNcMmU0LpTkQNCh5j03IrsJNopDIS2I4mj1nfnqI2XMUvdp5gT9JRzIecUeuixqxfKgeV4Ff+pglzU4Zc9X7pqztQLJUYWyaoMQuqLWcCl143NZhQNqEjXDAZaT4a29WUSmPmMlr6V5LasVnvsvC/7jW1w/vegsdJajFmbuK6YSp8q/yMyR9wjcyKuTOUae7+8dmYasqsI/c8xiuE2aVrUS3lSqjdty7SGnLcIFXMI1hHAHNXiCOjSBAcI7fMAneSaKTMnsb1og+c05rIi8/QDHd3/o</latexit>
slide-132
SLIDE 132

1 2Φ

  • x − x0

2

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LaYce dynamics in the harmonic approximaFon

M¨ x = −∂V ∂x

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M ¨ u = −Φu

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V (x0 + u) ≈ V (x0) + 1 2Φu2 + O

  • u3
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V (x)

<latexit sha1_base64="/sOwKCD6oaLAjk3cYIUN1n9Z58=">ABynicdY9LS8NAFIXv1FeNr6hLN4NFqJuQpKGtGym4ceGign2ALWUynTZDJw9mJmI3bl2q3/Dn+O/McG6qOhZHc45F+7nJ4IrbdufqLKxubW9U9019vYPDo/M45O+ilNJWY/GIpZDnygmeMR6mvBholkJPQFG/iLm7IfPDGpeBw96Cxh45DMIz7jlOgy6tefLydmzbau2k3Xa2Lbsu2W4zqlcVtew8NOkZSqwUrdifkxmsY0DVmkqSBK5URqTgVbGqNUsYTQBZmznIa+5PNAr6ckVCoL/SW+CIkO1O+uDP/qHlM9a49zHiWpZhEtJkU3SwXWMS6x8JRLRrXICkOo5MU/mAZEqoLeMoGH9A8P+m71pOw3LvVrnekVbhTM4hzo40IO3EIXekAhgFd4g3d0hyTKUP49raDVzSmsCb18AXbQgO8=</latexit>

x

<latexit sha1_base64="KCBxiMsuGj6/9+qinApIUdrJUdk=">ABx3icbY87TwJBFIXv4AvXF2ps5GYWJFdNHKkNhoB4k8EiFkdrjAhJmdzcwsgRAKa1v9If4c/42zug3gqU7OTe5X5QIbmwQfJPC1vbO7l5x3zs4PDo+KZ2etYxKNcMmU0LpTkQNCh5j03IrsJNopDIS2I4mj1nfnqI2XMUvdp5gT9JRzIecUeuixqxfKgeV4Ff+pglzU4Zc9X7pqztQLJUYWyaoMQuqLWcCl143NZhQNqEjXDAZaT4a29WUSmPmMlr6V5LasVnvsvC/7jW1w/vegsdJajFmbuK6YSp8q/yMyR9wjcyKuTOUae7+8dmYasqsI/c8xiuE2aVrUS3lSqjdty7SGnLcIFXMI1hHAHNXiCOjSBAcI7fMAneSaKTMnsb1og+c05rIi8/QDHd3/o</latexit>

u(t) = u(0) cos(ωt) + ˙ u(0) 1 ω sin(ωt)

<latexit sha1_base64="qVf50sMobymGZgbNMvtYEVyhCPg=">ACnicbZBLSwMxFIUz9VXrq+rSTbAIU4QyUwXdKEU3LivYBzilZNJMG5p5kNwIZeg/8Ie41Z24de1W/Temdha29awO57sJ91w/EVyB43xbuaXldW1/HphY3Nre6e4u9dUsZaUNWgsYtn2iWKCR6wBHARrJ5KR0Bes5Q+vJ7z1wKTicXQHo4R1QtKPeMApARN1i1fahvKFtp0y9misbC8OWZ9gKONjrxcDnhAvkISm7jidwrGnePRnsFsORXnV3jRuJkpoUz1bvHZfE1yCKgiVEgmcCjYueFqxhNAh6bOUhr7k/QHMpiRUahT6Y3wUEhioeTYJ/2P3GoLzTsqjRAOLqBkxLNACQ4wnZ8E9LhkFMTKGUMnNPpgOiOkN5niFgunozjdaNM1qxT2pVG9PS7XLrG0eHaBDZCMXnaEaukF1EAUPaEP9Im+rEfrxXq13qajOSt7s49mZL3/AEU1mDU=</latexit>

ω =

  • Φ

M

<latexit sha1_base64="xT6x9L7ekFgTGnHrHb8wgV2oHm8=">AB5HicbZBLS8NAFIVv6qvGV9SlLoJFcFWSVrAboeDGjVDBPsCUMplOkqEzSZyZCVk49qduHXtv4b/41TzatZ3U451643/VTRqVynG+jsra+sblV3TZ3dvf2D6zDo5MoFJFycsEQMfScJoTLqKkYGqSCI+4z0/cnNvO8/EyFpEj+oaUqGHIUxDShGSkcj69RLOAnRtSefhMq9QCce52IFvldUYysmlN3fmWvGrc0NSjVGVkzb5zgjJNYakzJFQFDNSmF4mSYrwBIUkx9wXNIzUYoq4lFPuF/Y5RyqSy908/K97zFTQGuY0TjNFYqxHdBdkzFaJPQe2x1QrNhUG4QF1fYOEKaU+m3mKZmdJeJVk2vUXeb9cb9Za3dKmrcAJncAEuXEbqEDXcDwAp8wgy8jMF6N+P9b7RilDvHsCDj4wfvb4wb</latexit>
slide-133
SLIDE 133

1 2Φ

  • x − x0

2

<latexit sha1_base64="JfNCI5Rqnx/jaACAwzXJ+5TExs=">AB7XicbZA7T8MwFIWd8irhFWBksagqlYHICVUfC6rEwlgk+pBIqRzXba06D9kOahXlFzCzIVbmrvBL+DckbRkKnOnc+6V/NkNOZMKoS8t7G5tb2T39X39g8Oj4zjk7YMIkFoiwQ8EF0XS8qZT1uKU67oaDYczntuJObrO8USFZ4N+rWUh7Hh75bMgIVmnUN4rOUGASW0lsJ05zByXjTgsTS+nfQSzQcCLR7tvFJCJalalbkFkVmyrWrVTg6xyvV6DlokWKoCVmn1j7gwCEnUV4RjKWMsFCOcJroTSRpiMsEjGhPFWw0Vusp9qSceW4Cix5WY/m7y8L/uodIDWu9mPlhpKhP0pW0G0YcqgBm5HDABCWKz1KDiWDpeyAZ45Repf+j60vGheBf8PYtk3ryrTvyoXG9Yo2D87AOSgBC1RBA9yCJmgBAp7BHyATy3QXrRX7W25mtNWN6dgTdr7N70XjgY=</latexit>

LaYce dynamics in the harmonic approximaFon

M¨ x = −∂V ∂x

<latexit sha1_base64="FgtUFvHAdG+6g57nYZM4bl6HdU=">AB8HicbZBLS8NAFIVv6qvWV9Slm8EidGNJqAbpeDGjVDBPsCUMplM2qGTBzMTaQn5C67diVt3glv9H/4bJxqQtp7V4Zx7L/ONG3MmlWV9GaWl5ZXVtfJ6ZWNza3vH3N3ryCgRhLZJxCPRc7GknIW0rZjitBcLigOX0647vsr7gMVkXhnZrGtB/gYch8RrDS0cCs3TieFyk0QRfHji8wSZ0YC8UwR53sz0+ygVm16taP0KxC1OFQq2B+eZ4EUkCGirCsZRpfopwmlWcRNIYkzEe0pQErmDkZpNcSDlNHAzdBRgNZLzXR7+190nyj/vpyME0VDokd05ycqQjl8MhjghLFp9pgIph+DyIjrKmV/qJKRTPa80SLptOo2yf1xu1ptXlZ0JbhA6hBjacQROuoQVtIPAI7/ABn4Ywnoxn4+V3tGQUO/swI+P1Gxz1kCU=</latexit>

M ¨ u = −Φu

<latexit sha1_base64="CXAeNIm8gnVr5+5yrUEeCXM98Uk=">AB2nicbZA7SwNBFIXv+ozra42lzWAQbAy7UdBGCdjYCBHMA9wQZmcn2SGzD2buiCGksRNba1t/Tn+Gze6TRJP9XHOvXDPDTIpNLrut7W0vLK6tl7asDe3tnd2nb1yS6dGMd5kqUxVJ6CaS5HwJgqUvJMpTuNA8nYwvJ7m7UeutEiTexlvBvTQSL6glHMrZ5TvXDMEViyCU58RuRIKbnVNyq+yuyCF4BFSjU6DlfpgyE/MEmaRaj6lCwSf2L7RPKNsSAd8zOJAiUGEsy6NtR7FwYQcxRQjPZ9Nzf+yB4P9i+5YJlBnrB8JM/6RhJMybQmCYXiDOUoB8qUyO8hLKMsyfYdt5R2+0SK0alXvtFq7O6vUr4q2JTiAQzgGD86hDjfQgCYweIJ3+IBPy7erRfr9W90ySp29mFG1tsPDPuF8A=</latexit>

V (x0 + u) ≈ V (x0) + 1 2Φu2 + O

  • u3
<latexit sha1_base64="ZjTL3R/+HKbhLj5KYn/qGUOSnY=">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</latexit>

V (x)

<latexit sha1_base64="/sOwKCD6oaLAjk3cYIUN1n9Z58=">ABynicdY9LS8NAFIXv1FeNr6hLN4NFqJuQpKGtGym4ceGign2ALWUynTZDJw9mJmI3bl2q3/Dn+O/McG6qOhZHc45F+7nJ4IrbdufqLKxubW9U9019vYPDo/M45O+ilNJWY/GIpZDnygmeMR6mvBholkJPQFG/iLm7IfPDGpeBw96Cxh45DMIz7jlOgy6tefLydmzbau2k3Xa2Lbsu2W4zqlcVtew8NOkZSqwUrdifkxmsY0DVmkqSBK5URqTgVbGqNUsYTQBZmznIa+5PNAr6ckVCoL/SW+CIkO1O+uDP/qHlM9a49zHiWpZhEtJkU3SwXWMS6x8JRLRrXICkOo5MU/mAZEqoLeMoGH9A8P+m71pOw3LvVrnekVbhTM4hzo40IO3EIXekAhgFd4g3d0hyTKUP49raDVzSmsCb18AXbQgO8=</latexit>

x

<latexit sha1_base64="KCBxiMsuGj6/9+qinApIUdrJUdk=">ABx3icbY87TwJBFIXv4AvXF2ps5GYWJFdNHKkNhoB4k8EiFkdrjAhJmdzcwsgRAKa1v9If4c/42zug3gqU7OTe5X5QIbmwQfJPC1vbO7l5x3zs4PDo+KZ2etYxKNcMmU0LpTkQNCh5j03IrsJNopDIS2I4mj1nfnqI2XMUvdp5gT9JRzIecUeuixqxfKgeV4Ff+pglzU4Zc9X7pqztQLJUYWyaoMQuqLWcCl143NZhQNqEjXDAZaT4a29WUSmPmMlr6V5LasVnvsvC/7jW1w/vegsdJajFmbuK6YSp8q/yMyR9wjcyKuTOUae7+8dmYasqsI/c8xiuE2aVrUS3lSqjdty7SGnLcIFXMI1hHAHNXiCOjSBAcI7fMAneSaKTMnsb1og+c05rIi8/QDHd3/o</latexit>

u(t) = u(0) cos(ωt) + ˙ u(0) 1 ω sin(ωt)

<latexit sha1_base64="qVf50sMobymGZgbNMvtYEVyhCPg=">ACnicbZBLSwMxFIUz9VXrq+rSTbAIU4QyUwXdKEU3LivYBzilZNJMG5p5kNwIZeg/8Ie41Z24de1W/Temdha29awO57sJ91w/EVyB43xbuaXldW1/HphY3Nre6e4u9dUsZaUNWgsYtn2iWKCR6wBHARrJ5KR0Bes5Q+vJ7z1wKTicXQHo4R1QtKPeMApARN1i1fahvKFtp0y9misbC8OWZ9gKONjrxcDnhAvkISm7jidwrGnePRnsFsORXnV3jRuJkpoUz1bvHZfE1yCKgiVEgmcCjYueFqxhNAh6bOUhr7k/QHMpiRUahT6Y3wUEhioeTYJ/2P3GoLzTsqjRAOLqBkxLNACQ4wnZ8E9LhkFMTKGUMnNPpgOiOkN5niFgunozjdaNM1qxT2pVG9PS7XLrG0eHaBDZCMXnaEaukF1EAUPaEP9Im+rEfrxXq13qajOSt7s49mZL3/AEU1mDU=</latexit>

ω =

  • Φ

M

<latexit sha1_base64="xT6x9L7ekFgTGnHrHb8wgV2oHm8=">AB5HicbZBLS8NAFIVv6qvGV9SlLoJFcFWSVrAboeDGjVDBPsCUMplOkqEzSZyZCVk49qduHXtv4b/41TzatZ3U451643/VTRqVynG+jsra+sblV3TZ3dvf2D6zDo5MoFJFycsEQMfScJoTLqKkYGqSCI+4z0/cnNvO8/EyFpEj+oaUqGHIUxDShGSkcj69RLOAnRtSefhMq9QCce52IFvldUYysmlN3fmWvGrc0NSjVGVkzb5zgjJNYakzJFQFDNSmF4mSYrwBIUkx9wXNIzUYoq4lFPuF/Y5RyqSy908/K97zFTQGuY0TjNFYqxHdBdkzFaJPQe2x1QrNhUG4QF1fYOEKaU+m3mKZmdJeJVk2vUXeb9cb9Za3dKmrcAJncAEuXEbqEDXcDwAp8wgy8jMF6N+P9b7RilDvHsCDj4wfvb4wb</latexit>

V (R + u) ≈ V (R) + 1 2

  • ijαβ

Φjβ

iαuiαujβ + O(u3)

<latexit sha1_base64="qvHV5AcZCByS1Ymgjp0HIky3BMA=">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</latexit>

Mi ¨ uiα = −

Φjβ

iαujβ

<latexit sha1_base64="MBSYap2qKEoce9y+IxicK+9qh8c=">ACD3icbVDLSsNAFJ34rPEVdelmsAhuLEkVdKMUXOhGqGAfYGqYTKbN2JkzEMoIR/h7hw5U7cunYl6L+YalTaelbnMu3HP9hFGpbPvNmJqemZ2bLy2Yi0vLK6vW2npTxlpg0sAxi0XbR5IwGpGoqRdiI4j4jLb9/MvRbt0RIGkeXapCQDke9iHYpRiqXPOv03KPQDYJYQe2l1EUsCVEGj+CuKzX30hvXJwplbj2k1z/DX07/BjyrbFfsL8BJ4hSkDArUPevBDWKsOYkUZkjKFAlFMSOZ6WpJEoT7qEdSzH1Be6EaVRGXcsD9DG5zpEI57g3F/7wrbqHnZRGiVYkwnk97qaQRXD4W9gQAXBig1ygrCg+T0Qh0grPIPmbe0RlvNEma1YqzV6le7Jdrx0XbEtgEW2AHOA1MAZqIMGwOAevIJ38GHcGY/Gk/H8HZ0yip0NMALj5RORZJ3p</latexit>
slide-134
SLIDE 134

1 2Φ

  • x − x0

2

<latexit sha1_base64="JfNCI5Rqnx/jaACAwzXJ+5TExs=">AB7XicbZA7T8MwFIWd8irhFWBksagqlYHICVUfC6rEwlgk+pBIqRzXba06D9kOahXlFzCzIVbmrvBL+DckbRkKnOnc+6V/NkNOZMKoS8t7G5tb2T39X39g8Oj4zjk7YMIkFoiwQ8EF0XS8qZT1uKU67oaDYczntuJObrO8USFZ4N+rWUh7Hh75bMgIVmnUN4rOUGASW0lsJ05zByXjTgsTS+nfQSzQcCLR7tvFJCJalalbkFkVmyrWrVTg6xyvV6DlokWKoCVmn1j7gwCEnUV4RjKWMsFCOcJroTSRpiMsEjGhPFWw0Vusp9qSceW4Cix5WY/m7y8L/uodIDWu9mPlhpKhP0pW0G0YcqgBm5HDABCWKz1KDiWDpeyAZ45Repf+j60vGheBf8PYtk3ryrTvyoXG9Yo2D87AOSgBC1RBA9yCJmgBAp7BHyATy3QXrRX7W25mtNWN6dgTdr7N70XjgY=</latexit>

LaYce dynamics in the harmonic approximaFon

M¨ x = −∂V ∂x

<latexit sha1_base64="FgtUFvHAdG+6g57nYZM4bl6HdU=">AB8HicbZBLS8NAFIVv6qvWV9Slm8EidGNJqAbpeDGjVDBPsCUMplM2qGTBzMTaQn5C67diVt3glv9H/4bJxqQtp7V4Zx7L/ONG3MmlWV9GaWl5ZXVtfJ6ZWNza3vH3N3ryCgRhLZJxCPRc7GknIW0rZjitBcLigOX0647vsr7gMVkXhnZrGtB/gYch8RrDS0cCs3TieFyk0QRfHji8wSZ0YC8UwR53sz0+ygVm16taP0KxC1OFQq2B+eZ4EUkCGirCsZRpfopwmlWcRNIYkzEe0pQErmDkZpNcSDlNHAzdBRgNZLzXR7+190nyj/vpyME0VDokd05ycqQjl8MhjghLFp9pgIph+DyIjrKmV/qJKRTPa80SLptOo2yf1xu1ptXlZ0JbhA6hBjacQROuoQVtIPAI7/ABn4Ywnoxn4+V3tGQUO/swI+P1Gxz1kCU=</latexit>

M ¨ u = −Φu

<latexit sha1_base64="CXAeNIm8gnVr5+5yrUEeCXM98Uk=">AB2nicbZA7SwNBFIXv+ozra42lzWAQbAy7UdBGCdjYCBHMA9wQZmcn2SGzD2buiCGksRNba1t/Tn+Gze6TRJP9XHOvXDPDTIpNLrut7W0vLK6tl7asDe3tnd2nb1yS6dGMd5kqUxVJ6CaS5HwJgqUvJMpTuNA8nYwvJ7m7UeutEiTexlvBvTQSL6glHMrZ5TvXDMEViyCU58RuRIKbnVNyq+yuyCF4BFSjU6DlfpgyE/MEmaRaj6lCwSf2L7RPKNsSAd8zOJAiUGEsy6NtR7FwYQcxRQjPZ9Nzf+yB4P9i+5YJlBnrB8JM/6RhJMybQmCYXiDOUoB8qUyO8hLKMsyfYdt5R2+0SK0alXvtFq7O6vUr4q2JTiAQzgGD86hDjfQgCYweIJ3+IBPy7erRfr9W90ySp29mFG1tsPDPuF8A=</latexit>

V (x0 + u) ≈ V (x0) + 1 2Φu2 + O

  • u3
<latexit sha1_base64="ZjTL3R/+HKbhLj5KYn/qGUOSnY=">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</latexit>

V (x)

<latexit sha1_base64="/sOwKCD6oaLAjk3cYIUN1n9Z58=">ABynicdY9LS8NAFIXv1FeNr6hLN4NFqJuQpKGtGym4ceGign2ALWUynTZDJw9mJmI3bl2q3/Dn+O/McG6qOhZHc45F+7nJ4IrbdufqLKxubW9U9019vYPDo/M45O+ilNJWY/GIpZDnygmeMR6mvBholkJPQFG/iLm7IfPDGpeBw96Cxh45DMIz7jlOgy6tefLydmzbau2k3Xa2Lbsu2W4zqlcVtew8NOkZSqwUrdifkxmsY0DVmkqSBK5URqTgVbGqNUsYTQBZmznIa+5PNAr6ckVCoL/SW+CIkO1O+uDP/qHlM9a49zHiWpZhEtJkU3SwXWMS6x8JRLRrXICkOo5MU/mAZEqoLeMoGH9A8P+m71pOw3LvVrnekVbhTM4hzo40IO3EIXekAhgFd4g3d0hyTKUP49raDVzSmsCb18AXbQgO8=</latexit>

x

<latexit sha1_base64="KCBxiMsuGj6/9+qinApIUdrJUdk=">ABx3icbY87TwJBFIXv4AvXF2ps5GYWJFdNHKkNhoB4k8EiFkdrjAhJmdzcwsgRAKa1v9If4c/42zug3gqU7OTe5X5QIbmwQfJPC1vbO7l5x3zs4PDo+KZ2etYxKNcMmU0LpTkQNCh5j03IrsJNopDIS2I4mj1nfnqI2XMUvdp5gT9JRzIecUeuixqxfKgeV4Ff+pglzU4Zc9X7pqztQLJUYWyaoMQuqLWcCl143NZhQNqEjXDAZaT4a29WUSmPmMlr6V5LasVnvsvC/7jW1w/vegsdJajFmbuK6YSp8q/yMyR9wjcyKuTOUae7+8dmYasqsI/c8xiuE2aVrUS3lSqjdty7SGnLcIFXMI1hHAHNXiCOjSBAcI7fMAneSaKTMnsb1og+c05rIi8/QDHd3/o</latexit>

u(t) = u(0) cos(ωt) + ˙ u(0) 1 ω sin(ωt)

<latexit sha1_base64="qVf50sMobymGZgbNMvtYEVyhCPg=">ACnicbZBLSwMxFIUz9VXrq+rSTbAIU4QyUwXdKEU3LivYBzilZNJMG5p5kNwIZeg/8Ie41Z24de1W/Temdha29awO57sJ91w/EVyB43xbuaXldW1/HphY3Nre6e4u9dUsZaUNWgsYtn2iWKCR6wBHARrJ5KR0Bes5Q+vJ7z1wKTicXQHo4R1QtKPeMApARN1i1fahvKFtp0y9misbC8OWZ9gKONjrxcDnhAvkISm7jidwrGnePRnsFsORXnV3jRuJkpoUz1bvHZfE1yCKgiVEgmcCjYueFqxhNAh6bOUhr7k/QHMpiRUahT6Y3wUEhioeTYJ/2P3GoLzTsqjRAOLqBkxLNACQ4wnZ8E9LhkFMTKGUMnNPpgOiOkN5niFgunozjdaNM1qxT2pVG9PS7XLrG0eHaBDZCMXnaEaukF1EAUPaEP9Im+rEfrxXq13qajOSt7s49mZL3/AEU1mDU=</latexit>

ω =

  • Φ

M

<latexit sha1_base64="xT6x9L7ekFgTGnHrHb8wgV2oHm8=">AB5HicbZBLS8NAFIVv6qvGV9SlLoJFcFWSVrAboeDGjVDBPsCUMplOkqEzSZyZCVk49qduHXtv4b/41TzatZ3U451643/VTRqVynG+jsra+sblV3TZ3dvf2D6zDo5MoFJFycsEQMfScJoTLqKkYGqSCI+4z0/cnNvO8/EyFpEj+oaUqGHIUxDShGSkcj69RLOAnRtSefhMq9QCce52IFvldUYysmlN3fmWvGrc0NSjVGVkzb5zgjJNYakzJFQFDNSmF4mSYrwBIUkx9wXNIzUYoq4lFPuF/Y5RyqSy908/K97zFTQGuY0TjNFYqxHdBdkzFaJPQe2x1QrNhUG4QF1fYOEKaU+m3mKZmdJeJVk2vUXeb9cb9Za3dKmrcAJncAEuXEbqEDXcDwAp8wgy8jMF6N+P9b7RilDvHsCDj4wfvb4wb</latexit>

V (R + u) ≈ V (R) + 1 2

  • ijαβ

Φjβ

iαuiαujβ + O(u3)

<latexit sha1_base64="qvHV5AcZCByS1Ymgjp0HIky3BMA=">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</latexit>

Mi ¨ uiα = −

Φjβ

iαujβ

<latexit sha1_base64="MBSYap2qKEoce9y+IxicK+9qh8c=">ACD3icbVDLSsNAFJ34rPEVdelmsAhuLEkVdKMUXOhGqGAfYGqYTKbN2JkzEMoIR/h7hw5U7cunYl6L+YalTaelbnMu3HP9hFGpbPvNmJqemZ2bLy2Yi0vLK6vW2npTxlpg0sAxi0XbR5IwGpGoqRdiI4j4jLb9/MvRbt0RIGkeXapCQDke9iHYpRiqXPOv03KPQDYJYQe2l1EUsCVEGj+CuKzX30hvXJwplbj2k1z/DX07/BjyrbFfsL8BJ4hSkDArUPevBDWKsOYkUZkjKFAlFMSOZ6WpJEoT7qEdSzH1Be6EaVRGXcsD9DG5zpEI57g3F/7wrbqHnZRGiVYkwnk97qaQRXD4W9gQAXBig1ygrCg+T0Qh0grPIPmbe0RlvNEma1YqzV6le7Jdrx0XbEtgEW2AHOA1MAZqIMGwOAevIJ38GHcGY/Gk/H8HZ0yip0NMALj5RORZJ3p</latexit>

det

  • Φjβ

iα − ω2Miδijδαβ

  • = 0
<latexit sha1_base64="FYvPMFAkrJ0IxcKCU3gHvNX1xUo=">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</latexit>
slide-135
SLIDE 135

Periodic systems

R

i = τ i + di

<latexit sha1_base64="pQVQhZLoUMvKY4UnjL70WR7BHko=">AB/nicbVDLSsNAFJ34rPEVdelmsAiCUJIq6kYpuHFZxT7A1DC5mbRDJw9mJkIJBT/EtTsRd67d6g/4N040m7aezZw5145/opZ1LZ9rcxN7+wuLRcWTFX19Y3Nq2t7bZMgG0BQlPRNcnknIW05ZitNuKiJfE47/vCy8DsPVEiWxLdqlNJeRPoxCxkQpSXPOnEjogZ+iG/uXWACPIbPsesnPJCjSD+5q0g21uohLifzQH89q2rX7F/gWeKUpIpKND3rzQ0SyCIaK+BEypwIxYDTselmkqYEhqRPc4h8wfoDNamSBZhxni/SCnvUL8z7vLVHjWy1mcZorGoEe0F2YcqwQXp8ABExQUH2lCQDCdB8OACAJKH8w0dUdnutEsadrzlGtfn1cbVyUbStoF+2hA+SgU9RAV6iJWgjQE/pAn+jLeDSejRfj9W90zih3dtAEjPcfWOKV6Q=</latexit>

GaAs d1

<latexit sha1_base64="e7K0yRVKE0zimAixIPyb+vPDaV8=">AB03icbZBLT8JAFIVv8YX1hbp0M5GYuGpaJLRsDIkbl5hYIBFCZoYpTJg+nJmakIaNcevarf4Ef47/xhZwgXpWJ+fcm9zvkRwpW37yhtbG5t75R3zb39g8OjyvFJR8WpMynsYhlj2DFBI+Yr7kWrJdIhkMiWJdMb4q+8Sk4nF0r2cJG4R4HPGAU6zaID6IdYTEmSj+dAZVq25dZtx/WQbdW9htcsjF1rug0XOZa9UBVWag8rn/1RTNOQRZoKrFSGpeZUsLnZTxVLMJ3iMctoSCQfT/R6ikOlZiGZo4viAvW7K8L/uodUB94g41GSahbRfCTvglQgHaOCD424ZFSLW4wlTy/B9EJlpjq/AumuWBcgqC/5oexU7OcK6t2V6+2rle0ZTiDc7gEB1xowS20wQcKj/AG7/Bh+EZmPBsvy9GSsdo5hTUZr981YUk</latexit>

d2

<latexit sha1_base64="cZtfmxgQolCBNCw6qST3u7c9VA0=">AB03icdVA7SwNBGPzWZzxfUubxSBYHXd5p5GAjWUELwmYEPY2e8mSvYe7e0I40oita3+BH+O/8Y9jUVEpxpm5oOZz08EV9pxPtDa+sbm1nZhx9rd2z84LB4d1WcSso8GotY9n2imOAR8zTXgvUTyUjoC9bzZ1e53tgUvE4utXzhA1DMol4wCnRhriQUj01A+y8WJUHhVLjl1p1htOBRvSaracqiG1at2t1bFrO18owRKdUfF9MI5pGrJIU0GUyojUnAq2sAapYgmhMzJhGQ19ySdTvaqSUKl56C/wed5A/fZy8S/vLtVBc5jxKEk1i6iJGC9IBdYxzvfhMZeMajE3hFDJTR9Mp0QSqs0XLMts/BmC/yfdsu1W7PJNtdS+XK4twCmcwQW40IA2XEMHPKBwDy/wCm/IQxl6RE/f0TW0vDmBFaDnTyYIhRo=</latexit>
slide-136
SLIDE 136

Periodic systems

R

i = τ i + di

<latexit sha1_base64="pQVQhZLoUMvKY4UnjL70WR7BHko=">AB/nicbVDLSsNAFJ34rPEVdelmsAiCUJIq6kYpuHFZxT7A1DC5mbRDJw9mJkIJBT/EtTsRd67d6g/4N040m7aezZw5145/opZ1LZ9rcxN7+wuLRcWTFX19Y3Nq2t7bZMgG0BQlPRNcnknIW05ZitNuKiJfE47/vCy8DsPVEiWxLdqlNJeRPoxCxkQpSXPOnEjogZ+iG/uXWACPIbPsesnPJCjSD+5q0g21uohLifzQH89q2rX7F/gWeKUpIpKND3rzQ0SyCIaK+BEypwIxYDTselmkqYEhqRPc4h8wfoDNamSBZhxni/SCnvUL8z7vLVHjWy1mcZorGoEe0F2YcqwQXp8ABExQUH2lCQDCdB8OACAJKH8w0dUdnutEsadrzlGtfn1cbVyUbStoF+2hA+SgU9RAV6iJWgjQE/pAn+jLeDSejRfj9W90zih3dtAEjPcfWOKV6Q=</latexit>

ωn = ων(q)

<latexit sha1_base64="m0f5wbNhXQ0rJn0Bh+fuqPg/Xg0=">AB6HicbZA7T8MwFIVvyquEV4CRxaJCKkuVAhIsoEosjEWiD4lUkeM6qantBNtBquzGyIlZmFAf4L/4YEwtCWM306515dHwcJZ9q47pdVWlhcWl4pr9pr6xubW872TlvHqSK0RWIeq26ANeVM0pZhtNuoigWAaedYHiZ50HqjSL5Y0ZJbQncCRZyAg2meU7yIsFjbAv0fkfejKtegKbQRCi+0Pfqbg190doHuoFVKBQ03fevX5MUkGlIRxrPcbKMLpxPZSTRNMhjiYyICxaKBmXax0Hokgk6yO/r2Sw3/8tuUxOe9cZMJqmhkmQjWRamHJkY5aVRnylKDB9lgIli2XsQGWCFicm+xrazjvXZRvPQPqrVj2tH1yeVxkXRtgx7sA9VqMpNOAKmtACAo/wBh/wad1ZT9az9fI7WrKnV2YkvX6DYvJjDk=</latexit>
  • P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni, Phys. Rev. B 43, 7231 (1991)

GaAs d1

<latexit sha1_base64="e7K0yRVKE0zimAixIPyb+vPDaV8=">AB03icbZBLT8JAFIVv8YX1hbp0M5GYuGpaJLRsDIkbl5hYIBFCZoYpTJg+nJmakIaNcevarf4Ef47/xhZwgXpWJ+fcm9zvkRwpW37yhtbG5t75R3zb39g8OjyvFJR8WpMynsYhlj2DFBI+Yr7kWrJdIhkMiWJdMb4q+8Sk4nF0r2cJG4R4HPGAU6zaID6IdYTEmSj+dAZVq25dZtx/WQbdW9htcsjF1rug0XOZa9UBVWag8rn/1RTNOQRZoKrFSGpeZUsLnZTxVLMJ3iMctoSCQfT/R6ikOlZiGZo4viAvW7K8L/uodUB94g41GSahbRfCTvglQgHaOCD424ZFSLW4wlTy/B9EJlpjq/AumuWBcgqC/5oexU7OcK6t2V6+2rle0ZTiDc7gEB1xowS20wQcKj/AG7/Bh+EZmPBsvy9GSsdo5hTUZr981YUk</latexit>

d2

<latexit sha1_base64="cZtfmxgQolCBNCw6qST3u7c9VA0=">AB03icdVA7SwNBGPzWZzxfUubxSBYHXd5p5GAjWUELwmYEPY2e8mSvYe7e0I40oita3+BH+O/8Y9jUVEpxpm5oOZz08EV9pxPtDa+sbm1nZhx9rd2z84LB4d1WcSso8GotY9n2imOAR8zTXgvUTyUjoC9bzZ1e53tgUvE4utXzhA1DMol4wCnRhriQUj01A+y8WJUHhVLjl1p1htOBRvSaracqiG1at2t1bFrO18owRKdUfF9MI5pGrJIU0GUyojUnAq2sAapYgmhMzJhGQ19ySdTvaqSUKl56C/wed5A/fZy8S/vLtVBc5jxKEk1i6iJGC9IBdYxzvfhMZeMajE3hFDJTR9Mp0QSqs0XLMts/BmC/yfdsu1W7PJNtdS+XK4twCmcwQW40IA2XEMHPKBwDy/wCm/IQxl6RE/f0TW0vDmBFaDnTyYIhRo=</latexit>
slide-137
SLIDE 137

Anharmonic lifeFmes

uνuνuν ∼

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • <latexit sha1_base64="P/OJqC9FB+HE2fuLsoVs/qIO5s=">ACk3ichVHZSsNAFJ3EPW5V8cmXwSKtCVRQUGQoj74IihYWzAab6bTdOhkYRahHyU3+EX+DcmNYLWqvdlzpwF7uH6CWdS2fabYU5Nz8zOzS9Yi0vLK6uVtfU7GWtBaIvEPBYdHyTlLKItxRSnURQCH1O2/7gvNDbz1RIFke3apjQhxCiPUYAZVTXuVe26ksfbS/KlnyBHrmQhdn0WcFzHbh8UhpGW4T2cfvyzR7cLQUBFKRugXexNSlX+zVY+y/5R/Rr1qtU7Y9GvwTOCWonKuvcqL242JDmkCAcpUxCKEU4zy9WSJkAGENCUhL5gQV9ZyGUchj6Gd4JQfXluFaQk7R7rXrHDymLEq1oRHJLrvU0xyrGxX1wlwlKFB/mAIhg+T6Y9EAUfkVraKjM97oJ7jbzgHjf2bw2rztGw7j7bQNqojBx2hJrpE16iFiOEYbePJAHPTPDHPzIsPq2mUmQ30bcyrd6bBw8=</latexit>
slide-138
SLIDE 138

Anharmonic lifeFmes

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ a†

ν

<latexit sha1_base64="uZvHs+qEn9qBzBYOQjaJS3n8CkA=">AB3icdZA7T8MwFIWd8irhFWBgYLGoExVWq2bBUsjEWiD4mU6MZ1E6vOQ7aDVEWZ2RArMysM/Bz+DS6EoQjO9Omce6V7rpdwJpVtfxilpeWV1bXyurmxubW9Y+3u9WcCkJ7JOaxGHogKWcR7SmOB0mgkLocTrwpfzfHBPhWRxdKNmCR2F4EdswgobnWAXYCUBjunDH4PhVu5kTpSY5dq2JX7War1WhgDV/S0KzVm+1zXCucCirUda13ZxyTNKSRIhykzEAoRjNTSeVNAEyBZ9mJPQE8wO16EIo5Sz0cnwcgrk72xu/pXdpmrSHmUsSlJFI6JHdDZJOVYxnfFYyYoUXymAYhg+h5MAhBAlP6IaeqOP0Xw/9CvV2tn1fp1o9K5KNqW0SE6Qqeohlqog65QF/UQTl6Qa/ozQDjwXg0nr5HS0axs48WZDx/AtPoiPg=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ aν

<latexit sha1_base64="4Mtub4Bhzvs4wlmoWkHwhsG5aik=">AB13icdZA7TwJBFIXv4gvXB6uWNhOJ0YrsIgHsiDaWmMjDuGQzOwzshNlH5mFCNsTO2Frb6g/w5/hvHBQLjJ7qyzn3JvfcMONMKtf9sAorq2vrG8VNe2t7Z7fk7O13ZaoFoR2S8lT0QywpZwntKY47WeC4jktBdOLud5754KydLkRk0zOojxOGEjRrAyVuCUkB9hXCQ+4k+maHAKbsVt95o1GrIwJcM1L1qvXmOvIVThoXagfPuD1OiY5owrGUORaKEU5ntq8lzTCZ4DHNSRwKNo7UsotjKadxOEPHMVaR/J3Nzb+yO61GzUHOkwrmhAzYrKR5kilaN4RDZmgRPGpAUwEM/cgEmGBiTKfsG3T8acI+h+61Yp3Vqle18qti0XbIhzCEZyCBw1owRW0oQMENLzAK7xZt9aD9Wg9fY8WrMXOASzJev4EGv+FhA=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

uνuνuν ∼

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • <latexit sha1_base64="P/OJqC9FB+HE2fuLsoVs/qIO5s=">ACk3ichVHZSsNAFJ3EPW5V8cmXwSKtCVRQUGQoj74IihYWzAab6bTdOhkYRahHyU3+EX+DcmNYLWqvdlzpwF7uH6CWdS2fabYU5Nz8zOzS9Yi0vLK6uVtfU7GWtBaIvEPBYdHyTlLKItxRSnURQCH1O2/7gvNDbz1RIFke3apjQhxCiPUYAZVTXuVe26ksfbS/KlnyBHrmQhdn0WcFzHbh8UhpGW4T2cfvyzR7cLQUBFKRugXexNSlX+zVY+y/5R/Rr1qtU7Y9GvwTOCWonKuvcqL242JDmkCAcpUxCKEU4zy9WSJkAGENCUhL5gQV9ZyGUchj6Gd4JQfXluFaQk7R7rXrHDymLEq1oRHJLrvU0xyrGxX1wlwlKFB/mAIhg+T6Y9EAUfkVraKjM97oJ7jbzgHjf2bw2rztGw7j7bQNqojBx2hJrpE16iFiOEYbePJAHPTPDHPzIsPq2mUmQ30bcyrd6bBw8=</latexit>
slide-139
SLIDE 139

Anharmonic lifeFmes

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ a†

ν

<latexit sha1_base64="uZvHs+qEn9qBzBYOQjaJS3n8CkA=">AB3icdZA7T8MwFIWd8irhFWBgYLGoExVWq2bBUsjEWiD4mU6MZ1E6vOQ7aDVEWZ2RArMysM/Bz+DS6EoQjO9Omce6V7rpdwJpVtfxilpeWV1bXyurmxubW9Y+3u9WcCkJ7JOaxGHogKWcR7SmOB0mgkLocTrwpfzfHBPhWRxdKNmCR2F4EdswgobnWAXYCUBjunDH4PhVu5kTpSY5dq2JX7War1WhgDV/S0KzVm+1zXCucCirUda13ZxyTNKSRIhykzEAoRjNTSeVNAEyBZ9mJPQE8wO16EIo5Sz0cnwcgrk72xu/pXdpmrSHmUsSlJFI6JHdDZJOVYxnfFYyYoUXymAYhg+h5MAhBAlP6IaeqOP0Xw/9CvV2tn1fp1o9K5KNqW0SE6Qqeohlqog65QF/UQTl6Qa/ozQDjwXg0nr5HS0axs48WZDx/AtPoiPg=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ aν

<latexit sha1_base64="4Mtub4Bhzvs4wlmoWkHwhsG5aik=">AB13icdZA7TwJBFIXv4gvXB6uWNhOJ0YrsIgHsiDaWmMjDuGQzOwzshNlH5mFCNsTO2Frb6g/w5/hvHBQLjJ7qyzn3JvfcMONMKtf9sAorq2vrG8VNe2t7Z7fk7O13ZaoFoR2S8lT0QywpZwntKY47WeC4jktBdOLud5754KydLkRk0zOojxOGEjRrAyVuCUkB9hXCQ+4k+maHAKbsVt95o1GrIwJcM1L1qvXmOvIVThoXagfPuD1OiY5owrGUORaKEU5ntq8lzTCZ4DHNSRwKNo7UsotjKadxOEPHMVaR/J3Nzb+yO61GzUHOkwrmhAzYrKR5kilaN4RDZmgRPGpAUwEM/cgEmGBiTKfsG3T8acI+h+61Yp3Vqle18qti0XbIhzCEZyCBw1owRW0oQMENLzAK7xZt9aD9Wg9fY8WrMXOASzJev4EGv+FhA=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

uνuνuν ∼

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • <latexit sha1_base64="P/OJqC9FB+HE2fuLsoVs/qIO5s=">ACk3ichVHZSsNAFJ3EPW5V8cmXwSKtCVRQUGQoj74IihYWzAab6bTdOhkYRahHyU3+EX+DcmNYLWqvdlzpwF7uH6CWdS2fabYU5Nz8zOzS9Yi0vLK6uVtfU7GWtBaIvEPBYdHyTlLKItxRSnURQCH1O2/7gvNDbz1RIFke3apjQhxCiPUYAZVTXuVe26ksfbS/KlnyBHrmQhdn0WcFzHbh8UhpGW4T2cfvyzR7cLQUBFKRugXexNSlX+zVY+y/5R/Rr1qtU7Y9GvwTOCWonKuvcqL242JDmkCAcpUxCKEU4zy9WSJkAGENCUhL5gQV9ZyGUchj6Gd4JQfXluFaQk7R7rXrHDymLEq1oRHJLrvU0xyrGxX1wlwlKFB/mAIhg+T6Y9EAUfkVraKjM97oJ7jbzgHjf2bw2rztGw7j7bQNqojBx2hJrpE16iFiOEYbePJAHPTPDHPzIsPq2mUmQ30bcyrd6bBw8=</latexit>

τ−1

i

= 2π

  • f

δ(Ei − Ef )

  • Vif
  • 2
<latexit sha1_base64="BqJfv1CPSQdWUJWTf8Kr3Lyi5k=">ACEXicbVDLSsNAFJ3UV42vqEs3wSLURUsSBd0oBSnoroJ9gNOGyXSDJ08mJkIJc1X+CHi0p24de3Cjf6KiXbT1rM695x74Z7jxIwKaRifSmlpeWV1rbyubmxube9ou3sdESUckzaOWMR7DhKE0ZC0JZWM9GJOUOAw0nVGV4XfSBc0Ci8k+OY9APkhdSlGMlcsrUbKFi0FaM7ML6HKEUwvGNEuh7yCeQZEtguHhElUbdq01rTdY+hQj+mTjp1SNysGrk8Glq1VjLrxC32RmFNSAVO0bO0ZDiOcBCSUmCEhUsQlxYxkKkwEiREeIY+kOHA49Xw5q6JAiHgZPpRgKQv5r1C/M+7T6R73k9pGCeShDhfyT03YbqM9KIdfUg5wZKNc4Iwp/k/OvZR3orMO1TVPKM5n2iRdKy6eVK3bk8rjctp2jI4AIegCkxwBhrgGrRAG2DwBD7AF/hWHpUX5V5+1stKdObfTAD5f0H2GOdxw=</latexit>
slide-140
SLIDE 140

Anharmonic lifeFmes

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ a†

ν

<latexit sha1_base64="uZvHs+qEn9qBzBYOQjaJS3n8CkA=">AB3icdZA7T8MwFIWd8irhFWBgYLGoExVWq2bBUsjEWiD4mU6MZ1E6vOQ7aDVEWZ2RArMysM/Bz+DS6EoQjO9Omce6V7rpdwJpVtfxilpeWV1bXyurmxubW9Y+3u9WcCkJ7JOaxGHogKWcR7SmOB0mgkLocTrwpfzfHBPhWRxdKNmCR2F4EdswgobnWAXYCUBjunDH4PhVu5kTpSY5dq2JX7War1WhgDV/S0KzVm+1zXCucCirUda13ZxyTNKSRIhykzEAoRjNTSeVNAEyBZ9mJPQE8wO16EIo5Sz0cnwcgrk72xu/pXdpmrSHmUsSlJFI6JHdDZJOVYxnfFYyYoUXymAYhg+h5MAhBAlP6IaeqOP0Xw/9CvV2tn1fp1o9K5KNqW0SE6Qqeohlqog65QF/UQTl6Qa/ozQDjwXg0nr5HS0axs48WZDx/AtPoiPg=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ aν

<latexit sha1_base64="4Mtub4Bhzvs4wlmoWkHwhsG5aik=">AB13icdZA7TwJBFIXv4gvXB6uWNhOJ0YrsIgHsiDaWmMjDuGQzOwzshNlH5mFCNsTO2Frb6g/w5/hvHBQLjJ7qyzn3JvfcMONMKtf9sAorq2vrG8VNe2t7Z7fk7O13ZaoFoR2S8lT0QywpZwntKY47WeC4jktBdOLud5754KydLkRk0zOojxOGEjRrAyVuCUkB9hXCQ+4k+maHAKbsVt95o1GrIwJcM1L1qvXmOvIVThoXagfPuD1OiY5owrGUORaKEU5ntq8lzTCZ4DHNSRwKNo7UsotjKadxOEPHMVaR/J3Nzb+yO61GzUHOkwrmhAzYrKR5kilaN4RDZmgRPGpAUwEM/cgEmGBiTKfsG3T8acI+h+61Yp3Vqle18qti0XbIhzCEZyCBw1owRW0oQMENLzAK7xZt9aD9Wg9fY8WrMXOASzJev4EGv+FhA=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

uνuνuν ∼

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • <latexit sha1_base64="P/OJqC9FB+HE2fuLsoVs/qIO5s=">ACk3ichVHZSsNAFJ3EPW5V8cmXwSKtCVRQUGQoj74IihYWzAab6bTdOhkYRahHyU3+EX+DcmNYLWqvdlzpwF7uH6CWdS2fabYU5Nz8zOzS9Yi0vLK6uVtfU7GWtBaIvEPBYdHyTlLKItxRSnURQCH1O2/7gvNDbz1RIFke3apjQhxCiPUYAZVTXuVe26ksfbS/KlnyBHrmQhdn0WcFzHbh8UhpGW4T2cfvyzR7cLQUBFKRugXexNSlX+zVY+y/5R/Rr1qtU7Y9GvwTOCWonKuvcqL242JDmkCAcpUxCKEU4zy9WSJkAGENCUhL5gQV9ZyGUchj6Gd4JQfXluFaQk7R7rXrHDymLEq1oRHJLrvU0xyrGxX1wlwlKFB/mAIhg+T6Y9EAUfkVraKjM97oJ7jbzgHjf2bw2rztGw7j7bQNqojBx2hJrpE16iFiOEYbePJAHPTPDHPzIsPq2mUmQ30bcyrd6bBw8=</latexit>

1

ν

= 2

  • νν
  • Vννν2 ×
  • (ν − ν − ν)nν(nν + 1)(nν + 1)+

(ν + ν − ν)nνnν(nν + 1)

  • <latexit sha1_base64="EZAkO0TV1/oPrBQ1yZdzpDTN1ws=">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</latexit>

τ−1

i

= 2π

  • f

δ(Ei − Ef )

  • Vif
  • 2
<latexit sha1_base64="BqJfv1CPSQdWUJWTf8Kr3Lyi5k=">ACEXicbVDLSsNAFJ3UV42vqEs3wSLURUsSBd0oBSnoroJ9gNOGyXSDJ08mJkIJc1X+CHi0p24de3Cjf6KiXbT1rM695x74Z7jxIwKaRifSmlpeWV1rbyubmxube9ou3sdESUckzaOWMR7DhKE0ZC0JZWM9GJOUOAw0nVGV4XfSBc0Ci8k+OY9APkhdSlGMlcsrUbKFi0FaM7ML6HKEUwvGNEuh7yCeQZEtguHhElUbdq01rTdY+hQj+mTjp1SNysGrk8Glq1VjLrxC32RmFNSAVO0bO0ZDiOcBCSUmCEhUsQlxYxkKkwEiREeIY+kOHA49Xw5q6JAiHgZPpRgKQv5r1C/M+7T6R73k9pGCeShDhfyT03YbqM9KIdfUg5wZKNc4Iwp/k/OvZR3orMO1TVPKM5n2iRdKy6eVK3bk8rjctp2jI4AIegCkxwBhrgGrRAG2DwBD7AF/hWHpUX5V5+1stKdObfTAD5f0H2GOdxw=</latexit>
slide-141
SLIDE 141

Anharmonic lifeFmes

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ a†

ν

<latexit sha1_base64="uZvHs+qEn9qBzBYOQjaJS3n8CkA=">AB3icdZA7T8MwFIWd8irhFWBgYLGoExVWq2bBUsjEWiD4mU6MZ1E6vOQ7aDVEWZ2RArMysM/Bz+DS6EoQjO9Omce6V7rpdwJpVtfxilpeWV1bXyurmxubW9Y+3u9WcCkJ7JOaxGHogKWcR7SmOB0mgkLocTrwpfzfHBPhWRxdKNmCR2F4EdswgobnWAXYCUBjunDH4PhVu5kTpSY5dq2JX7War1WhgDV/S0KzVm+1zXCucCirUda13ZxyTNKSRIhykzEAoRjNTSeVNAEyBZ9mJPQE8wO16EIo5Sz0cnwcgrk72xu/pXdpmrSHmUsSlJFI6JHdDZJOVYxnfFYyYoUXymAYhg+h5MAhBAlP6IaeqOP0Xw/9CvV2tn1fp1o9K5KNqW0SE6Qqeohlqog65QF/UQTl6Qa/ozQDjwXg0nr5HS0axs48WZDx/AtPoiPg=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ aν

<latexit sha1_base64="4Mtub4Bhzvs4wlmoWkHwhsG5aik=">AB13icdZA7TwJBFIXv4gvXB6uWNhOJ0YrsIgHsiDaWmMjDuGQzOwzshNlH5mFCNsTO2Frb6g/w5/hvHBQLjJ7qyzn3JvfcMONMKtf9sAorq2vrG8VNe2t7Z7fk7O13ZaoFoR2S8lT0QywpZwntKY47WeC4jktBdOLud5754KydLkRk0zOojxOGEjRrAyVuCUkB9hXCQ+4k+maHAKbsVt95o1GrIwJcM1L1qvXmOvIVThoXagfPuD1OiY5owrGUORaKEU5ntq8lzTCZ4DHNSRwKNo7UsotjKadxOEPHMVaR/J3Nzb+yO61GzUHOkwrmhAzYrKR5kilaN4RDZmgRPGpAUwEM/cgEmGBiTKfsG3T8acI+h+61Yp3Vqle18qti0XbIhzCEZyCBw1owRW0oQMENLzAK7xZt9aD9Wg9fY8WrMXOASzJev4EGv+FhA=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

uνuνuν ∼

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • <latexit sha1_base64="P/OJqC9FB+HE2fuLsoVs/qIO5s=">ACk3ichVHZSsNAFJ3EPW5V8cmXwSKtCVRQUGQoj74IihYWzAab6bTdOhkYRahHyU3+EX+DcmNYLWqvdlzpwF7uH6CWdS2fabYU5Nz8zOzS9Yi0vLK6uVtfU7GWtBaIvEPBYdHyTlLKItxRSnURQCH1O2/7gvNDbz1RIFke3apjQhxCiPUYAZVTXuVe26ksfbS/KlnyBHrmQhdn0WcFzHbh8UhpGW4T2cfvyzR7cLQUBFKRugXexNSlX+zVY+y/5R/Rr1qtU7Y9GvwTOCWonKuvcqL242JDmkCAcpUxCKEU4zy9WSJkAGENCUhL5gQV9ZyGUchj6Gd4JQfXluFaQk7R7rXrHDymLEq1oRHJLrvU0xyrGxX1wlwlKFB/mAIhg+T6Y9EAUfkVraKjM97oJ7jbzgHjf2bw2rztGw7j7bQNqojBx2hJrpE16iFiOEYbePJAHPTPDHPzIsPq2mUmQ30bcyrd6bBw8=</latexit>

1

ν

= 2

  • νν
  • Vννν2 ×
  • (ν − ν − ν)nν(nν + 1)(nν + 1)+

(ν + ν − ν)nνnν(nν + 1)

  • <latexit sha1_base64="EZAkO0TV1/oPrBQ1yZdzpDTN1ws=">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</latexit>

Germanium Γ = 0.7 cm-1 Γ L U X W K L W

U W K X L Γ max Alberto Debernardi, Stefano Baroni, and Elisa Molinari, Phys. Rev. Lett. 75, 1819 (1995)

slide-142
SLIDE 142

Heat transport from laYce dynamics

J =

  • n

˙ Rnen + Rn ˙ en

  • d
<latexit sha1_base64="AvA/Wt+aTpBFH/a/oAROPgota/s=">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</latexit>
slide-143
SLIDE 143

Heat transport from laYce dynamics

  • =
  • n

R

n ˙

en + d dt

  • n

unen

<latexit sha1_base64="AvA/Wt+aTpBFH/a/oAROPgota/s=">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</latexit>

Rn = R

n + un

<latexit sha1_base64="Vn+N/R9vaS62E3uDWHRx/V4RZU=">AB+HicbVDLSgNBEOyNr7i+Vj16GQyCIRNFMxFCHjxGMU8wMRltjObDJl9MDMrxJDf8OxNPOrZq/6Df+NEFyWJdaqu6oaq9hPBlXbdTyu3sLi0vJftdfWNza3nO2dhopTiayOsYhly6eKCR6xuZasFYiGQ19wZr+4HziN+YVDyOrvUwYZ2Q9iIecKTaSJ7jEtIOqe7AbnyInL2N92kUs02tGvlprJcwpu0f0GmSeljBQgQ81zXtrdGNOQRoFVWpEpeYo2Nhup4olFAe0x0Y+pL3+npapaFSw9Afk4NJAjXrTcT/vJtUB5XOiEdJqlmEZsV4QSqIjsnkCaTLJUMthoZQlNzkIdinkqI2r7Jt07E02ieNMrF0nGxfHlSqFaytnYg304hBKcQhUuoAZ1QHiAN3iHD+verSerOef1ZyV3ezCFKzXL+bqkV4=</latexit>

J =

  • n

˙ Rnen + Rn ˙ en

  • d
<latexit sha1_base64="AvA/Wt+aTpBFH/a/oAROPgota/s=">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</latexit>
slide-144
SLIDE 144

Heat transport from laYce dynamics

  • =
  • n

R

n ˙

en + d dt

  • n

unen

<latexit sha1_base64="AvA/Wt+aTpBFH/a/oAROPgota/s=">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</latexit>

Rn = R

n + un

<latexit sha1_base64="Vn+N/R9vaS62E3uDWHRx/V4RZU=">AB+HicbVDLSgNBEOyNr7i+Vj16GQyCIRNFMxFCHjxGMU8wMRltjObDJl9MDMrxJDf8OxNPOrZq/6Df+NEFyWJdaqu6oaq9hPBlXbdTyu3sLi0vJftdfWNza3nO2dhopTiayOsYhly6eKCR6xuZasFYiGQ19wZr+4HziN+YVDyOrvUwYZ2Q9iIecKTaSJ7jEtIOqe7AbnyInL2N92kUs02tGvlprJcwpu0f0GmSeljBQgQ81zXtrdGNOQRoFVWpEpeYo2Nhup4olFAe0x0Y+pL3+npapaFSw9Afk4NJAjXrTcT/vJtUB5XOiEdJqlmEZsV4QSqIjsnkCaTLJUMthoZQlNzkIdinkqI2r7Jt07E02ieNMrF0nGxfHlSqFaytnYg304hBKcQhUuoAZ1QHiAN3iHD+verSerOef1ZyV3ezCFKzXL+bqkV4=</latexit>

J =

  • n

˙ Rnen + Rn ˙ en

  • d
<latexit sha1_base64="AvA/Wt+aTpBFH/a/oAROPgota/s=">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</latexit>
slide-145
SLIDE 145

Heat transport from laYce dynamics

  • =
  • n

R

n ˙

en + d dt

  • n

unen

<latexit sha1_base64="AvA/Wt+aTpBFH/a/oAROPgota/s=">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</latexit>

Rn = R

n + un

<latexit sha1_base64="Vn+N/R9vaS62E3uDWHRx/V4RZU=">AB+HicbVDLSgNBEOyNr7i+Vj16GQyCIRNFMxFCHjxGMU8wMRltjObDJl9MDMrxJDf8OxNPOrZq/6Df+NEFyWJdaqu6oaq9hPBlXbdTyu3sLi0vJftdfWNza3nO2dhopTiayOsYhly6eKCR6xuZasFYiGQ19wZr+4HziN+YVDyOrvUwYZ2Q9iIecKTaSJ7jEtIOqe7AbnyInL2N92kUs02tGvlprJcwpu0f0GmSeljBQgQ81zXtrdGNOQRoFVWpEpeYo2Nhup4olFAe0x0Y+pL3+npapaFSw9Afk4NJAjXrTcT/vJtUB5XOiEdJqlmEZsV4QSqIjsnkCaTLJUMthoZQlNzkIdinkqI2r7Jt07E02ieNMrF0nGxfHlSqFaytnYg304hBKcQhUuoAZ1QHiAN3iHD+verSerOef1ZyV3ezCFKzXL+bqkV4=</latexit>

Φjγ

iβ =

∂2E ∂uiβ∂ujγ

  • u=0
<latexit sha1_base64="YkLqhQkrRa0LwZ3RneaFcE2Laq0=">ACQHicbVDLSgMxFE18W7VWXboJFsFVmVbBbgRBJcVbBWMHe7ETBubzAxJRizjfJS/4F+4ce1O3LoyU6to9a7OPe+zg0SKYz1vCc8NT0zOze/sFhaWl4pr1bW1jsmTjXjbRbLWF8EYLgUEW9bYSW/SDQHFUh+HgyOCv38lmsj4ujMDhN+paAXiVAwsI7yK3e01Rd+JmjALeTd7Ib2QCnIyQGhkoeW1AgNbCMJqCtANltkGOSf6ck/W7+SX2NyQnVote35N7PqALbD0KSutle7leqXs0bBfkL6mNQReNo+ZUHeh2zVPHIMgnGZMU2JnleoqnhCbAB9HjGVDBa+JsFZcxQBTnZLm4wk1pB/qdpjZsXmUiSlLI+ZKnBamktiYFM8k10JzZuXQAWBauHsI64N7l3UvL5Wcx/qko7+g06jVd2uN073qYXPsdgFtoi20g+poHx2iE9RCbcTQM57BK7iMH/ELfsVvn6VTeNyzgX4Ffv8A9XGuAQ=</latexit>

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

J =

  • n

˙ Rnen + Rn ˙ en

  • d
<latexit sha1_base64="AvA/Wt+aTpBFH/a/oAROPgota/s=">ACbXicbVHZSgMxFM2MWx23qvikyMXihlA6VAUpeCL+KRiteDUkzbWhmIckIZjP8QP8HL/CXzBTx6WtFwKHc869uSdxI86kqlTeDXNicmp6pjBrzc0vLC4Vl1ceZBgLQusk5KFouFhSzgJaV0x2ogExb7L6aPbu8z0xcqJAuDe9WPaNPHnYB5jGClqVbx1fGx6roeXMOnIMjY78VgOyDoc9cNqhAki+PXep1qg+B/BDZe6Bi+Z9AvbBcay/437dzw5hgz16FmewCRp0lbpd8L5A3x14WtYqlSrgwKxoGdgxLK6ZVfNMXkNingSIcS5lgoRjhNLWcWNIkx7u0IT4rmCdrhpmsS9l3dT2M6WkKNaRv6nPcXKO2kmLIhiRQOiLVrzYg4qhOzloc0EJYr3NcBEML0PkC7W0ZX+H8vSGe3ROPgoVq2D8vV26NS7SJPW0DraAvtIRsdoxq6QjeojogxZ9jGqXFmfJhr5oa5+WU1jbxnFQ2VufsJ5dK1Cw=</latexit>
slide-146
SLIDE 146

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

κ = 1 V

  • nm

cnm

  • vnm

nm

<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>
  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255
slide-147
SLIDE 147

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

v α

nm =

1 2√ωnωm

  • ijβγ

R

iα − R jα

  • MiMj

Φjγ

iβeiβ n ejγ m

τ

nm =

γn + γm (γn + γm)2 + (ωn − ωm)2 cnm = ωnωm T n(ωm) − n(ωn) ωm − ωn

<latexit sha1_base64="IQSrUBxf8BZ0mu4Xn2OnBomcWOQ=">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</latexit>

n(ω) = 1 e

ω kBT − 1

<latexit sha1_base64="xgR9+nbT9/d9H/dBRdLs2NLuo=">AC3icbVBLS8NAGNz4rPEV9ehlsQj1YEmqYC9CsRePFfoCU8Nmu0mW7iZhdyOUkJ/gD/Eo3sSrZ6/235jYXNo6p2FmPvhm3JhRqUxzpq2tb2xubVd29N29/YND4+i4L6NEYNLDEYvE0EWSMBqSnqKkWEsCOIuIwN30i78wTMRkZhV01jMuLID6lHMVK5BjtsGZHnPjoAt5C2xMIp1aWQpsjFQiekuwpnat24CIB59ksnTh3sJtl8BJaMHOMqlk3/wBXiVWSKijRcYw3exzhJNQYakTJFQFDOS6XYiSYzwBPkxdwV1A/Uoq4lFPuZvC8eFEue4X4n/eYK85SmkYJ4qEOI/knpcwqCJY7ALHVBCs2DQnCAua/wNxgPLmKl9P1/O1nKjVdJv1K2reuPhutpqlm0r4BScgRqwA1ogXvQAT2AwSv4Bj9gpr1o79qH9jmPrmnlzQlYgPb1C7JMmkU=</latexit>

cnn = kB ωn kBT 2 eωn/kBT (eωn/kBT − 1)2

<latexit sha1_base64="dJXfV5eqzq/WVsc5CUYa/VeqP4E=">ACRXicbVDNSgMxGEz8rfWv2qOXYBHqwdqtgr0IohePCq0Kri7Z+G0bmSXJCuUJQ/lU/gMnj17E6+a6l6qzmn4ZhJmJs4EN7bdfsEzs3PzC4uVperyuraem1j8qkuWbQZ6lI9U1MDQiuoG+5FXCTaAyFnAdj84m+vUjaMNT1bPjDO4kHSiecEatP0U1x6JCKXc8ik5DAYklTRImrIiHMZUh6mEAY2UK7xOeo6Emg+Gluzed0ob3E87972x51zRDCW1Qy0LcP869gL/h4tqjXar/Q3ylwQlaASF1HtKXxIWS5BWSaoMQXVljMBrhrmBjLKRnQABZPxd87pK5XGjGXsyM4kmvmtTY7/abe5Tbp3BVdZbkExb/FakgtiUzJZlDxwDcyKsSeUae7zEDakfhzrd69Wfcfgd6O/5KrTCg5ancvDxkm3bFtBW2gbNVGAjtAJOkcXqI8YesVLeBPX8TN+w+/48c6g8s3dTQF/PkFxJev2A=</latexit>

κ = 1 V

  • nm

cnm

  • vnm

nm

<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>
  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255
slide-148
SLIDE 148

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

κ = 1 V

  • nm

cnm

  • vnm

nm

<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>
  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255

a-Si

slide-149
SLIDE 149

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

κ = 1 V

  • nm

cnm

  • vnm

nm

<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>
  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255

a-Si

slide-150
SLIDE 150

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

κ = 1 V

  • nm

cnm

  • vnm

nm

<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>
  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255

a-Si a-Si

slide-151
SLIDE 151

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

κ = 1 V

  • nm

cnm

  • vnm

nm

<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>

In a periodic system

vnn = δννδqq

<latexit sha1_base64="vZSeAhclBjrOb91gpl3A4Ydp94=">AB/HicbVBNS8NAEJ3Urxq/oh69BIvU0lqQS9KwYvHCvYDTAmb7aZdutnE3U2hPhDPHsTD148e9V/4L8x0VBo62MOb96bgXnjRYxKZVnfWmldW19o7ypb23v7O4Z+wcdGcYCkzYOWSh6HpKEU7aipGepEgKPAY6Xrj69zvToiQNOR3ahqRfoCGnPoUI5VJrtGYuAn1fTSGRCmkJs4PM6qms76AKmR5sPM1JNXaNi1axfmMvELkgFCrRc480ZhDgOCFeYISkTJBTFjKS6E0sSITxGQ5LgwBN0OFLzKgqknAZeap7kB8hFLxf/8+5j5V/0E8qjWBGOs5HM82NmqtDMH2EOqCBYsWlGEBY0u8fEIyQVtm7dD3LaC8mWiades0+q9VvG5XmVZG2DEdwDKdgwzk04QZa0AYMT/ABn/ClPWrP2ov2+jda0oqdQ5iD9v4D1MGVsA=</latexit>
  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255
slide-152
SLIDE 152

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

κ = 1 V

  • nm

cnm

  • vnm

nm

<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>

In a periodic system

vnn = δννδqq

<latexit sha1_base64="vZSeAhclBjrOb91gpl3A4Ydp94=">AB/HicbVBNS8NAEJ3Urxq/oh69BIvU0lqQS9KwYvHCvYDTAmb7aZdutnE3U2hPhDPHsTD148e9V/4L8x0VBo62MOb96bgXnjRYxKZVnfWmldW19o7ypb23v7O4Z+wcdGcYCkzYOWSh6HpKEU7aipGepEgKPAY6Xrj69zvToiQNOR3ahqRfoCGnPoUI5VJrtGYuAn1fTSGRCmkJs4PM6qms76AKmR5sPM1JNXaNi1axfmMvELkgFCrRc480ZhDgOCFeYISkTJBTFjKS6E0sSITxGQ5LgwBN0OFLzKgqknAZeap7kB8hFLxf/8+5j5V/0E8qjWBGOs5HM82NmqtDMH2EOqCBYsWlGEBY0u8fEIyQVtm7dD3LaC8mWiades0+q9VvG5XmVZG2DEdwDKdgwzk04QZa0AYMT/ABn/ClPWrP2ov2+jda0oqdQ5iD9v4D1MGVsA=</latexit>

κ = 1 V

cν(q)vν(q)2τν(q)

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  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255
slide-153
SLIDE 153

BTE

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

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κ = 1 V

  • nm

cnm

  • vnm

nm

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In a periodic system

vnn = δννδqq

<latexit sha1_base64="vZSeAhclBjrOb91gpl3A4Ydp94=">AB/HicbVBNS8NAEJ3Urxq/oh69BIvU0lqQS9KwYvHCvYDTAmb7aZdutnE3U2hPhDPHsTD148e9V/4L8x0VBo62MOb96bgXnjRYxKZVnfWmldW19o7ypb23v7O4Z+wcdGcYCkzYOWSh6HpKEU7aipGepEgKPAY6Xrj69zvToiQNOR3ahqRfoCGnPoUI5VJrtGYuAn1fTSGRCmkJs4PM6qms76AKmR5sPM1JNXaNi1axfmMvELkgFCrRc480ZhDgOCFeYISkTJBTFjKS6E0sSITxGQ5LgwBN0OFLzKgqknAZeap7kB8hFLxf/8+5j5V/0E8qjWBGOs5HM82NmqtDMH2EOqCBYsWlGEBY0u8fEIyQVtm7dD3LaC8mWiades0+q9VvG5XmVZG2DEdwDKdgwzk04QZa0AYMT/ABn/ClPWrP2ov2+jda0oqdQ5iD9v4D1MGVsA=</latexit>

κ = 1 V

cν(q)vν(q)2τν(q)

<latexit sha1_base64="CIrRL/c3f1YcWZ3VBtIqAyPw7D8=">ACI3icbVC7TsMwFHXKq4RXgJElokIqS9UJFhAlVgYi0QfUl0ix3Vaq84DPypVUT6HD+ATmNkQCwMzK/wBDkRITmDdXzOvbLP8WJGhazX34zS0vLK6lp53dzY3NresXb3OiJSHJM2jljEex4ShNGQtCWVjPRiTlDgMdL1JleZ350SLmgU3spZTAYBGoXUpxhJLblWH05QHKML6HOEydNOikUKnATGCA59nz7HoYqxa4+q3/S8bRwv2vYUCJVUF2rUq/Vf2AvEicnFZCj5VqPcBhFZBQYoaESBCXFDOSmlAJEiM8QSOS4MDjdDSW8yoKhJgFXmofZe+LopeJ/3l9Jf3zQULDWEkSYj2iPV8xW0Z21pc9pJxgyWaIMyp/o+Nx0iXJXWrpqkzOsVEi6TqDkntcbNaV5mactgwNwCKrAWegCa5BC7QBk/gA3yCL+PBeDZejNf0ZKR7+yDORjv37shpPk=</latexit>
  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255
slide-154
SLIDE 154

BTE

κ = 1 V

cν(q)vν(q)2τν(q)

<latexit sha1_base64="CIrRL/c3f1YcWZ3VBtIqAyPw7D8=">ACI3icbVC7TsMwFHXKq4RXgJElokIqS9UJFhAlVgYi0QfUl0ix3Vaq84DPypVUT6HD+ATmNkQCwMzK/wBDkRITmDdXzOvbLP8WJGhazX34zS0vLK6lp53dzY3NresXb3OiJSHJM2jljEex4ShNGQtCWVjPRiTlDgMdL1JleZ350SLmgU3spZTAYBGoXUpxhJLblWH05QHKML6HOEydNOikUKnATGCA59nz7HoYqxa4+q3/S8bRwv2vYUCJVUF2rUq/Vf2AvEicnFZCj5VqPcBhFZBQYoaESBCXFDOSmlAJEiM8QSOS4MDjdDSW8yoKhJgFXmofZe+LopeJ/3l9Jf3zQULDWEkSYj2iPV8xW0Z21pc9pJxgyWaIMyp/o+Nx0iXJXWrpqkzOsVEi6TqDkntcbNaV5mactgwNwCKrAWegCa5BC7QBk/gA3yCL+PBeDZejNf0ZKR7+yDORjv37shpPk=</latexit>

Heat transport from laYce dynamics

Jα = 1 2

  • ijβγ
  • R

iα − R jα

  • Φjγ

iβuiβ ˙

ujγ,

<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>

In a periodic system

  • L. Isaeva, G. Barbalinardo, D. Donadio, and S. Baroni, Nature Commun. in press (2019), hups://arxiv.org/abs/1904.02255
slide-155
SLIDE 155

laYce dynamics from density-funcFonal perturbaFon theory

slide-156
SLIDE 156

laYce dynamics

R’ R u(R) u(R’)

V (r) = V0(r) + X

R

u(R) · ∂v(r − R) ∂R + · · ·

slide-157
SLIDE 157

laYce dynamics

R’ R u(R) u(R’)

V (r) = V0(r) + X

R

u(R) · ∂v(r − R) ∂R + · · · E = E0 + 1 2 X

R,R0

u(R) · ∂2E ∂u(R)∂u(R0) · u(R0) + · · ·

− ∂F(R) ∂u(R0)

slide-158
SLIDE 158

laYce dynamics

R’ R u(R) u(R’)

DFT DPFT

V (r) = V0(r) + X

R

u(R) · ∂v(r − R) ∂R + · · · E = E0 + 1 2 X

R,R0

u(R) · ∂2E ∂u(R)∂u(R0) · u(R0) + · · ·

− ∂F(R) ∂u(R0)

slide-159
SLIDE 159

laYce dynamics

R’ R u(R) u(R’)

det  ∂2E ∂u(R)∂u(R0) − ω2M(R)δR,R0

  • = 0

DFT DPFT

V (r) = V0(r) + X

R

u(R) · ∂v(r − R) ∂R + · · · E = E0 + 1 2 X

R,R0

u(R) · ∂2E ∂u(R)∂u(R0) · u(R0) + · · ·

− ∂F(R) ∂u(R0)

slide-160
SLIDE 160

Vλ(r) = V0(r) + X

i

λivi(r)

density-funcFonal perturbaFon theory

slide-161
SLIDE 161

Vλ(r) = V0(r) + X

i

λivi(r)

density-funcFonal perturbaFon theory

E(λ) = min

n

✓ F[n] + Z Vλ(r)n(r) ◆ Z n(r)dr = N DFT

slide-162
SLIDE 162

Vλ(r) = V0(r) + X

i

λivi(r)

density-funcFonal perturbaFon theory

∂E(λ) ∂λi = Z nλ(r)vi(r)dr HF E(λ) = min

n

✓ F[n] + Z Vλ(r)n(r) ◆ Z n(r)dr = N DFT

slide-163
SLIDE 163

Vλ(r) = V0(r) + X

i

λivi(r)

density-funcFonal perturbaFon theory

∂E(λ) ∂λi = Z nλ(r)vi(r)dr HF E(λ) = min

n

✓ F[n] + Z Vλ(r)n(r) ◆ Z n(r)dr = N DFT DFPT ∂2E(λ) ∂λi∂λj = Z ∂nλ(r) ∂λj vi(r)dr

slide-164
SLIDE 164

calculaFng the response

n(r) = X

v

|φv(r)|2 n0(r) = 2Re X

v

φ⇤

v (r)φ0 v(r)

slide-165
SLIDE 165

⇥0

v =

X

c

c

⇥⇥

c|V 0|⇥ v⇤

  • v

c

n0(r) = 2Re X

v

φ⇤

v (r)φ0 v(r)

= 2Re X

cv

ρ0

vcφ⇤ v (r)φ c(r)

calculaFng the response

n(r) = X

v

|φv(r)|2 n0(r) = 2Re X

v

φ⇤

v (r)φ0 v(r)

slide-166
SLIDE 166

(H −

v)⇥0 v = −PcV 0⇥ v

⇥0

v =

X

c

c

⇥⇥

c|V 0|⇥ v⇤

  • v

c

n0(r) = 2Re X

v

φ⇤

v (r)φ0 v(r)

= 2Re X

cv

ρ0

vcφ⇤ v (r)φ c(r)

calculaFng the response

n(r) = X

v

|φv(r)|2 n0(r) = 2Re X

v

φ⇤

v (r)φ0 v(r)

slide-167
SLIDE 167

calculaFng the response

n0(r) = 2Re X

v

φ⇤

v (r)φ0 v(r)

(H −

v)⇥0 v = −PcV 0⇥ v

slide-168
SLIDE 168

V0(r) n(r)

DFPT: the equaFons

  • −∆ + VSCF (r)

⇥ ⇥v(r) = v⇥v(r) n(r) =

  • v<EF

|φv(r)|2 VSCF (r) = V0(r) +

  • n(r)

|r − r|dr + µxc(r)

DFT

slide-169
SLIDE 169

V0(r) n(r)

DFPT: the equaFons

V (r) n(r) DFPT

V

SCF (r) = V (r) +

  • n(r)

|r − r|dr + µ

xc(r)

n⇥(r) = 2 Re

  • v<EF

φ

v(r)φ⇥ v(r)

  • −∆ + VSCF (r)

⇥ ⇥v(r) = v⇥v(r) n(r) =

  • v<EF

|φv(r)|2 VSCF (r) = V0(r) +

  • n(r)

|r − r|dr + µxc(r)

DFT

  • −∆ + VSCF (r) − v

⇥ ⇥

v(r) = PcV SCF (r)⇥v(r)

SB, P. Giannozzi, and A. Testa, Phys. Rev. Leu. 58, 1861 (1987)

slide-170
SLIDE 170

phonons from DFPT

  • P. Giannozzi, S. de Gironcoli, P. Pavone, and SB, Phys. Rev. B 43, 7231 (1991)
slide-171
SLIDE 171

Φ = Φ0 + O() ⇒ E = E0 + O(2) Φ = Φ0 +

n

X

l=1

⇥lΦ(l) + O(⇥n+1) ⇒ E = E0 +

2n+1

X

l=1

⇥lE(l) + O(⇥2n+2)

the “2n+1” theorem

slide-172
SLIDE 172

Φ = Φ0 + O() ⇒ E = E0 + O(2) Φ = Φ0 +

n

X

l=1

⇥lΦ(l) + O(⇥n+1) ⇒ E = E0 +

2n+1

X

l=1

⇥lE(l) + O(⇥2n+2)

the “2n+1” theorem

slide-173
SLIDE 173

Φ = Φ0 + O() ⇒ E = E0 + O(2) Φ = Φ0 +

n

X

l=1

⇥lΦ(l) + O(⇥n+1) ⇒ E = E0 +

2n+1

X

l=1

⇥lE(l) + O(⇥2n+2)

the “2n+1” theorem

slide-174
SLIDE 174

Φ = Φ0 + O() ⇒ E = E0 + O(2) Φ = Φ0 +

n

X

l=1

⇥lΦ(l) + O(⇥n+1) ⇒ E = E0 +

2n+1

X

l=1

⇥lE(l) + O(⇥2n+2)

the “2n+1” theorem

E = hΦ0 + Φ0|(H0 + V 0)|Φ0 + Φ0i hΦ0 + Φ0|Φ0 + Φ0i + O(V 04)

slide-175
SLIDE 175

Φ = Φ0 + O() ⇒ E = E0 + O(2) Φ = Φ0 +

n

X

l=1

⇥lΦ(l) + O(⇥n+1) ⇒ E = E0 +

2n+1

X

l=1

⇥lE(l) + O(⇥2n+2)

the “2n+1” theorem

E = hΦ0 + Φ0|(H0 + V 0)|Φ0 + Φ0i hΦ0 + Φ0|Φ0 + Φ0i + O(V 04) E(3) = hΦ0|V 0|Φ0i hΦ0|Φ0ihΦ0|V 0|Φ0i

slide-176
SLIDE 176

Anharmonic lifeFmes

ˆ aν

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ˆ a†

ν

<latexit sha1_base64="uZvHs+qEn9qBzBYOQjaJS3n8CkA=">AB3icdZA7T8MwFIWd8irhFWBgYLGoExVWq2bBUsjEWiD4mU6MZ1E6vOQ7aDVEWZ2RArMysM/Bz+DS6EoQjO9Omce6V7rpdwJpVtfxilpeWV1bXyurmxubW9Y+3u9WcCkJ7JOaxGHogKWcR7SmOB0mgkLocTrwpfzfHBPhWRxdKNmCR2F4EdswgobnWAXYCUBjunDH4PhVu5kTpSY5dq2JX7War1WhgDV/S0KzVm+1zXCucCirUda13ZxyTNKSRIhykzEAoRjNTSeVNAEyBZ9mJPQE8wO16EIo5Sz0cnwcgrk72xu/pXdpmrSHmUsSlJFI6JHdDZJOVYxnfFYyYoUXymAYhg+h5MAhBAlP6IaeqOP0Xw/9CvV2tn1fp1o9K5KNqW0SE6Qqeohlqog65QF/UQTl6Qa/ozQDjwXg0nr5HS0axs48WZDx/AtPoiPg=</latexit>

ˆ a†

ν

<latexit sha1_base64="sK9q1ZqacMKZvs0eXN3eBpcAog=">AB4HicdZBLS8NAFIUn9VXjK+pGcDNYpK5KWktbd0U3LivYB5gabqbTZOjkwcxEKGu3Ylb124V/Dn+G6caFxU9q49z7oV7rpdwJpVtfxiFpeWV1bXiurmxubW9Y+3u9WScCkK7JOaxGHgKWcR7SqmOB0kgkLocdr3JhfzvH9HhWRxdK2mCR2G4EdszAgobnWAXYCUBhunRH4PhVu5kRpuTzDrlWyK3aj2azXsYvaWhUa43WGa7mTgnl6rjWuzOKSRrSBEOUmYgFCOczkwnlTQBMgGfZiT0BPMDtehCKOU09Gb4OAQVyN/Z3Pwru0nVuDXMWJSkikZEj+hsnHKsYjwvi0dMUKL4VAMQwfQ9mAQgCj9EtPUHX+K4P+hV6tUTyu1q3qpfZ63LaJDdIROUBU1URtdog7qIoLu0Qt6RW+GZzwYj8bT92jByHf20YKM5083Bokp</latexit>

ˆ aν

<latexit sha1_base64="wiZCqpHGlpFywcoWCnxP+NXfY0s=">AB1HicdZA7TwJBFIXv4gvXF2pM5GYWJFdJIAd0cYSE3kLpLZYWAnzD4yc9eErFTG1tpW/4E/x3/joGuB0VN9Oefe5J7rJ1JodJwPq7Cyura+Udy0t7Z3dvdK+wdHaeK8Q6LZaz6PtVcioh3UKDk/URxGvqS9/zp5SLv3XOlRzd4Czhg5BOIjEWjKx7ogXUCR0mHlROifDUtmpOPVGo1YjBr5koO5W681z4uZOGXK1h6V3bxSzNOQRMkm1zqhCwSf216qeULZlE54xkJfiUmAy4NtZ6F/pychBQD/TtbmH9ltymOm4NMREmKPGJmxGTjVBKMyaIgGQnFGcqZAcqUMPcQFlBFGZo32Lbp+FOE/A/dasU9q1Sva+XWRd62CEdwDKfgQgNacAVt6ADBS/wCm9W13qwHq2n79GCle8cwpKs509Hp4Ui</latexit>

ˆ aν

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ˆ a†

ν

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uνuνuν ∼

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

  • ˆ

aν + ˆ a†

ν

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1

ν

= 2

  • νν
  • Vννν2 ×
  • (ν − ν − ν)nν(nν + 1)(nν + 1)+

(ν + ν − ν)nνnν(nν + 1)

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Germanium Γ = 0.7 cm-1 Γ L U X W K L W

U W K X L Γ max Alberto Debernardi, Stefano Baroni, and Elisa Molinari, Phys. Rev. Lett. 75, 1819 (1995)

slide-177
SLIDE 177

some references

  • Aris Marcolongo, Paolo Umari, and Stefano Baroni, Microscopic theory and quantum simula?on of atomic heat

transport, Nat. Phys. 12, 80-U111 (2016), hup://dx.doi.org/10.1038/NPHYS3509

  • Loris Ercole, Aris Marcolongo, Paolo Umari, and Stefano Baroni, Gauge Invariance of Thermal Transport Coefficients, J.

Low Temp. Phys. 185, 79 (2016), hup://dx.doi.org/10.1007/s10909-016-1617-6

  • Loris Ercole, Aris Marcolongo, and Stefano Baroni, Accurate thermal conduc?vi?es from op?mally short molecular

dynamics simula?ons, Sci. Rep. 7, 15835 (2017), hup://dx.doi.org/10.1038/s41598-017-15843-2

  • Stefano Baroni, Riccardo Bertossa, Loris Ercole, Federico Grasselli, and Aris Marcolongo, Heat transport in insulators

from ab ini?o Green-Kubo theory, in Handbook of Materials Modeling: Applica?ons: Current and Emerging Materials, edited by Wanda Andreoni and Sidney Yip (Springer InternaFonal Publishing, Cham, 2018) pp. 1–36, 2nd ed., hup:// dx.doi.org/10.1007/978-3-319-50257-1_12-1, hup://arxiv.org/abs/1802.08006

  • Riccardo Bertossa, Federico Grasselli, Loris Ercole, and Stefano Baroni, Theory and numerical simulaFon of heat

transport in mulF-component systems, Phys. Rev. Leu. in press (2019), hup://arxiv.org/abs/1808.03341

  • Leyla Isaeva, Giuseppe Barbalinardo, Davide Donadio, Stefano Baroni, Modeling heat transport in crystals and glasses

from a unified laYce-dynamical approach, Nat. Commun. in press (2019), hups://arxiv.org/abs/1904.02255

  • Michele Simoncelli, Nicola Marzari, and Francesco Mauri, Unified theory of thermal transport in crystals and glasses.
  • Nat. Phys. (2019), hups://doi.org/10.1038/s41567-019-0520-x
  • For an overview of various simulaFon approaches to heat transport, see, e.g.: Giorgia Fugallo and Luciano Colombo,

Calcula?ng laHce thermal conduc?vity: a synopsis, Phys. Scr. 93, 043002, (2018), hup://doi.org/10.1088/1402-4896/ aaa6f3

  • Stefano Baroni, working dra} of the notes for a short course on Theory and numerical simula?on of heat, mass, and

charge transport in condensed maJer, hup://stefano.baroni.me/hydrodynamic-fluctuaFons

slide-178
SLIDE 178

QUANTUM ESPRESSO FOUNDATION

slide-179
SLIDE 179

Aris Marcolongo, SISSA now @IBM Zürich

thanks to:

Federico Grasselli, SISSA Riccardo Bertossa, SISSA

Loris Ercole, SISSA now @EPFL Lausanne Davide Donadio, UC Davis

Leyla Isaeva, SISSA

slide-180
SLIDE 180

these slides at hup://talks.baroni.me