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
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
lectures given at the ICTP Caribbean School on Materials for Clean Energy, Cartagena de Indias, May 30 — June 5, 2019
Scuola Internazionale Superiore di Studi AvanzaF, Trieste
coefficients;
cepstral analysis
allows).
T2 T1
T2 T1
T2 T1
T2 T1
T2 T1
energy saving
heat dissipaFon
heat shielding
heat shielding
energy conversion
planetary sciences
Uranus & Neptune
… energy saving and heat dissipaFon heat shielding energy conversion earth and planetary sciences …
…
E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2]
<latexit sha1_base64="QXrV1fvnKSHO2Orzd+KfVpHMt+U=">ACx3icfVHbatAEF0pvbhumjNY1+WmJRAi6NLUycPgdBSJ/qQGUHJEWs1mt58a4kdlchRuihn9dP6Bf0N7ryhdZW6MAOh5lzZnZm4pxRqSzrl2HuPHn67HnrRfvl7qu9/c7B6HMCoGJhzOWidsYScJoSjxFSO3uSCIx4yM4tnOj+6J0LSLP2u5jkJOUpSOqEYKR2KOj8DjtRU8PJL5ZfBop4vkjgsrZ7jvLd6/b52lMF3zhJUGRXAS7yLaL10dWkM7f2F675jpV+Pay/d+if3uv64cVfAe3RY0GDZ0TVlGnq5kLg01gr0AXrGwQdX4H4wXnKQKMySlb1u5CkskFMWMVO2gkCRHeIYS4muYIk5kWC4+VsFjHRnDSb0SxVcRP9VlJjHgiZTtVGnRFzKOY+1vh5Bbufq4GM5v1CT87CkaV4okuJl+0nBoMpgfVU4poJgxeYaICyongDiKRIK317vRl7ew9NMHR6tzbj50rz6tdtQCb8AROAE26IMrcA0GwAPYODU8486IzK9mZt6bD0uqaw0h2DzB9/ACF61oU=</latexit>E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2]
<latexit sha1_base64="QXrV1fvnKSHO2Orzd+KfVpHMt+U=">ACx3icfVHbatAEF0pvbhumjNY1+WmJRAi6NLUycPgdBSJ/qQGUHJEWs1mt58a4kdlchRuihn9dP6Bf0N7ryhdZW6MAOh5lzZnZm4pxRqSzrl2HuPHn67HnrRfvl7qu9/c7B6HMCoGJhzOWidsYScJoSjxFSO3uSCIx4yM4tnOj+6J0LSLP2u5jkJOUpSOqEYKR2KOj8DjtRU8PJL5ZfBop4vkjgsrZ7jvLd6/b52lMF3zhJUGRXAS7yLaL10dWkM7f2F675jpV+Pay/d+if3uv64cVfAe3RY0GDZ0TVlGnq5kLg01gr0AXrGwQdX4H4wXnKQKMySlb1u5CkskFMWMVO2gkCRHeIYS4muYIk5kWC4+VsFjHRnDSb0SxVcRP9VlJjHgiZTtVGnRFzKOY+1vh5Bbufq4GM5v1CT87CkaV4okuJl+0nBoMpgfVU4poJgxeYaICyongDiKRIK317vRl7ew9NMHR6tzbj50rz6tdtQCb8AROAE26IMrcA0GwAPYODU8486IzK9mZt6bD0uqaw0h2DzB9/ACF61oU=</latexit>E[Ω] =
(r)dr
<latexit sha1_base64="nJ7O7wYz86gCjOrm4vPZQP5AzFw=">ACMXicbVDLSgMxFM34flt16SZYBN2UGSvoRiK4E4FawudoWTSO20wyQzJHaEMfojf4Qe41U9wJ25c+BOmD8TXgcDJOTlJ7okzKSz6/os3MTk1PTM7N7+wuLS8slpaW7+2aW41HkqU9OMmQUpNRoIRmZoCpWEIjvjkZ+I1bMFak+gr7GUSKdbVIBGfopHapetoKzxV0WXQUCo3t0YaGkFkhU70TKoa9OKFmt/NF26WyX/GHoH9JMCZlMsZFu/QedlKeK9DIJbO2FfgZRgUzKLiEu4Uwt5AxfsO60HJUMwU2KobD3dFtp3Rokhq3NKh+j1RcBUb0e3hj3sKpqztq9jlB9+2v72B+J/XyjE5jAqhsxB89HzS4pnTQH+0IAxl3xHGjXATUN5jhnF0Lbtmgt89/CXe5WgWtm73C/XjscdzZFNskV2SEAOSI2ckQtSJ5zck0fyRJ69B+/Fe/XeRkcnvHFmg/yA9/EJzgmqnQ=</latexit>dE(Ω, t) dt = −
j(r, t)·ˆ n dS +
σ(r, t)dΩ
<latexit sha1_base64="XbuOEKse8wLAGTzqaROYHRn8QM=">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</latexit>Ω
<latexit sha1_base64="BaZ+tB0g2WIFPY2faH4+JbaEXls=">ABzHicbY9LS8NAFIVvfNb4qrp0M1gEV2GS1tjuCiK40gr2AbaUyXSajp1JwsxEKFb1271X/hz/DemDxdVz+pwzrlwvyARXBuMv6y19Y3Nre3Cjr27t39wWDw6buk4VZQ1aSxi1QmIZoJHrGm4EayTKEZkIFg7GF/P+vYLU5rH0aOZJKwnSRjxIafE5FGrey9ZSPrFEnZqNVxfYSdS4w9v5YbXPaqvo9cB89VgqUa/eJndxDTVLIUEG0zogynAo2tbupZgmhYxKyjMpA8XBkVlMitZ7IYIrOJTEj/bubhf91T6kZVnsZj5LUsIjmk7wbpgKZGM3A0IArRo2Y5IZQxfN/EB0RajJ8W17zrgAQX/ND2PLc9y4z1USvWbJW0BTuEMLsCFK6jDLTSgCRSe4Q3e4cO6s4yVWdPFdM1a3pzAiqzXb2lfgjc=</latexit>∂Ω
<latexit sha1_base64="9XU30fqENgeU7YCusNCRICny7I=">AB1HicbZBLT8JAFIVv8YX1hbp0M5GYuGqmBSvsSIyJOzERJLFIpsNQJkwfmZmaEGRl3Lp2q/An+O/sQVcoJ7VyTn3sw3fiK40h/GYWV1bX1jeKmubW9s7tX2j9oqziVlLVoLGLZ8YligkespbkWrJNIRkJfsDt/dJH3d49MKh5Ht3qcsG5IgogPOCU6ix68hEjNifCuQxaQXqmMrXodV20XYesMY8etZwZXnJrItvCM5VhoWav9On1Y5qGLNJUEKUm+TUq2NT0UsUSQkckYBMa+pIHQ72cklCpcehP0UlI9FD97vLwv+4+1YNad8KjJNUsotlI1g1SgXSMckDU5JRLcaZIVTy7D2IDokVGfYJozxjkI+mt+GNuOZVcs56ZablwuaItwBMdwCjacQwOuoAktoCDhDd7hw2gbT8az8TIfLRiLnUNYkvH6DT2qhdA=</latexit>ˆ n
<latexit sha1_base64="63Fug9Nb8/YzHaBtAyeHRF+w5s=">AB13icbZA7T8MwFIVvyquERwOMLBYVElPkpCW0WwULY5FoC6JV5bhOa9V5yHaQqhiQ6zMrPAD+Dn8G9IHQ4EzHZ1zr64/+4ngSmP8ZRTW1jc2t4rb5s7u3n7JOjhsqziVlLVoLGJ5xPFBI9YS3Mt2F0iGQl9wTr+GrWdx6ZVDyObvUkYb2QDCMecEp0HvWtUndEdNYNiR75AYqmfauM7XodVx0PYfscY9er5wZX3JrnIcfGc5VhqWbf+uwOYpqGLNJUEKUyIjWngk3NbqpYQuiYDFlGQ1/y4UivpiRUahL6U3Q6u69+d7Pwv+4h1UGtl/EoSTWLaD6Sd0EqkI7RjBENuGRUi0luCJU8fw+iIyIJ1flPmOacQGC/pofxrZrOxXbvamWG5dL2iIcwmcgQMX0IBraEILKTwBu/wYdwbT8az8bIYLRjLnSNYkfH6DV6lhmc=</latexit>dE(Ω, t) dt = −
j(r, t)·ˆ n dS +
σ(r, t)dΩ
<latexit sha1_base64="XbuOEKse8wLAGTzqaROYHRn8QM=">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</latexit>˙ (r, t)dΩ =
· j(r, t)dΩ
<latexit sha1_base64="lfl9TnmoqkcgN5qMe9dsPKyiUF8=">ACNHicbVA9SwNBFNyNXzF+nVraLAYhgoZLFLRAjZ2RjAf4IWwt9m7rNnbO3bfCSHkR9n4Q+wEO7G1tnQTr0nMa97wZgZmnp9IYcB13FuaXldS2/XtjY3NrecXb3miZONeMNFstYt31quBSKN0CA5O1Ecxr5krf8wc2Ebz1zbUSsHmCY8E5EQyUCwSjYU9eRnlDQ9e4iHlLi9WIgHk+MkLEiJS+i0PcDok/gmPQyzRU5JTMeRX1pF5t6M8fTYnPXKbpldzrkP6hkoIiyqXedFxuJpRFXwCQ1ZkQ1Cb5uOClhieUDWjIRyzytQj7MHulkTHDyB+To0kSM89Njou4xSCy85IqCQFrpiVWC5IJYGYTB5IekJzBnJoAWVa2DyE9amDOybCwXbsTLf6D9oVsuVs3L1/rxYu87a5tEBOkQlVEXqIZuUR01ENv6AdjnMOv+AN/4q8/aQ5n0M/j7FzvQpvY=</latexit>Ω
<latexit sha1_base64="BaZ+tB0g2WIFPY2faH4+JbaEXls=">ABzHicbY9LS8NAFIVvfNb4qrp0M1gEV2GS1tjuCiK40gr2AbaUyXSajp1JwsxEKFb1271X/hz/DemDxdVz+pwzrlwvyARXBuMv6y19Y3Nre3Cjr27t39wWDw6buk4VZQ1aSxi1QmIZoJHrGm4EayTKEZkIFg7GF/P+vYLU5rH0aOZJKwnSRjxIafE5FGrey9ZSPrFEnZqNVxfYSdS4w9v5YbXPaqvo9cB89VgqUa/eJndxDTVLIUEG0zogynAo2tbupZgmhYxKyjMpA8XBkVlMitZ7IYIrOJTEj/bubhf91T6kZVnsZj5LUsIjmk7wbpgKZGM3A0IArRo2Y5IZQxfN/EB0RajJ8W17zrgAQX/ND2PLc9y4z1USvWbJW0BTuEMLsCFK6jDLTSgCRSe4Q3e4cO6s4yVWdPFdM1a3pzAiqzXb2lfgjc=</latexit>∂Ω
<latexit sha1_base64="9XU30fqENgeU7YCusNCRICny7I=">AB1HicbZBLT8JAFIVv8YX1hbp0M5GYuGqmBSvsSIyJOzERJLFIpsNQJkwfmZmaEGRl3Lp2q/An+O/sQVcoJ7VyTn3sw3fiK40h/GYWV1bX1jeKmubW9s7tX2j9oqziVlLVoLGLZ8YligkespbkWrJNIRkJfsDt/dJH3d49MKh5Ht3qcsG5IgogPOCU6ix68hEjNifCuQxaQXqmMrXodV20XYesMY8etZwZXnJrItvCM5VhoWav9On1Y5qGLNJUEKUm+TUq2NT0UsUSQkckYBMa+pIHQ72cklCpcehP0UlI9FD97vLwv+4+1YNad8KjJNUsotlI1g1SgXSMckDU5JRLcaZIVTy7D2IDokVGfYJozxjkI+mt+GNuOZVcs56ZablwuaItwBMdwCjacQwOuoAktoCDhDd7hw2gbT8az8TIfLRiLnUNYkvH6DT2qhdA=</latexit>ˆ n
<latexit sha1_base64="63Fug9Nb8/YzHaBtAyeHRF+w5s=">AB13icbZA7T8MwFIVvyquERwOMLBYVElPkpCW0WwULY5FoC6JV5bhOa9V5yHaQqhiQ6zMrPAD+Dn8G9IHQ4EzHZ1zr64/+4ngSmP8ZRTW1jc2t4rb5s7u3n7JOjhsqziVlLVoLGJ5xPFBI9YS3Mt2F0iGQl9wTr+GrWdx6ZVDyObvUkYb2QDCMecEp0HvWtUndEdNYNiR75AYqmfauM7XodVx0PYfscY9er5wZX3JrnIcfGc5VhqWbf+uwOYpqGLNJUEKUyIjWngk3NbqpYQuiYDFlGQ1/y4UivpiRUahL6U3Q6u69+d7Pwv+4h1UGtl/EoSTWLaD6Sd0EqkI7RjBENuGRUi0luCJU8fw+iIyIJ1flPmOacQGC/pofxrZrOxXbvamWG5dL2iIcwmcgQMX0IBraEILKTwBu/wYdwbT8az8bIYLRjLnSNYkfH6DV6lhmc=</latexit>dE(Ω, t) dt = −
j(r, t)·ˆ n dS +
σ(r, t)dΩ
<latexit sha1_base64="XbuOEKse8wLAGTzqaROYHRn8QM=">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</latexit>˙ (r, t)dΩ =
· j(r, t)dΩ
<latexit sha1_base64="lfl9TnmoqkcgN5qMe9dsPKyiUF8=">ACNHicbVA9SwNBFNyNXzF+nVraLAYhgoZLFLRAjZ2RjAf4IWwt9m7rNnbO3bfCSHkR9n4Q+wEO7G1tnQTr0nMa97wZgZmnp9IYcB13FuaXldS2/XtjY3NrecXb3miZONeMNFstYt31quBSKN0CA5O1Ecxr5krf8wc2Ebz1zbUSsHmCY8E5EQyUCwSjYU9eRnlDQ9e4iHlLi9WIgHk+MkLEiJS+i0PcDok/gmPQyzRU5JTMeRX1pF5t6M8fTYnPXKbpldzrkP6hkoIiyqXedFxuJpRFXwCQ1ZkQ1Cb5uOClhieUDWjIRyzytQj7MHulkTHDyB+To0kSM89Njou4xSCy85IqCQFrpiVWC5IJYGYTB5IekJzBnJoAWVa2DyE9amDOybCwXbsTLf6D9oVsuVs3L1/rxYu87a5tEBOkQlVEXqIZuUR01ENv6AdjnMOv+AN/4q8/aQ5n0M/j7FzvQpvY=</latexit>Ω
<latexit sha1_base64="BaZ+tB0g2WIFPY2faH4+JbaEXls=">ABzHicbY9LS8NAFIVvfNb4qrp0M1gEV2GS1tjuCiK40gr2AbaUyXSajp1JwsxEKFb1271X/hz/DemDxdVz+pwzrlwvyARXBuMv6y19Y3Nre3Cjr27t39wWDw6buk4VZQ1aSxi1QmIZoJHrGm4EayTKEZkIFg7GF/P+vYLU5rH0aOZJKwnSRjxIafE5FGrey9ZSPrFEnZqNVxfYSdS4w9v5YbXPaqvo9cB89VgqUa/eJndxDTVLIUEG0zogynAo2tbupZgmhYxKyjMpA8XBkVlMitZ7IYIrOJTEj/bubhf91T6kZVnsZj5LUsIjmk7wbpgKZGM3A0IArRo2Y5IZQxfN/EB0RajJ8W17zrgAQX/ND2PLc9y4z1USvWbJW0BTuEMLsCFK6jDLTSgCRSe4Q3e4cO6s4yVWdPFdM1a3pzAiqzXb2lfgjc=</latexit>∂Ω
<latexit sha1_base64="9XU30fqENgeU7YCusNCRICny7I=">AB1HicbZBLT8JAFIVv8YX1hbp0M5GYuGqmBSvsSIyJOzERJLFIpsNQJkwfmZmaEGRl3Lp2q/An+O/sQVcoJ7VyTn3sw3fiK40h/GYWV1bX1jeKmubW9s7tX2j9oqziVlLVoLGLZ8YligkespbkWrJNIRkJfsDt/dJH3d49MKh5Ht3qcsG5IgogPOCU6ix68hEjNifCuQxaQXqmMrXodV20XYesMY8etZwZXnJrItvCM5VhoWav9On1Y5qGLNJUEKUm+TUq2NT0UsUSQkckYBMa+pIHQ72cklCpcehP0UlI9FD97vLwv+4+1YNad8KjJNUsotlI1g1SgXSMckDU5JRLcaZIVTy7D2IDokVGfYJozxjkI+mt+GNuOZVcs56ZablwuaItwBMdwCjacQwOuoAktoCDhDd7hw2gbT8az8TIfLRiLnUNYkvH6DT2qhdA=</latexit>ˆ n
<latexit sha1_base64="63Fug9Nb8/YzHaBtAyeHRF+w5s=">AB13icbZA7T8MwFIVvyquERwOMLBYVElPkpCW0WwULY5FoC6JV5bhOa9V5yHaQqhiQ6zMrPAD+Dn8G9IHQ4EzHZ1zr64/+4ngSmP8ZRTW1jc2t4rb5s7u3n7JOjhsqziVlLVoLGJ5xPFBI9YS3Mt2F0iGQl9wTr+GrWdx6ZVDyObvUkYb2QDCMecEp0HvWtUndEdNYNiR75AYqmfauM7XodVx0PYfscY9er5wZX3JrnIcfGc5VhqWbf+uwOYpqGLNJUEKUyIjWngk3NbqpYQuiYDFlGQ1/y4UivpiRUahL6U3Q6u69+d7Pwv+4h1UGtl/EoSTWLaD6Sd0EqkI7RjBENuGRUi0luCJU8fw+iIyIJ1flPmOacQGC/pofxrZrOxXbvamWG5dL2iIcwmcgQMX0IBraEILKTwBu/wYdwbT8az8bIYLRjLnSNYkfH6DV6lhmc=</latexit>continuity equation ˙ (r, t) + · j(r, t) = 0
<latexit sha1_base64="wgHQ/yxv482WmNPQ/lehZPhCKc=">ACDnicbVC7SgNBFL0bXzG+Vi1tBoMQUcJuFLRAoJYRjAPcEOYmcwmY2YfzMwKIeQf/BAbGzuxtbYT/Rhn4zZJPNXhnHO591wSC6043xZuYXFpeWV/GphbX1jc8ve3moKJGU1WkItkiWDHBQ1bXAvWiXDARGsSQZXqd98ZFLxKLzTw5i1A9wLuc8p1kbq2NfI60YaeSxWXEQhKnkB1n3iI3msD9ER8kJMBEYenaQy72EmduF07KJTdiZA8TNSBEy1Dr2i9lLk4CFmgqs1AhLzalg4KXKBZjOsA9NqIBkbzX19MqDpQaBmSMDtIj1KyXiv9594n2z9sjHsaJZiE1EeP5iUA6QulrUJdLRrUYGoKp5OYeRPtYqrNAwsF09GdbTRPGpWye1Ku3J4Wq5dZ2zswT6UwIUzqMIN1KAOFJ7hE7hx3qyXq036/0vmrOymV2YgvXxC7dmV4=</latexit>˙ (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>˙ (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
˙ (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
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>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>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>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>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, 0)
<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</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, 0)
<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit>T(r, t) = 1 (4παt)
3 2
4αt T(r, 0)dr
<latexit sha1_base64="QezugKTriq0vDXB6FuodfXuCJ4E=">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</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, 0)
<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit>T(r, t) = 1 (4παt)
3 2
4αt T(r, 0)dr
<latexit sha1_base64="QezugKTriq0vDXB6FuodfXuCJ4E=">ACknicbVHbhMxEPUuBUq5BcpbX0ZEqAlqo90EqgSqKgvfeChSE1bKZtEXsebWLV3V/YsUuT6L/g5voB/4Akn2d4ZydLxHJ8Zz5m0lMJgFP0Owkdrj58X+28fzFy1evG2/enpqi0oz3WSELfZ5Sw6XIeR8FSn5eak5VKvlZenG4M9+cm1EkZ/gvORDRae5yASj6FPjxq+TVqIoztIM9A624UuSacps7Gyrl5QCEirLGQVsj1bEnrNd5xKRIyx1WlnuRnZ3xV7Wtax2sAtXha3edpejrO963LOwa3G2ztRewI313GjGXWiZcBDENegSeo4Hjf+JOCVYrnyCQ1ZhBHJQ4t1SiY5G4jqQwvKbugUz7wMKeKm6Fdufg89MICu0P36sZfa2wjKVajGd4Z06lipj5ir1+sW/zX1ukfwfN6gw+zy0Ii8r5Dlbtc8qCVjAYkEwEZozlHMPKNPCTwBsRr276NfonYnv+/AQnHY78V6n+6PXPhae7ROtsh70iIx+UQOyBE5Jn3CyN8AgnbwMXwX7ofwsPV0zCoNZvkToTf/wHdLchp</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>∂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, 0)
<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit>T(r, t) = 1 (4παt)
3 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>P conserved (current) densities:
P conserved quantities:
P conserved (current) densities:
P conserved quantities:
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>P conserved (current) densities:
P conserved quantities:
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>P conserved (current) densities:
P conserved quantities:
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
λjAj
δS δai = αi(r) = λi
<latexit sha1_base64="3sMxC2Co5d01PHJMOmG3M6dPTaI=">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</latexit>P conserved (current) densities:
P conserved quantities:
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
λjAj
δS δai = αi(r) = λi
<latexit sha1_base64="3sMxC2Co5d01PHJMOmG3M6dPTaI=">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</latexit>ji =
Λijfj f = α
<latexit sha1_base64="M6y7rALkPqCoKP31PCfXrdEF4Hg=">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</latexit>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>P conserved (current) densities:
P conserved quantities:
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
λjAj
δS δai = αi(r) = λi
<latexit sha1_base64="3sMxC2Co5d01PHJMOmG3M6dPTaI=">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</latexit>ji =
Λijfj f = α
<latexit sha1_base64="M6y7rALkPqCoKP31PCfXrdEF4Hg=">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</latexit>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 =
Λijfj f = α
<latexit sha1_base64="M6y7rALkPqCoKP31PCfXrdEF4Hg=">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</latexit>Λij = Ω kB ∞ Ji(t)Jj(0)dt
<latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit>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>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>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) =
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>T t T t J(t + τ, Γ0)J(τ, Γ0)dτ
<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>J(t)J(0) =
Einstein (1905)
=
v(t)dt
2Dt D = v(t)v(0)dt
<latexit sha1_base64="gxqmbM4YXxOWgvLt4/qMElHRs=">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</latexit>Einstein (1905)
=
v(t)dt
2Dt D = v(t)v(0)dt
<latexit sha1_base64="gxqmbM4YXxOWgvLt4/qMElHRs=">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</latexit>Helfand (1960)
J(t)dt
2Λt Λ = J(t)J(0)dt
<latexit sha1_base64="QaXBtrlPDlz6zV/YrGvBPDfA9c=">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</latexit>J = 1 Ω
j(r)dr
<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>J = 1 Ω
j(r)dr
<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>· j(r) = ˙ (r)
J = 1 Ω
j(r)dr
<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit>·
r
r˙ (r)dr
· j(r) = ˙ (r)
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
r˙ (r)dr
· j(r) = ˙ (r)
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=">ACVXicbVHLSgMxFE3H+n606tJNsAi6KTMq6EYQ3YgbK1grdErJpHfaYJIZkjtCGfpfoe4duFGP0Ewfj2QuBwzrn3JidRKoVF38seFPF6ZnZufmFxaXlVJ5de3aJpnhUOeJTMxNxCxIoaGOAiXcpAaYiQ0otvTod64A2NFoq+wn0JLsa4WseAMHdUuN0LFsBfF9Jwe0TA2jOfBIA8vFHTZIBQa2NM6YfR0LCTYAipFTLRdPuT36GdL0+7XPGr/qjoXxBMQIVMqtYuP7uxPFOgkUtmbTPwU2zlzKDgEgYLYWYhZfyWdaHpoGYKbCsfBTCgW47p0Dgx7mikI/Z7R85VZES3hz/m5ExZ21eR6x/e2/7WhuR/WjPD+LCVC51mCJqP18eZpJjQYca0IwxwlH0HGDfCvYDyHnPZovsJl0zwO4e/4Hq3GuxVdy/3K8cnk4zmyAbZJNskIAfkmJyRGqkTu7JE3khr4WHwptX9GbGVq8w6VknP8orvQPECrU/</latexit>j(r)dr =
r˙ (r)dr + surface terms
<latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit>·
r
r˙ (r)dr
· j(r) = ˙ (r)
J = 1 Ω
r˙ (r)dr
<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit>(r, t) =
J = 1 Ω
r˙ (r)dr
<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit>(r, t) =
eI(R, V) = 1 2MIV2
I + 1
2
v(|RI − RJ|)
<latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit>J =
˙ RIeI + RI ˙ eI
J = 1 Ω
r˙ (r)dr
<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit>(r, t) =
eI(R, V) = 1 2MIV2
I + 1
2
v(|RI − RJ|)
<latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit>J =
˙ RIeI + RI ˙ eI
J = 1 Ω
r˙ (r)dr
<latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit>(r, t) =
eI(R, V) = 1 2MIV2
I + 1
2
v(|RI − RJ|)
<latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit>J =
eIVI + 1 2
(VI · FIJ)(RI − RJ)
<latexit sha1_base64="yZWQm8/jgK1ERHROkp6KXIRx+Q8=">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</latexit>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.
Je =
IVI + 1 2
(VI · FIJ)(RI − RJ) I(R, V) = 1 2MIV2
I + 1
2
v(|RI − RJ|) E =
I(R, V) =
I(R, V) = I(R, V) = 1 2MIV2
I + 1
2
v(|RI − RJ|)(1 + ΓIJ) Je =
IVI + 1 2
(VI · FIJ)(RI − RJ) + 1 2
ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]
Je =
IVI + 1 2
(VI · FIJ)(RI − RJ) + 1 2
ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]
Je =
IVI + 1 2
(VI · FIJ)(RI − RJ) + 1 2
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
Je =
IVI + 1 2
(VI · FIJ)(RI − RJ) + 1 2
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
Je =
IVI + 1 2
(VI · FIJ)(RI − RJ) + 1 2
ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]
˙ P = d dt 1 4
ΓIJ v(|RI − RJ|)(RI − RI)
Je =
IVI + 1 2
(VI · FIJ)(RI − RJ) + 1 2
ΓIJ [VIv(|RI − RJ|) + (VI · FIJ)(RI − RI)]
var
t J(t)dt
<latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit>κ ∼ 1 2tvar
D(t) = D(t)+P(t) − P(0)
<latexit sha1_base64="x2k2eGrhD4eEPVy5YoCeftyBFc=">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</latexit>var
t J(t)dt
<latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">ACnXicbZHfaxQxEMezq61/dGr4pMPBg/74+XYLYK+CMUqiIhU8HqFy3lkc7PXcEk2JLOFI+y/Kfi/+GD2eojXOhD48pkZuab0irpMc9/Jemdu1vb93buZw8ePnq829t7cu7rxgkYilrV7qLkHpQ0MESJCi6sA65LBaNycdrlR1fgvKzNd1xamGg+N7KSgmNE05lC24tp/uUeakpqxwXoWjDMbZMc7x0Olx17JSzhUd0xUrq/ChPcSjDjo6oYxlG5zuv2PS4DT/gX8bPsfEwdEMD6a9fj7IV0Fvi2It+mQdZ9O95Bmb1aLRYFAo7v24yC1OAncohYI2Y40Hy8WCz2EcpeEa/CSsrGnpq0hmtKpdfAbpiv7bEbj2fqnLWNlt6m/mOvi/3LjB6u0kSGMbBCOuB1WNoljTzmc6kw4EqmUXDgZd6Xikd3Mf7GxpRuMW9BxEs8oObSdCSciVLJzfuC2AaLRF0m2XRyOKmbfF+fGgyAfFt9f9k/drS3fIc/KSHJKCvCEn5BM5I0MiyE/yO9lKtMX6cf0S/r1ujRN1j1PyUakoz/OE8zW</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">ACnXicbZHfaxQxEMezq61/dGr4pMPBg/74+XYLYK+CMUqiIhU8HqFy3lkc7PXcEk2JLOFI+y/Kfi/+GD2eojXOhD48pkZuab0irpMc9/Jemdu1vb93buZw8ePnq829t7cu7rxgkYilrV7qLkHpQ0MESJCi6sA65LBaNycdrlR1fgvKzNd1xamGg+N7KSgmNE05lC24tp/uUeakpqxwXoWjDMbZMc7x0Olx17JSzhUd0xUrq/ChPcSjDjo6oYxlG5zuv2PS4DT/gX8bPsfEwdEMD6a9fj7IV0Fvi2It+mQdZ9O95Bmb1aLRYFAo7v24yC1OAncohYI2Y40Hy8WCz2EcpeEa/CSsrGnpq0hmtKpdfAbpiv7bEbj2fqnLWNlt6m/mOvi/3LjB6u0kSGMbBCOuB1WNoljTzmc6kw4EqmUXDgZd6Xikd3Mf7GxpRuMW9BxEs8oObSdCSciVLJzfuC2AaLRF0m2XRyOKmbfF+fGgyAfFt9f9k/drS3fIc/KSHJKCvCEn5BM5I0MiyE/yO9lKtMX6cf0S/r1ujRN1j1PyUakoz/OE8zW</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit>κ ∼ 1 2tvar
D(t) = D(t)+P(t) − P(0)
<latexit sha1_base64="x2k2eGrhD4eEPVy5YoCeftyBFc=">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</latexit>var
var
t J(t)dt
<latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit>κ ∼ 1 2tvar
D(t) = D(t)+P(t) − P(0)
<latexit sha1_base64="x2k2eGrhD4eEPVy5YoCeftyBFc=">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</latexit>var
var
t J(t)dt
<latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit><latexit sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit>κ ∼ 1 2tvar
var
var
· P
E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
E[Ω] =
e(r)dr
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
E[Ω] =
e(r)dr
E′[Ω] = E[Ω] + O[∂Ω]
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
E[Ω] =
e(r)dr
E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r)
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
˙ e′(r, t) = −∇ ·
p(r, t)
e(r)dr
E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r)
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
˙ e′(r, t) = −∇ ·
p(r, t)
e(r)dr
E′[Ω] = E[Ω] + O[∂Ω] e′(r) = e(r) − ∇ · p(r) j′(r, t) = j(r, t) + ˙ p(r, t)
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
˙ e′(r, t) = −∇ ·
p(r, 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)
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
˙ e′(r, t) = −∇ ·
p(r, 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)
∂𝝯
E ∪ E E
?
= E[Ω1] + E[Ω2] E[Ω1 ∪ Ω2] = E[Ω1] + E[Ω2] + [∂Ω]
˙ e′(r, t) = −∇ ·
p(r, 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)
∂𝝯
EDF T = 1 2
MIV2
I + e2
2
ZIZJ RIJ +
v−1 2EH + XC(r) − µXC(r)
EDF T = 1 2
MIV2
I + e2
2
ZIZJ RIJ +
v−1 2EH + XC(r) − µXC(r)
eDF T (r) = e0(r) + eKS(r) + eH(r) + eXC(r) r
R
Re r r r r r r r r r
EDF T = 1 2
MIV2
I + e2
2
ZIZJ RIJ +
v−1 2EH + XC(r) − µXC(r)
r r r r r e0(r) =
(r − RI) 1 2MIV 2
I + wI
∗
v(r)
ˆ HKSv(r)
2(r)vH(r) eXC(r) = (XC(r) − vXC(r)) (r)
eDF T (r) = e0(r) + eKS(r) + eH(r) + eXC(r) r
R
Re r r r r r r r r r
JDF T =
eDF T (r, t)dr = JKS + JH + J
0 + J0 + JXC
JKS =
HKS| ˙ v + v ˙ v|r|v
4
vH(r)vH(r)dr J
0 =
v |(r RI) (VI · Iˆ v0)| v J0 =
I +
(RI RL) (VL · LwI)
(r)GGA(r)dr (GGA) JDF T =
eDF T (r, t)dr = JKS + JH + J
0 + J0 + JXC
JKS =
HKS| ˙ v + v ˙ v|r|v
r r J r R V J V R R V J (LDA) r r r r (GGA) J r r
4
vH(r)vH(r)dr J
0 =
v |(r RI) (VI · Iˆ v0)| v J0 =
I +
(RI RL) (VL · LwI)
(r)GGA(r)dr (GGA)
ϕv and ˆ HKS| ˙ ϕv orthogonal to the
Pcr|ϕv computed from standard DFPT
JDF T =
eDF T (r, t)dr = JKS + JH + J
0 + J0 + JXC
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✏
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✏
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✏
1 3V kBT 2 t J(t) · J(0)dt
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]
64 molecules, T=385 K expt density @ac ω S(ω) = ∞
−∞
J(t) · J(0) eiωtdt
1 3V kBT 2 t J(t) · J(0)dt
1 2 t[ps]
64 molecules, T=385 K expt density @ac t 3V kBT 2 t J(t) · J(0)dt 1 6V kBT 2
J(t)dt
≈ Einstein’s relaFon
1 6V kBT 2
J(t)dt
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]
1 6V kBT 2
J(t)dt
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)
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 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
κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt
κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt
J(t)J(0)dt
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<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=">ACMXicbVBNSyNBEO1xV1fjx+qKJy+NQfAUZtRN4i2Qi0cXTBSIDWdirbp7m6axbCkP/gdfdv7K/xtnj1T2xPEsHs7oOCx3tVNVLrZKe4vg5WvnwcXt0/pGZXNre+fz7t6Xrs9yJ7AjMpW52xQ8KmwQ5IU3lqHoFOFN+m4Xfo39F5mZlrmlgcaLg3ciQFUJC6/TFYC3e71biW1OvNrwmPa43Gaf28EUh80TyLE57U4hmqbIGru73oD/MRK7RkFDgfS+JLQ0KcCSFwmln3u0IMZwj71ADWj0g2J27pQfB2XIR5kLZYjP1PcTBWjvJzoNnRrowf/tleL/vF5Oo+agkMbmhEbMF41yxSnj5e98KB0KUpNAQDgZbuXiARwICgktbSkP8xZF+MQjaZCmVIo2KJk6ufRfgSbXklBPK5VZkPO0+L/kLcjuaS0JeX87r7ai0jX2SE7YicsYQ3WYpfsinWYI/sif1gP6Nf0XP0O3qZt65Ei5l9toTo9Q/l26p2</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
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=">ACMXicbVBNSyNBEO1xV1fjx+qKJy+NQfAUZtRN4i2Qi0cXTBSIDWdirbp7m6axbCkP/gdfdv7K/xtnj1T2xPEsHs7oOCx3tVNVLrZKe4vg5WvnwcXt0/pGZXNre+fz7t6Xrs9yJ7AjMpW52xQ8KmwQ5IU3lqHoFOFN+m4Xfo39F5mZlrmlgcaLg3ciQFUJC6/TFYC3e71biW1OvNrwmPa43Gaf28EUh80TyLE57U4hmqbIGru73oD/MRK7RkFDgfS+JLQ0KcCSFwmln3u0IMZwj71ADWj0g2J27pQfB2XIR5kLZYjP1PcTBWjvJzoNnRrowf/tleL/vF5Oo+agkMbmhEbMF41yxSnj5e98KB0KUpNAQDgZbuXiARwICgktbSkP8xZF+MQjaZCmVIo2KJk6ufRfgSbXklBPK5VZkPO0+L/kLcjuaS0JeX87r7ai0jX2SE7YicsYQ3WYpfsinWYI/sif1gP6Nf0XP0O3qZt65Ei5l9toTo9Q/l26p2</latexit><latexit sha1_base64="pSv+DUAk6YfLYmEj4xBlOjmR/nA=">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</latexit>κ = 1 3V kBT 2 ∞ Jq(t) · Jq(0)dt
J(t)dt
2t J(t)J(0)dt
<latexit sha1_base64="wLZhjaWaiCPS2hmyYBewowj2U=">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</latexit>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=">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>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
1 2
J(t)dt
κGK(T) = T v(t)v(0)dt ∞ v t v 0
T t dt
Cv
T
d
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
κGK(T) = T v(t)v(0)dt = ∞ v(t)v(0)ΘT(t)dt ∞ Cv
T
d κE(T) = 1 2T
v(t)dt
∞ v t v 0
T t dt
Cv
T
d
ΘT(t)
T 1
ΛT(t) = 1 − t T
T 1
ΘT(t)
κGK(T) = T v(t)v(0)dt = ∞ v(t)v(0)ΘT(t)dt ∞ Cv
T
d κE(T) = 1 2T
v(t)dt
∞ v t v 0
T t dt
Cv
T
d
E T
1 2T
v t dt
= ∞ v(t)v(0)ΛT(t)dt ∞ Cv
T
d
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
v(t)dt
= ∞ 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)
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>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 →∞
T
2
− T
2
J(t)eiωt
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 →∞
T
2
− T
2
J(t)eiωt
in pracFce: S (k) = N
Jne−i 2πnk
N
S(k) = N
J(k)
2S(k) × 2
2
S(k) = N
J(k)
2S(k) × 2
2
S(k) = N
J(k)
2S(k) × 2
2
S(ω) log
S(k) = S(ωk)ˆ ξk
ˆ S(k)
ˆ ξk
S(ω) log
S(k) = S(ωk)ˆ ξk
ˆ S(k)
ˆ ξk
log ˆ S(k)
ˆ ξk
Cn 1 N
N−1
log ˆ S(k)
N
= Cn + white noise
log ˆ S(k)
ˆ ξk
Cn 1 N
N−1
log ˆ S(k)
N
= Cn + white noise
log
P ∗−1
Cnei 2πkn
N
+ less noise
n
Cn log
N
N−1
log ˆ S(k)
N
= Cn + white noise log ˆ S(k)
ˆ ξk
log
P ∗−1
Cnei 2πkn
N
+ less noise
n
Cn log
N
N−1
log ˆ S(k)
N
= Cn + white noise log ˆ S(k)
ˆ ξk
log
P ∗−1
Cnei 2πkn
N
+ less noise
n
Cn log
N
N−1
log ˆ S(k)
N
= Cn + white noise log ˆ S(k)
ˆ ξk
log
P ∗−1
Cnei 2πkn
N
+ less noise
n
Cn log
N
N−1
log ˆ S(k)
N
= Cn + white noise log ˆ S(k)
ˆ ξk
log
P ∗−1
Cnei 2πkn
N
+ less noise
log(κ) = λ + C0 + 2
P ∗−1
Cn ± σ
N∗
constants independent of the Fme series being sampled
log
P ∗−1
Cnei 2πkn
N
+ less noise
log(κ) = λ + C0 + 2
P ∗−1
Cn ± σ
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
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
molecular dynamics is less and less ergodic as temperature decreases
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
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
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
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
M¨ x = −∂V ∂x
<latexit sha1_base64="FgtUFvHAdG+6g57nYZM4bl6HdU=">AB8HicbZBLS8NAFIVv6qvWV9Slm8EidGNJqAbpeDGjVDBPsCUMplM2qGTBzMTaQn5C67diVt3glv9H/4bJxqQtp7V4Zx7L/ONG3MmlWV9GaWl5ZXVtfJ6ZWNza3vH3N3ryCgRhLZJxCPRc7GknIW0rZjitBcLigOX0647vsr7gMVkXhnZrGtB/gYch8RrDS0cCs3TieFyk0QRfHji8wSZ0YC8UwR53sz0+ygVm16taP0KxC1OFQq2B+eZ4EUkCGirCsZRpfopwmlWcRNIYkzEe0pQErmDkZpNcSDlNHAzdBRgNZLzXR7+190nyj/vpyME0VDokd05ycqQjl8MhjghLFp9pgIph+DyIjrKmV/qJKRTPa80SLptOo2yf1xu1ptXlZ0JbhA6hBjacQROuoQVtIPAI7/ABn4Ywnoxn4+V3tGQUO/swI+P1Gxz1kCU=</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>1 2Φ
2
<latexit sha1_base64="JfNCI5Rqnx/jaACAwzXJ+5TExs=">AB7XicbZA7T8MwFIWd8irhFWBksagqlYHICVUfC6rEwlgk+pBIqRzXba06D9kOahXlFzCzIVbmrvBL+DckbRkKnOnc+6V/NkNOZMKoS8t7G5tb2T39X39g8Oj4zjk7YMIkFoiwQ8EF0XS8qZT1uKU67oaDYczntuJObrO8USFZ4N+rWUh7Hh75bMgIVmnUN4rOUGASW0lsJ05zByXjTgsTS+nfQSzQcCLR7tvFJCJalalbkFkVmyrWrVTg6xyvV6DlokWKoCVmn1j7gwCEnUV4RjKWMsFCOcJroTSRpiMsEjGhPFWw0Vusp9qSceW4Cix5WY/m7y8L/uodIDWu9mPlhpKhP0pW0G0YcqgBm5HDABCWKz1KDiWDpeyAZ45Repf+j60vGheBf8PYtk3ryrTvyoXG9Yo2D87AOSgBC1RBA9yCJmgBAp7BHyATy3QXrRX7W25mtNWN6dgTdr7N70XjgY=</latexit>M¨ x = −∂V ∂x
<latexit sha1_base64="FgtUFvHAdG+6g57nYZM4bl6HdU=">AB8HicbZBLS8NAFIVv6qvWV9Slm8EidGNJqAbpeDGjVDBPsCUMplM2qGTBzMTaQn5C67diVt3glv9H/4bJxqQtp7V4Zx7L/ONG3MmlWV9GaWl5ZXVtfJ6ZWNza3vH3N3ryCgRhLZJxCPRc7GknIW0rZjitBcLigOX0647vsr7gMVkXhnZrGtB/gYch8RrDS0cCs3TieFyk0QRfHji8wSZ0YC8UwR53sz0+ygVm16taP0KxC1OFQq2B+eZ4EUkCGirCsZRpfopwmlWcRNIYkzEe0pQErmDkZpNcSDlNHAzdBRgNZLzXR7+190nyj/vpyME0VDokd05ycqQjl8MhjghLFp9pgIph+DyIjrKmV/qJKRTPa80SLptOo2yf1xu1ptXlZ0JbhA6hBjacQROuoQVtIPAI7/ABn4Ywnoxn4+V3tGQUO/swI+P1Gxz1kCU=</latexit>V (x0 + u) ≈ V (x0) + 1 2Φu2 + O
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>1 2Φ
2
<latexit sha1_base64="JfNCI5Rqnx/jaACAwzXJ+5TExs=">AB7XicbZA7T8MwFIWd8irhFWBksagqlYHICVUfC6rEwlgk+pBIqRzXba06D9kOahXlFzCzIVbmrvBL+DckbRkKnOnc+6V/NkNOZMKoS8t7G5tb2T39X39g8Oj4zjk7YMIkFoiwQ8EF0XS8qZT1uKU67oaDYczntuJObrO8USFZ4N+rWUh7Hh75bMgIVmnUN4rOUGASW0lsJ05zByXjTgsTS+nfQSzQcCLR7tvFJCJalalbkFkVmyrWrVTg6xyvV6DlokWKoCVmn1j7gwCEnUV4RjKWMsFCOcJroTSRpiMsEjGhPFWw0Vusp9qSceW4Cix5WY/m7y8L/uodIDWu9mPlhpKhP0pW0G0YcqgBm5HDABCWKz1KDiWDpeyAZ45Repf+j60vGheBf8PYtk3ryrTvyoXG9Yo2D87AOSgBC1RBA9yCJmgBAp7BHyATy3QXrRX7W25mtNWN6dgTdr7N70XjgY=</latexit>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
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>1 2Φ
2
<latexit sha1_base64="JfNCI5Rqnx/jaACAwzXJ+5TExs=">AB7XicbZA7T8MwFIWd8irhFWBksagqlYHICVUfC6rEwlgk+pBIqRzXba06D9kOahXlFzCzIVbmrvBL+DckbRkKnOnc+6V/NkNOZMKoS8t7G5tb2T39X39g8Oj4zjk7YMIkFoiwQ8EF0XS8qZT1uKU67oaDYczntuJObrO8USFZ4N+rWUh7Hh75bMgIVmnUN4rOUGASW0lsJ05zByXjTgsTS+nfQSzQcCLR7tvFJCJalalbkFkVmyrWrVTg6xyvV6DlokWKoCVmn1j7gwCEnUV4RjKWMsFCOcJroTSRpiMsEjGhPFWw0Vusp9qSceW4Cix5WY/m7y8L/uodIDWu9mPlhpKhP0pW0G0YcqgBm5HDABCWKz1KDiWDpeyAZ45Repf+j60vGheBf8PYtk3ryrTvyoXG9Yo2D87AOSgBC1RBA9yCJmgBAp7BHyATy3QXrRX7W25mtNWN6dgTdr7N70XjgY=</latexit>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
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 =
M
<latexit sha1_base64="xT6x9L7ekFgTGnHrHb8wgV2oHm8=">AB5HicbZBLS8NAFIVv6qvGV9SlLoJFcFWSVrAboeDGjVDBPsCUMplOkqEzSZyZCVk49qduHXtv4b/41TzatZ3U451643/VTRqVynG+jsra+sblV3TZ3dvf2D6zDo5MoFJFycsEQMfScJoTLqKkYGqSCI+4z0/cnNvO8/EyFpEj+oaUqGHIUxDShGSkcj69RLOAnRtSefhMq9QCce52IFvldUYysmlN3fmWvGrc0NSjVGVkzb5zgjJNYakzJFQFDNSmF4mSYrwBIUkx9wXNIzUYoq4lFPuF/Y5RyqSy908/K97zFTQGuY0TjNFYqxHdBdkzFaJPQe2x1QrNhUG4QF1fYOEKaU+m3mKZmdJeJVk2vUXeb9cb9Za3dKmrcAJncAEuXEbqEDXcDwAp8wgy8jMF6N+P9b7RilDvHsCDj4wfvb4wb</latexit>1 2Φ
2
<latexit sha1_base64="JfNCI5Rqnx/jaACAwzXJ+5TExs=">AB7XicbZA7T8MwFIWd8irhFWBksagqlYHICVUfC6rEwlgk+pBIqRzXba06D9kOahXlFzCzIVbmrvBL+DckbRkKnOnc+6V/NkNOZMKoS8t7G5tb2T39X39g8Oj4zjk7YMIkFoiwQ8EF0XS8qZT1uKU67oaDYczntuJObrO8USFZ4N+rWUh7Hh75bMgIVmnUN4rOUGASW0lsJ05zByXjTgsTS+nfQSzQcCLR7tvFJCJalalbkFkVmyrWrVTg6xyvV6DlokWKoCVmn1j7gwCEnUV4RjKWMsFCOcJroTSRpiMsEjGhPFWw0Vusp9qSceW4Cix5WY/m7y8L/uodIDWu9mPlhpKhP0pW0G0YcqgBm5HDABCWKz1KDiWDpeyAZ45Repf+j60vGheBf8PYtk3ryrTvyoXG9Yo2D87AOSgBC1RBA9yCJmgBAp7BHyATy3QXrRX7W25mtNWN6dgTdr7N70XjgY=</latexit>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
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 =
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
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>1 2Φ
2
<latexit sha1_base64="JfNCI5Rqnx/jaACAwzXJ+5TExs=">AB7XicbZA7T8MwFIWd8irhFWBksagqlYHICVUfC6rEwlgk+pBIqRzXba06D9kOahXlFzCzIVbmrvBL+DckbRkKnOnc+6V/NkNOZMKoS8t7G5tb2T39X39g8Oj4zjk7YMIkFoiwQ8EF0XS8qZT1uKU67oaDYczntuJObrO8USFZ4N+rWUh7Hh75bMgIVmnUN4rOUGASW0lsJ05zByXjTgsTS+nfQSzQcCLR7tvFJCJalalbkFkVmyrWrVTg6xyvV6DlokWKoCVmn1j7gwCEnUV4RjKWMsFCOcJroTSRpiMsEjGhPFWw0Vusp9qSceW4Cix5WY/m7y8L/uodIDWu9mPlhpKhP0pW0G0YcqgBm5HDABCWKz1KDiWDpeyAZ45Repf+j60vGheBf8PYtk3ryrTvyoXG9Yo2D87AOSgBC1RBA9yCJmgBAp7BHyATy3QXrRX7W25mtNWN6dgTdr7N70XjgY=</latexit>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
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 =
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
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
iα − ω2Miδijδαβ
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>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>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>uνuνuν ∼
aν + ˆ a†
ν
aν + ˆ a†
ν
aν + ˆ a†
ν
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 ∼
aν + ˆ a†
ν
aν + ˆ a†
ν
aν + ˆ a†
ν
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 ∼
aν + ˆ a†
ν
aν + ˆ a†
ν
aν + ˆ a†
ν
τ−1
i
= 2π
δ(Ei − Ef )
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 ∼
aν + ˆ a†
ν
aν + ˆ a†
ν
aν + ˆ a†
ν
1
ν
= 2
(ν + ν − ν)nνnν(nν + 1)
τ−1
i
= 2π
δ(Ei − Ef )
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 ∼
aν + ˆ a†
ν
aν + ˆ a†
ν
aν + ˆ a†
ν
1
ν
= 2
(ν + ν − ν)nνnν(nν + 1)
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)
J =
˙ Rnen + Rn ˙ en
R
n ˙
en + d dt
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 =
˙ Rnen + Rn ˙ en
R
n ˙
en + d dt
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 =
˙ Rnen + Rn ˙ en
R
n ˙
en + d dt
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
Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">ACZnicbZHfSuQwFMbTrn9HV8cV8cKbsIPgig7TUVhvFMEb8WqUHRWMltOYaTMmbUnShSH0VXwH8c38DFMtSP+O1cfv+87h5yTKBdcm07n0fN/TExOTc/MNubmfy4sNpd+neusUJT1aSYydRmBZoKnrG+4EewyVwxkJNhFdHdU+Rf/mdI8S/+ZUc6uJcQpH3AKxqGweX8SWgIiT6DE+xiTgQJqg9J2S0x0IUPLhyRiBkgMUroMiXgs8MbZjSWUK1q6wLh9G7+jwzGtGhT+Q3oJr7LVrPLG2fU8W5RvGJPbzLyAsb0VNluduel8FcR1KF6uqFzQc3hBaSpYK0NqCMpwKVjZIoVkO9A5iZqmMFI8T85GC1HokoxKvSzCJ/uxV8DvqjCDvWvL07wLKUu4rxBIbDJcHVyfMsVo0aMnACquHsPpgm4Qxv3MY2G2zH4vNFXcd5tBzvt7ulu6/Cg3nYGraHfaAMF6C86RMeoh/qIehPeprfj7XpP/oK/4q+Rn2v7lGH8rHz+VueQ=</latexit>J =
˙ Rnen + Rn ˙ en
Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>κ = 1 V
cnm
2τ
nm
<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>v α
nm =
1 2√ωnωm
R
iα − R jα
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
cnm
2τ
nm
<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>κ = 1 V
cnm
2τ
nm
<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>a-Si
Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>κ = 1 V
cnm
2τ
nm
<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>a-Si
Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>κ = 1 V
cnm
2τ
nm
<latexit sha1_base64="Gri+Kn9ozGy65ibXem9Bu1Yos04=">ACF3icbVC7TsMwFHXKq4RXgZHFokIqS9UJFhAlVjYKBJ9SKSNnFu3tWonke1UqJ8CJ/BwMyGWBkRK/wHLs3SlrP43HPulXyOH3GmdKXyaeVWVtfWN/Kb9tb2zu5eYf+gqcJYAm1AyEPZ9ominAW0oZnmtB1JSoTPacsf3Uz91phKxcLgQU8i2hFkELA+A6KN5BXu3BGJInKF3b4kDhp0kyxq2LhJYFIMcwe12cDjkt4bEaczRKfdqvY1STusAkzCyvUKyUK3/Ay8TJSBFlqHuFZ7cXQixoIETpRIiNQNOU9uNFY0IjMiAJiB8yQZDPa8SodRE+Ck+EUQP1aI3Ff/zHmPdv+wkLIhiTQMwK8brxzrE87wj0mKWg+MYSAZOY/GIbE9KNk7ZtMjqLiZJs1p2zsrV+/Ni7TpLm0dH6BiVkIMuUA3dojpqIEAv6At9ox/ryXq13qz32WrOym4O0Rysj18Kop9Q</latexit>a-Si a-Si
Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>κ = 1 V
cnm
2τ
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>Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>κ = 1 V
cnm
2τ
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)
<latexit sha1_base64="CIrRL/c3f1YcWZ3VBtIqAyPw7D8=">ACI3icbVC7TsMwFHXKq4RXgJElokIqS9UJFhAlVgYi0QfUl0ix3Vaq84DPypVUT6HD+ATmNkQCwMzK/wBDkRITmDdXzOvbLP8WJGhazX34zS0vLK6lp53dzY3NresXb3OiJSHJM2jljEex4ShNGQtCWVjPRiTlDgMdL1JleZ350SLmgU3spZTAYBGoXUpxhJLblWH05QHKML6HOEydNOikUKnATGCA59nz7HoYqxa4+q3/S8bRwv2vYUCJVUF2rUq/Vf2AvEicnFZCj5VqPcBhFZBQYoaESBCXFDOSmlAJEiM8QSOS4MDjdDSW8yoKhJgFXmofZe+LopeJ/3l9Jf3zQULDWEkSYj2iPV8xW0Z21pc9pJxgyWaIMyp/o+Nx0iXJXWrpqkzOsVEi6TqDkntcbNaV5mactgwNwCKrAWegCa5BC7QBk/gA3yCL+PBeDZejNf0ZKR7+yDORjv37shpPk=</latexit>BTE
Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>κ = 1 V
cnm
2τ
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)
<latexit sha1_base64="CIrRL/c3f1YcWZ3VBtIqAyPw7D8=">ACI3icbVC7TsMwFHXKq4RXgJElokIqS9UJFhAlVgYi0QfUl0ix3Vaq84DPypVUT6HD+ATmNkQCwMzK/wBDkRITmDdXzOvbLP8WJGhazX34zS0vLK6lp53dzY3NresXb3OiJSHJM2jljEex4ShNGQtCWVjPRiTlDgMdL1JleZ350SLmgU3spZTAYBGoXUpxhJLblWH05QHKML6HOEydNOikUKnATGCA59nz7HoYqxa4+q3/S8bRwv2vYUCJVUF2rUq/Vf2AvEicnFZCj5VqPcBhFZBQYoaESBCXFDOSmlAJEiM8QSOS4MDjdDSW8yoKhJgFXmofZe+LopeJ/3l9Jf3zQULDWEkSYj2iPV8xW0Z21pc9pJxgyWaIMyp/o+Nx0iXJXWrpqkzOsVEi6TqDkntcbNaV5mactgwNwCKrAWegCa5BC7QBk/gA3yCL+PBeDZejNf0ZKR7+yDORjv37shpPk=</latexit>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>Jα = 1 2
iα − R jα
iβuiβ ˙
ujγ,
<latexit sha1_base64="rJMBjAnMuCtRmtsVZSvR6oYxEi4=">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</latexit>In a periodic system
R’ R u(R) u(R’)
V (r) = V0(r) + X
R
u(R) · ∂v(r − R) ∂R + · · ·
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)
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)
R’ R u(R) u(R’)
det ∂2E ∂u(R)∂u(R0) − ω2M(R)δR,R0
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)
Vλ(r) = V0(r) + X
i
λivi(r)
Vλ(r) = V0(r) + X
i
λivi(r)
E(λ) = min
n
✓ F[n] + Z Vλ(r)n(r) ◆ Z n(r)dr = N DFT
Vλ(r) = V0(r) + X
i
λivi(r)
∂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
Vλ(r) = V0(r) + X
i
λivi(r)
∂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
n(r) = X
v
|φv(r)|2 n0(r) = 2Re X
v
φ⇤
v (r)φ0 v(r)
⇥0
v =
X
c
⇥
c
⇥⇥
c|V 0|⇥ v⇤
c
n0(r) = 2Re X
v
φ⇤
v (r)φ0 v(r)
= 2Re X
cv
ρ0
vcφ⇤ v (r)φ c(r)
n(r) = X
v
|φv(r)|2 n0(r) = 2Re X
v
φ⇤
v (r)φ0 v(r)
(H −
v)⇥0 v = −PcV 0⇥ v
⇥0
v =
X
c
⇥
c
⇥⇥
c|V 0|⇥ v⇤
c
n0(r) = 2Re X
v
φ⇤
v (r)φ0 v(r)
= 2Re X
cv
ρ0
vcφ⇤ v (r)φ c(r)
n(r) = X
v
|φv(r)|2 n0(r) = 2Re X
v
φ⇤
v (r)φ0 v(r)
n0(r) = 2Re X
v
φ⇤
v (r)φ0 v(r)
(H −
v)⇥0 v = −PcV 0⇥ v
V0(r) n(r)
⇥ ⇥v(r) = v⇥v(r) n(r) =
|φv(r)|2 VSCF (r) = V0(r) +
|r − r|dr + µxc(r)
DFT
V0(r) n(r)
V (r) n(r) DFPT
V
SCF (r) = V (r) +
|r − r|dr + µ
xc(r)
n⇥(r) = 2 Re
φ
v(r)φ⇥ v(r)
⇥ ⇥v(r) = v⇥v(r) n(r) =
|φv(r)|2 VSCF (r) = V0(r) +
|r − r|dr + µxc(r)
DFT
⇥ ⇥
v(r) = PcV SCF (r)⇥v(r)
SB, P. Giannozzi, and A. Testa, Phys. Rev. Leu. 58, 1861 (1987)
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)
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)
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)
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)
E = hΦ0 + Φ0|(H0 + V 0)|Φ0 + Φ0i hΦ0 + Φ0|Φ0 + Φ0i + O(V 04)
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)
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
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 ∼
aν + ˆ a†
ν
aν + ˆ a†
ν
aν + ˆ a†
ν
1
ν
= 2
(ν + ν − ν)nνnν(nν + 1)
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)
transport, Nat. Phys. 12, 80-U111 (2016), hup://dx.doi.org/10.1038/NPHYS3509
Low Temp. Phys. 185, 79 (2016), hup://dx.doi.org/10.1007/s10909-016-1617-6
dynamics simula?ons, Sci. Rep. 7, 15835 (2017), hup://dx.doi.org/10.1038/s41598-017-15843-2
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
transport in mulF-component systems, Phys. Rev. Leu. in press (2019), hup://arxiv.org/abs/1808.03341
from a unified laYce-dynamical approach, Nat. Commun. in press (2019), hups://arxiv.org/abs/1904.02255
Calcula?ng laHce thermal conduc?vity: a synopsis, Phys. Scr. 93, 043002, (2018), hup://doi.org/10.1088/1402-4896/ aaa6f3
charge transport in condensed maJer, hup://stefano.baroni.me/hydrodynamic-fluctuaFons
Aris Marcolongo, SISSA now @IBM Zürich
Federico Grasselli, SISSA Riccardo Bertossa, SISSA
Loris Ercole, SISSA now @EPFL Lausanne Davide Donadio, UC Davis
Leyla Isaeva, SISSA