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

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<latexit sha1_base64="pHzR0Evo+6LDEThSgi+oOGqwbI=">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</latexit> <latexit sha1_base64="wgHQ/yxv482WmNPQ/lehZPhCKc=">ACDnicbVC7SgNBFL0bXzG+Vi1tBoMQUcJuFLRAoJYRjAPcEOYmcwmY2YfzMwKIeQf/BAbGzuxtbYT/Rhn4zZJPNXhnHO591wSC6043xZuYXFpeWV/GphbX1jc8ve3moKJGU1WkItkiWDHBQ1bXAvWiXDARGsSQZXqd98ZFLxKLzTw5i1A9wLuc8p1kbq2NfI60YaeSxWXEQhKnkB1n3iI3msD9ER8kJMBEYenaQy72EmduF07KJTdiZA8TNSBEy1Dr2i9lLk4CFmgqs1AhLzalg4KXKBZjOsA9NqIBkbzX19MqDpQaBmSMDtIj1KyXiv9594n2z9sjHsaJZiE1EeP5iUA6QulrUJdLRrUYGoKp5OYeRPtYqrNAwsF09GdbTRPGpWye1Ku3J4Wq5dZ2zswT6UwIUzqMIN1KAOFJ7hE7hx3qyXq036/0vmrOymV2YgvXxC7dmV4=</latexit> adiabaFc decoupling of conserved densiFes � ( r , t ) + � · j ( r , t ) = 0 ˙ ˙ ˜ � ( k , t ) = k · ˜ � ( k , t )

<latexit sha1_base64="pHzR0Evo+6LDEThSgi+oOGqwbI=">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</latexit> <latexit sha1_base64="wgHQ/yxv482WmNPQ/lehZPhCKc=">ACDnicbVC7SgNBFL0bXzG+Vi1tBoMQUcJuFLRAoJYRjAPcEOYmcwmY2YfzMwKIeQf/BAbGzuxtbYT/Rhn4zZJPNXhnHO591wSC6043xZuYXFpeWV/GphbX1jc8ve3moKJGU1WkItkiWDHBQ1bXAvWiXDARGsSQZXqd98ZFLxKLzTw5i1A9wLuc8p1kbq2NfI60YaeSxWXEQhKnkB1n3iI3msD9ER8kJMBEYenaQy72EmduF07KJTdiZA8TNSBEy1Dr2i9lLk4CFmgqs1AhLzalg4KXKBZjOsA9NqIBkbzX19MqDpQaBmSMDtIj1KyXiv9594n2z9sjHsaJZiE1EeP5iUA6QulrUJdLRrUYGoKp5OYeRPtYqrNAwsF09GdbTRPGpWye1Ku3J4Wq5dZ2zswT6UwIUzqMIN1KAOFJ7hE7hx3qyXq036/0vmrOymV2YgvXxC7dmV4=</latexit> adiabaFc decoupling of conserved densiFes � ( r , t ) + � · j ( r , t ) = 0 ˙ ˙ ˜ � ( k , t ) = k · ˜ � ( k , t ) the smaller the wavevector, the slower the dynamics

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<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit> consFtuFve relaFons: the heat equaFon �� t = �� · j

<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit> <latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit> consFtuFve relaFons: the heat equaFon �� t = �� · j j = � κ � T

<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit> <latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit> <latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit> consFtuFve relaFons: the heat equaFon �� t = �� · j �� = c p �� T j = � κ � T

<latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit> <latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit> <latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit> <latexit sha1_base64="EPm/9eh/7c4A4uXHrDIw5pOo3CY=">ACHicbZDLSgMxGIUz9VbrepSkGARXJWZKuhGKbhxWcFeoDM/6SZTmhmJiQZoQzd+Rw+gFt9BHfiVvAJfA3Ty8K2/hA4nJOT8H+B4Exp2/62Ciura+sbxc3S1vbO7l5/6Cl0kwS2iQpT2UnAEU5S2hTM81pR0gKcBpOxjcjvP2I5WKpcmDHgrqxdBPWMgIaGP5WMXuIgAX7uhBJK7AxACRjnxhSujdOSXK3bVngxeFs5MVNBsGn75x+2lJItpogkHpbqOLbSXg9SMcDoquZmiAsgA+rRrZAIxV4+2WOET43Tw2EqzUk0nrh/GzmJA8n6kZ57J4dYqWEcmH4MOlKL2dj8L+tmOrzycpaITNOETL8PM451iseocI9JSjQfGgFEMrMBJhEYTNoANWScRQ7LolWrOufV2v1FpX4zY1RER+gEnSEHXaI6ukMN1EQEPaEX9IrerGfr3fqwPqdXC9asc4jmxvr6Ba0Rovk=</latexit> <latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit> consFtuFve relaFons: the heat equaFon �� t = �� · j �� = c p �� T j = � κ � T κ ∂ T α = ∂ t = α ∆ T c p ρ

<latexit sha1_base64="d+UF3wdRyVt3QEV1BGYB59NRIgs=">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</latexit> <latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit> <latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit> <latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit> <latexit sha1_base64="EPm/9eh/7c4A4uXHrDIw5pOo3CY=">ACHicbZDLSgMxGIUz9VbrepSkGARXJWZKuhGKbhxWcFeoDM/6SZTmhmJiQZoQzd+Rw+gFt9BHfiVvAJfA3Ty8K2/hA4nJOT8H+B4Exp2/62Ciura+sbxc3S1vbO7l5/6Cl0kwS2iQpT2UnAEU5S2hTM81pR0gKcBpOxjcjvP2I5WKpcmDHgrqxdBPWMgIaGP5WMXuIgAX7uhBJK7AxACRjnxhSujdOSXK3bVngxeFs5MVNBsGn75x+2lJItpogkHpbqOLbSXg9SMcDoquZmiAsgA+rRrZAIxV4+2WOET43Tw2EqzUk0nrh/GzmJA8n6kZ57J4dYqWEcmH4MOlKL2dj8L+tmOrzycpaITNOETL8PM451iseocI9JSjQfGgFEMrMBJhEYTNoANWScRQ7LolWrOufV2v1FpX4zY1RER+gEnSEHXaI6ukMN1EQEPaEX9IrerGfr3fqwPqdXC9asc4jmxvr6Ba0Rovk=</latexit> <latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit> consFtuFve relaFons: the heat equaFon �� t = �� · j �� = c p �� T j = � κ � T κ ∂ T α = ∂ t = α ∆ T c p ρ ∂ ˜ T ( k , t ) = − α k 2 ˜ T ( k , t ) ∂ t

<latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit> <latexit sha1_base64="gvryM1gA8N6U7bOtjGgw10wq7A=">ACHnicbVDLSgMxFM34tr5GXboJFsFVmVFBN4rgxmWFPoROKZn0tg1mkiG5I5ShW7/D3Crn+BO3OoX+Bum7Sy0euDC4Zx7Eu6JUyksBsGnNze/sLi0vLJaWlvf2Nzyt3caVmeGQ51rqc1tzCxIoaCOAiXcpgZYEktoxndXY795D8YKrWo4TKGdsL4SPcEZOqnj06gLElkEqRVSK3pOeSeNzEBPdVr+OWgEkxA/5KwIGVSoNrxv6Ku5lkCrlk1rbCIMV2zgwKLmFUijILKeN3rA8tRxVLwLbzySUjeuCULu1p40Yhnag/EzlPYiP6A/z1Ts4Sa4dJ7PIJw4Gd9cbif14rw95ZOxcqzRAUn37fyRFTcdl0a4wFEOHWHcCHcB5QNmGEdXqWsmnO3hL2kcVcLjytHNSfnyouhoheyRfXJIQnJKLsk1qZI64eSBPJFn8uI9eq/em/c+XZ3ziswu+QXv4xsUKaMQ</latexit> <latexit sha1_base64="d+UF3wdRyVt3QEV1BGYB59NRIgs=">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</latexit> <latexit sha1_base64="xYmdsD/ls5+Hb5Fo2LJYlso4miM=">ACOnicbZDLSgMxFIYzXu96tJNsAhuLDMq6EJFcONSwVahU8qZNGM8mQnBHK0JfxOXwAt7pz60bErQ9gpi14PRD4+f+cnJwvSqWw6PvP3tj4xOTUdGlmdm5+YXGpvLxStzozjNeYltpcRWC5FIrXUKDkV6nhkESX0Y3J0V+ecuNFVpdYC/lzQ6SsSCATqrVT4IYwMsD1MwKEDSkKdWSK36Xxb2D+lWqCSELK2RhomgN0optetcsWv+oOif0UwEhUyqrNW+TVsa5YlXCGTYG0j8FNs5sUgJnl/NswsT4HdQIc3nFSQcNvMB1v26YZz2jTWxh2FdOB+78hZEhnR6eKPd3JIrO0lkesvm1/Z4X5X9bIMN5v5kKlGXLFhuPjzOHQtABJ28JwhrLnBDAj3AaUdcHBRIfbkQl+c/gr6tvVYKe6fb5bOT4aMSqRNbJONklA9sgxOSVnpEYuSMP5JE8efei/fmvQ+vjnmjnlXyo7yPT7NgryQ=</latexit> <latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit> <latexit sha1_base64="z3Vr8mT+yPioCREyMtkIq5MAyQ=">ACGnicbVDNSgMxGMzWv1r/qh71ECyCF8tuFfSiFLx4rNA/6C7l2zTbxmazS5IVytKLz+EDeNVH8CZevfgEvoZpuwdbHQgM5mEGT/mTGnb/rJyS8srq2v59cLG5tb2TnF3r6miRBLaIBGPZNsHRTkTtKGZ5rQdSwqhz2nLH95M/NYDlYpFoq5HMfVC6AsWMALaSN3ioRuCHvgBvsdX+NQdQhwDdgX4HC9WyzZXsK/Jc4GSmhDLVu8dvtRSQJqdCEg1Idx461l4LUjHA6LriJojGQIfRpx1ABIVeOm0xsdG6eEgkuYIjafq70RKQl+y/kDPvZNCqNQo9E1+0kQtehPxP6+T6ODS5mIE0FmX0fJBzrCE+Gwj0mKdF8ZAgQyUwDTAYgWgzp1nGWdzhL2lWys5ZuXJ3XqpeZxvl0QE6QifIQReoim5RDTUQY/oGb2gV+vJerPerY/Z1ZyVZfbRHKzPHyeQoFc=</latexit> <latexit sha1_base64="EPm/9eh/7c4A4uXHrDIw5pOo3CY=">ACHicbZDLSgMxGIUz9VbrepSkGARXJWZKuhGKbhxWcFeoDM/6SZTmhmJiQZoQzd+Rw+gFt9BHfiVvAJfA3Ty8K2/hA4nJOT8H+B4Exp2/62Ciura+sbxc3S1vbO7l5/6Cl0kwS2iQpT2UnAEU5S2hTM81pR0gKcBpOxjcjvP2I5WKpcmDHgrqxdBPWMgIaGP5WMXuIgAX7uhBJK7AxACRjnxhSujdOSXK3bVngxeFs5MVNBsGn75x+2lJItpogkHpbqOLbSXg9SMcDoquZmiAsgA+rRrZAIxV4+2WOET43Tw2EqzUk0nrh/GzmJA8n6kZ57J4dYqWEcmH4MOlKL2dj8L+tmOrzycpaITNOETL8PM451iseocI9JSjQfGgFEMrMBJhEYTNoANWScRQ7LolWrOufV2v1FpX4zY1RER+gEnSEHXaI6ukMN1EQEPaEX9IrerGfr3fqwPqdXC9asc4jmxvr6Ba0Rovk=</latexit> consFtuFve relaFons: the heat equaFon �� t = �� · j �� = c p �� T j = � κ � T κ ∂ T α = ∂ t = α ∆ T c p ρ ∂ ˜ T ( k , t ) = − α k 2 ˜ T ( k , t ) ∂ t T ( k , t ) = e − α k 2 t � � T ( k , 0)

<latexit sha1_base64="QezugKTriq0vDXB6FuodfXuCJ4E=">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</latexit> <latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit> <latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit> consFtuFve relaFons: the heat equaFon ∂ T ∂ t = α ∆ T T ( k , t ) = e − α k 2 t � � T ( k , 0) 1 � e � | r � r � | 2 4 α t T ( r � , 0) d r � T ( r , t ) = 3 (4 πα t ) 2

<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> <latexit sha1_base64="1PGOyDRdV5ebL04hk7LE5OnMY=">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</latexit> <latexit sha1_base64="to9WkwJZ3IcO7M/jmX6wxaXS36M=">AB/XicbVDLSgMxFL3js9ZX1aWbwSK4KjNV0JU3LhsoS9oh5J0zY0yQzJHaEMxQ9wq5/gTtz6LX6Bv2HazsK2HgczslJ7j1hLhBz/t2Nja3tnd2c3v5/YPDo+PCyWnTRImrEjEel2SAwTXLEGchSsHWtGZChYKxw/zPzWE9OGR6qOk5gFkgwVH3BK0Eq1eq9Q9EreHO468TNShAzVXuGn249oIplCKogxHd+LMUiJRk4Fm+a7iWExoWMyZB1LFZHMBOl80Kl7aZW+O4i0PQrdufo3kVIZaj4c4dI7KZHGTGRo85LgyKx6M/E/r5Pg4C5IuYoTZIouvh8kwsXInXh9rlmFMXEkI1txu4dEQ0oWgbs834qz2sk2a5F+XyrWbYuU+6ygH53ABV+DLVTgEarQAoMXuAV3pxn5935cD4XVzecLHMGS3C+fgHYvZYT</latexit> <latexit sha1_base64="HpejWXidN7X0OM3maLTJatG7Qrk=">ACLXicbZDJSgNBEIZ7XGPcoh69NAbBU5yJgl6UgB48RsgGmRBqOj1Jk56F7hohDHkMn8MH8KqP4EQr3kNOwtoEgsafv6/q4r6vFgKjb9a2srq1vbGa2sts7u3v7uYPDmo4SxXiVRTJSDQ80lyLkVRQoeSNWHAJP8rXvxvn9SeutIjCg5i3gqgGwpfMEBjtXPnrq+ApW4MCgVIWhn+ahzSG+qCjHtA3XsuEWilncvbBXtSdFk4M5Ensyq3cyO3E7Ek4CEyCVo3HTvGVjpewSQfZt1E8xhYH7q8aWQIAdetdHLYkJ4ap0P9SJkXIp24fztSFnhKdHs4NyeFQOtB4Jn+ALCnF7Ox+V/WTNC/bqUijBPkIZu9xMDIqJjdrQjFGcoB0YAU8JcQFkPD80hA0Z5HDsqgVC85Fofh4mS/dzhlyDE5IWfEIVekRB5ImVQJI8/klbyRd+vF+rC+rO/p1xVr1nNE5soa/QCmFKj3</latexit> <latexit sha1_base64="QezugKTriq0vDXB6FuodfXuCJ4E=">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</latexit> consFtuFve relaFons: the heat equaFon ∂ T ∂ t = α ∆ T T ( k , t ) = e − α k 2 t � � T ( k , 0) 1 � e � | r � r � | 2 4 α t T ( r � , 0) d r � T ( r , t ) = 3 (4 πα t ) 2 T x

<latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit> <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit> consFtuFve relaFons: Onsager’s equaFons � � P conserved quantities: A 1 , A 2 , · · · A P � � � � P conserved (current) densities: a 1 ( r ) , · · · a P ( r ) , j 1 ( r ) , · · · j P ( r )

<latexit sha1_base64="XCViKXKeS7eUFse3lctwUm45QyU=">ACFXicbVC7SgNBFJ2NrxhfUQsLm8EgWISwGwVtlICNZSQmBpJluTuZJENmH8zcFcKS7/ADbPUT7MTW2i/wN5w8CpN4MLhnHtmuMePpdBo29WZmV1bX0ju5nb2t7Z3cvHzR0lCjG6ySkWr6oLkUIa+jQMmbseIQ+JI/+oPbsf/4xJUWUfiAw5i7AfRC0RUM0Ehe/qh2XWuB5xTBKxfbrBOhpuBVXS9fsEv2BHSZODNSIDNUvfxPuxOxJOAhMglatxw7RjcFhYJPsq1E81jYAPo8ZahIQRcu+nkgBE9NUqHdiNlJkQ6Uf8mUhb4SvT6OPdOCoHWw8A3+QCwrxe9sfif10qwe+WmIowT5CGbft9NJMWIjuiHaE4Qzk0BJgS5gLK+qCAoWnSNOMs9rBMGuWSc14q318UKjezjrLkmJyQM+KQS1Ihd6RK6oSREXkhr+TNerberQ/rc7qasWaZQzIH6+sXRzKeSw=</latexit> <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit> <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit> consFtuFve relaFons: Onsager’s equaFons � � P conserved quantities: A 1 , A 2 , · · · A P � � � � P conserved (current) densities: a 1 ( r ) , · · · a P ( r ) , j 1 ( r ) , · · · j P ( r ) S = S [ a 1 , a 2 , · · · a P ]

<latexit sha1_base64="1VSIyahDpqvwiWTLkDnFul+d4=">ACK3icbVDLSgMxFM34tr6qLt0Eq+DGMlMXiqAoblxWtFZoh5Jb9vQZGZM7kiHoV/hd/gBbvUTXCluxd8wrV1Y9UDgcE5Ocu8JYikMu6rMzE5NT0zOzefW1hcWl7Jr65dmyjRHCo8kpG+CZgBKUKoEAJN7EGpgIJ1aB7NvCrd6CNiMIrTGPwFWuHoiU4Qys18runSLEDWkXNGRKcAq3iZAi0CJRh3Tr8ojWEXqYKdbrbzXyBbfoDkH/Em9ECmSEciP/W9GPFEQIpfMmJrnxuhnTKPgEvq5emIgZrzL2lCz1E4Axs+Ga/XptlWatBVpe0KkQ/VnIuPKjtnu4Ng7GVPGpCqwecWwY357A/E/r5Zg68DPRBgnCH/r6VSIoRHTRHm0IDR5lawrgWdgPKO0wzjrZf24z3u4e/5LpU9PaKpYtS4eR41NEc2SCbZId4ZJ+ckHNSJhXCyT15JE/k2XlwXpw35/376oQzyqyTMTgfX25kqGg=</latexit> <latexit sha1_base64="ONEws8JcBdQOJdE9Avua23WGr9E=">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</latexit> <latexit sha1_base64="XCViKXKeS7eUFse3lctwUm45QyU=">ACFXicbVC7SgNBFJ2NrxhfUQsLm8EgWISwGwVtlICNZSQmBpJluTuZJENmH8zcFcKS7/ADbPUT7MTW2i/wN5w8CpN4MLhnHtmuMePpdBo29WZmV1bX0ju5nb2t7Z3cvHzR0lCjG6ySkWr6oLkUIa+jQMmbseIQ+JI/+oPbsf/4xJUWUfiAw5i7AfRC0RUM0Ehe/qh2XWuB5xTBKxfbrBOhpuBVXS9fsEv2BHSZODNSIDNUvfxPuxOxJOAhMglatxw7RjcFhYJPsq1E81jYAPo8ZahIQRcu+nkgBE9NUqHdiNlJkQ6Uf8mUhb4SvT6OPdOCoHWw8A3+QCwrxe9sfif10qwe+WmIowT5CGbft9NJMWIjuiHaE4Qzk0BJgS5gLK+qCAoWnSNOMs9rBMGuWSc14q318UKjezjrLkmJyQM+KQS1Ihd6RK6oSREXkhr+TNerberQ/rc7qasWaZQzIH6+sXRzKeSw=</latexit> <latexit sha1_base64="DFnPumzMUM7yHKOZtNp+7uKk=">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</latexit> consFtuFve relaFons: Onsager’s equaFons � � P conserved quantities: A 1 , A 2 , · · · A P � � � � P conserved (current) densities: a 1 ( r ) , · · · a P ( r ) , j 1 ( r ) , · · · j P ( r ) S = S [ a 1 , a 2 , · · · a P ] At thermodynamic equilibrium: S = max

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<latexit sha1_base64="M6y7rALkPqCoKP31PCfXrdEF4Hg=">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</latexit> <latexit sha1_base64="Wr/tm4lDRlhP4qFs0ULU4AxpquI=">ACGnicdVDLSgMxFM34rPU16lIXwSK4GmZadaYLoeDGhYsK9gHtUDJp2qbNPEgyQhlm43f4AW71E9yJWzd+gb9hpq1oRQ8ETs69J+EcL2JUSN81xYWl5ZXVnNr+fWNza1tfWe3LsKY1LDIQt50OCMBqQmqSkWbECfI9Rhre6CKbN24JFzQMbuQ4Iq6P+gHtUYykjr6AWxfqe0u6iR0mMLz7+uQph29YBp20bZPTWga5lm5XLYyUnKlgMtw5ygAGaodvSPdjfEsU8CiRkSomWZkXQTxCXFjKT5dixIhPAI9UlL0QD5RLjJEUKj5TShb2QqxNIOF/OhLse5z2B3LunQT5Qox9T/l9JAfi9ywT/5q1Ytlz3IQGUSxJgKf92IGZQizomCXcoIlGyuCMKcqAcQDxBGWqk7VzFd8+D+pFw2rZBSvTwoVZ9ZRDuyDQ3AMLGCDCrgEVADGNyB/AInrR7Vl70V6nqwvazLMH5qC9fQL5JqF3</latexit> consFtuFve relaFons: Onsager’s equaFons Onsager’s linear- � j i = Λ ij f j response equations Λ ij = Λ ji j f = � α α = ∂ S A ∂ A 1 E T p V T − μ i N i T

<latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit> Green-Kubo linear-response theory � ∞ Λ ij = Ω � J i ( t ) J j (0) � dt k B 0

<latexit sha1_base64="j1CMhF2qvqFg7ZQoOPL9+IsYeiI=">ACNXicbVDNSgMxGMz6W+vfqkcvwSLopeyqoBeh4EW8WMH+QLeUbJpto0l2Sb4VytJH8Tl8AK/6AB68iR59BbNtEbUOBCYzmSTfhIngBjzvxZmZnZtfWCwsFZdXVtfW3Y3NuolTVmNxiLWzZAYJrhiNeAgWDPRjMhQsEZ4e5b7jTumDY/VNQwS1pakp3jEKQErdzjC3yKg0gTmvnDLiUrEeGOAKOuMNvtkLJIF+G93/2mHbfklb0R8DTxJ6SEJqh23I+gG9NUMgVUEGNavpdAOyMaOBVsWAxSwxJCb0mPtSxVRDLTzkYDvGuVbo4irVdCvBI/ZnIqAw17/Xh1z0ZkcYMZGjz+bfNXy8X/NaKUQn7YyrJAWm6Pj5KBUYpx3iLtcMwpiYAmhmtsJMO0T2yHYpm0z/t8epkn9oOwflg+ujkqVk0lHBbSNdtAe8tExqBzVEU1RNE9ekRP6Nl5cF6dN+d9fHTGmWS20C84n1/x+auV</latexit> <latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit> Green-Kubo linear-response theory � ∞ Λ ij = Ω � J i ( t ) J j (0) � dt k B 0 J = 1 � j ( r ) d r Ω Ω

<latexit sha1_base64="j1CMhF2qvqFg7ZQoOPL9+IsYeiI=">ACNXicbVDNSgMxGMz6W+vfqkcvwSLopeyqoBeh4EW8WMH+QLeUbJpto0l2Sb4VytJH8Tl8AK/6AB68iR59BbNtEbUOBCYzmSTfhIngBjzvxZmZnZtfWCwsFZdXVtfW3Y3NuolTVmNxiLWzZAYJrhiNeAgWDPRjMhQsEZ4e5b7jTumDY/VNQwS1pakp3jEKQErdzjC3yKg0gTmvnDLiUrEeGOAKOuMNvtkLJIF+G93/2mHbfklb0R8DTxJ6SEJqh23I+gG9NUMgVUEGNavpdAOyMaOBVsWAxSwxJCb0mPtSxVRDLTzkYDvGuVbo4irVdCvBI/ZnIqAw17/Xh1z0ZkcYMZGjz+bfNXy8X/NaKUQn7YyrJAWm6Pj5KBUYpx3iLtcMwpiYAmhmtsJMO0T2yHYpm0z/t8epkn9oOwflg+ujkqVk0lHBbSNdtAe8tExqBzVEU1RNE9ekRP6Nl5cF6dN+d9fHTGmWS20C84n1/x+auV</latexit> <latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit> <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> Green-Kubo linear-response theory � ∞ Λ ij = Ω � J i ( t ) J j (0) � dt k B 0 J = 1 � j ( r ) d r Ω Ω J ( t ) = J ( Γ t ) = J ( t, Γ 0 )

<latexit sha1_base64="oN8l1/xSxXZw/0tCosyTRx3u4VU=">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</latexit> <latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">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</latexit> <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> <latexit sha1_base64="j1CMhF2qvqFg7ZQoOPL9+IsYeiI=">ACNXicbVDNSgMxGMz6W+vfqkcvwSLopeyqoBeh4EW8WMH+QLeUbJpto0l2Sb4VytJH8Tl8AK/6AB68iR59BbNtEbUOBCYzmSTfhIngBjzvxZmZnZtfWCwsFZdXVtfW3Y3NuolTVmNxiLWzZAYJrhiNeAgWDPRjMhQsEZ4e5b7jTumDY/VNQwS1pakp3jEKQErdzjC3yKg0gTmvnDLiUrEeGOAKOuMNvtkLJIF+G93/2mHbfklb0R8DTxJ6SEJqh23I+gG9NUMgVUEGNavpdAOyMaOBVsWAxSwxJCb0mPtSxVRDLTzkYDvGuVbo4irVdCvBI/ZnIqAw17/Xh1z0ZkcYMZGjz+bfNXy8X/NaKUQn7YyrJAWm6Pj5KBUYpx3iLtcMwpiYAmhmtsJMO0T2yHYpm0z/t8epkn9oOwflg+ujkqVk0lHBbSNdtAe8tExqBzVEU1RNE9ekRP6Nl5cF6dN+d9fHTGmWS20C84n1/x+auV</latexit> Green-Kubo linear-response theory � ∞ Λ ij = Ω � J i ( t ) J j (0) � dt k B 0 J = 1 � j ( r ) d r Ω Ω J ( t ) = J ( Γ t ) = J ( t, Γ 0 ) � � J ( t ) J (0) � = J ( t, Γ 0 ) J (0 , Γ 0 ) P � ( Γ 0 ) d Γ 0 �

<latexit sha1_base64="oN8l1/xSxXZw/0tCosyTRx3u4VU=">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</latexit> <latexit sha1_base64="MgyIJwRxPqmc5I5Wp5MmfGetjfc=">ACS3icbVBNaxRBEO1ZjcbEj1WPXposwuayzETBXISgFwmKCWSTwM5mqOmpme1sd8/QXSMsw/wqf0d+QK56yA/ITyk9+NgEguaerxXr5p6aWkozC8DoPHq49erz+ZGPz6bPnL7ovXx27srYCh6JUpT1NwaGSBockSeFpZRF0qvAknX6e6yc/0DpZmiOaVTjWUBiZSwHkqaT7Lf7qhzNIGnefoxzC6KJv2soG2myac2loaS8My3nGY8VmAKhXw/kX3a9u28H27z2C7ZjJuLxyEi+L3QbQCPbaqg6R7FWelqDUaEgqcG0VhReMGLEmhsN2Ia4cViCkUOPLQgEY3bhZnt/ytZzKel9Y/Q3zB/utohE6tLCZ0a08D2rmZTr1fA03cXW1O/k8b1ZTvjhtpqprQiOX3ea04lXyeLM+kRUFq5gEIK/0FXEzAB0o+f59MdDeH+B4ZxC9G+wcvu/t7a4yWmdv2Bbrs4h9YHvsCztgQybYT3bJfrHfwUVwHfwJ/i5HO8HK85rdqs7aDb8s8E=</latexit> <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> <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> <latexit sha1_base64="j1CMhF2qvqFg7ZQoOPL9+IsYeiI=">ACNXicbVDNSgMxGMz6W+vfqkcvwSLopeyqoBeh4EW8WMH+QLeUbJpto0l2Sb4VytJH8Tl8AK/6AB68iR59BbNtEbUOBCYzmSTfhIngBjzvxZmZnZtfWCwsFZdXVtfW3Y3NuolTVmNxiLWzZAYJrhiNeAgWDPRjMhQsEZ4e5b7jTumDY/VNQwS1pakp3jEKQErdzjC3yKg0gTmvnDLiUrEeGOAKOuMNvtkLJIF+G93/2mHbfklb0R8DTxJ6SEJqh23I+gG9NUMgVUEGNavpdAOyMaOBVsWAxSwxJCb0mPtSxVRDLTzkYDvGuVbo4irVdCvBI/ZnIqAw17/Xh1z0ZkcYMZGjz+bfNXy8X/NaKUQn7YyrJAWm6Pj5KBUYpx3iLtcMwpiYAmhmtsJMO0T2yHYpm0z/t8epkn9oOwflg+ujkqVk0lHBbSNdtAe8tExqBzVEU1RNE9ekRP6Nl5cF6dN+d9fHTGmWS20C84n1/x+auV</latexit> Green-Kubo linear-response theory � ∞ Λ ij = Ω � J i ( t ) J j (0) � dt k B 0 J = 1 � j ( r ) d r Ω Ω J ( t ) = J ( Γ t ) = J ( t, Γ 0 ) � � � J ( t ) J (0) � = J ( t, Γ 0 ) J (0 , Γ 0 ) P � ( Γ 0 ) d Γ 0 � T � t 1 � � J ( t + τ , Γ 0 ) J ( τ , Γ 0 ) d τ T � t 0

<latexit sha1_base64="gxqmbM4YXxOWgvLt4/qMElHRs=">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</latexit> Einstein-Helfand relaFons Einstein (1905) � t �� 2 � � | x ( t ) � x (0) | 2 � � � � v ( t � ) dt � = � � � � 0 � 2 Dt � � D = � v ( t ) v (0) � dt 0

<latexit sha1_base64="QaXBtrlPDlz6zV/YrGvBPDfA9c=">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</latexit> <latexit sha1_base64="gxqmbM4YXxOWgvLt4/qMElHRs=">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</latexit> Einstein-Helfand relaFons Einstein (1905) � t �� 2 � � | x ( t ) � x (0) | 2 � � � � v ( t � ) dt � = � � � � 0 � 2 Dt � � D = � v ( t ) v (0) � dt 0 Helfand (1960) � t �� 2 � � � � J ( t � ) dt � � 2 Λ t � � � � 0 � � Λ = � J ( t ) J (0) � dt 0

<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit> the classical energy current J = 1 � j ( r ) d r Ω Ω

<latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">ACRXicbZDLSgMxFIYz9VbvVZdugkWomzJTBd0IReKGxWsFjqlZNJMG5tkhuSMUIY+ks/hA7gSdO3KnbjV9KJo64HAl/9ckvMHseAGXPfJyUxNz8zOZecXFpeWV1Zza+vXJko0ZRUaiUhXA2KY4IpVgINg1VgzIgPBboLOcT9/c8e04ZG6gm7M6pK0FA85JWClRu7ElwTaQYjP8CH2Q01o6vVS/1yFulhnytoDC/4u/C28E16p/mDjVzeLbqDwJPgjSCPRnHRyL36zYgmkimghT89wY6inRwKlgvQU/MSwmtENarGZREclMPR0s3MPbVmniMNL2KMAD9XdHSmWgeasNf+akRBrTlYHt73/bjOf64n+5WgLhQT3lKk6AKTp8PkwEhgj3PcVNrhkF0bVAqOZ2A0zbxHoJ1nrjDfuwyRcl4rebrF0uZcvH408yqJNtIUKyEP7qIxO0QWqIru0SN6Ri/Og/PmvDsfw9KM+rZQH/C+fwCGFGyGw=</latexit> <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> the classical energy current J = 1 � j ( r ) d r Ω Ω � · j ( r ) = � ˙ � ( r ) � � � �

<latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> <latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit> <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> the classical energy current J = 1 � j ( r ) d r Ω Ω � · j ( r ) = � ˙ � ( r ) � · � � � � � � � � � � · j ( r ) d r = � r ˙ � ( r ) d r r Ω Ω � �

<latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> <latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit> <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> the classical energy current J = 1 � j ( r ) d r Ω Ω � · j ( r ) = � ˙ � ( r ) � · � � � � � � � � � � � � � � · j ( r ) d r = � r ˙ � ( r ) d r r Ω Ω � � � � j ( r ) d r = r ˙ � ( r ) d r + surface terms Ω Ω

<latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> <latexit sha1_base64="n47+Tng25YEVPz/U2qR7aWltxAU=">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</latexit> <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> <latexit sha1_base64="12CdjhV6gKdgMbgtfx8lL/UxD6g=">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</latexit> <latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit> the classical energy current J = 1 � j ( r ) d r Ω Ω � · j ( r ) = � ˙ � ( r ) � · � � � � � � � � � � � � � � · j ( r ) d r = � r ˙ � ( r ) d r r Ω Ω � � � � j ( r ) d r = r ˙ � ( r ) d r + surface terms Ω Ω J = 1 � r ˙ � ( r ) d r Ω Ω

<latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">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</latexit> <latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit> the classical energy current J = 1 � r ˙ � ( r ) d r Ω Ω � � � � � � ( r , t) = r − R I (t) R (t) , V (t) � e I I

<latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit> <latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit> <latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">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</latexit> the classical energy current J = 1 � r ˙ � ( r ) d r Ω Ω � � � � � � ( r , t) = r − R I (t) R (t) , V (t) � e I I e I ( R , V ) = 1 I + 1 � 2 M I V 2 v ( | R I − R J | ) 2 J � = I

<latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit> <latexit sha1_base64="hTV/28IlByLW+14TZOSP3HgwK5k=">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</latexit> <latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">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</latexit> <latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit> the classical energy current J = 1 � r ˙ � ( r ) d r Ω Ω � � � � � � ( r , t) = r − R I (t) R (t) , V (t) � e I I e I ( R , V ) = 1 I + 1 � 2 M I V 2 v ( | R I − R J | ) 2 J � = I � ˙ � � J = R I e I + R I ˙ e I I

<latexit sha1_base64="yZWQm8/jgK1ERHROkp6KXIRx+Q8=">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</latexit> <latexit sha1_base64="13CHDcGvThQUGB+JW2jrD3yGtNg=">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</latexit> <latexit sha1_base64="aF7Y3hKmeLcMCiN2SU8psK3CTVg=">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</latexit> <latexit sha1_base64="QH1I+pWfSIVASJyodBR4upyj+cI=">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</latexit> <latexit sha1_base64="hTV/28IlByLW+14TZOSP3HgwK5k=">ACUHicbVFNSyNBEK3JqvxK7sevTQGQRHCjAruZUHWi+tJxaiQCaGnU5M06Z4ZumsWwpD/tb9jb1686k/Ym/aMEYxaUPTjvarqrtdRpqQl37/zal/m5he+Li7Vl1dW19Yb375f2zQ3AtsiVam5jbhFJRNskySFt5lBriOFN9HopNRv/qCxMk2uaJxhV/NBImMpODmq17gMNadhFLMz9pOFNte93yxUGBPbYWE/peJVv5w4BV3usVmurKqE0MjBkNhur9H0W34V7CMIpqAJ0zjvNR7cDJFrTEgobm0n8DPqFtyQFAon9TC3mHEx4gPsOJhwjbZbVLtP2LZj+ixOjcuEWMW+7SiEjqHzcwpuLZ2rCPX+5i32sl+ZnWySn+0S1kuWEiXi5Ps4Vo5SV9rK+NChIjR3gwki3ARNDbrg9wnOmeC9Dx/B9X4rOGjtXxw2j39NPVqETdiCHQjgCI7hFM6hDQL+wj08wKP3z/vPdW8l9LXEzZgJmr1Z6+Ksos=</latexit> the classical energy current J = 1 � r ˙ � ( r ) d r Ω Ω � � � � � � ( r , t) = r − R I (t) R (t) , V (t) � e I I e I ( R , V ) = 1 I + 1 � 2 M I V 2 v ( | R I − R J | ) 2 J � = I e I V I + 1 � ˙ � � � � J = ( V I · F IJ )( R I − R J ) J = R I e I + R I ˙ e I 2 I I � = J I

hurdles towards an ab iniFo Green-Kubo theory week ending P H Y S I C A L R E V I E W L E T T E R S PRL 104, 208501 (2010) 21 MAY 2010 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 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.

insights from classical mechanics � � I ( R , V ) E = I = �� I ( R , V ) = 1 I + 1 2 M I V 2 � v ( | R I − R J | ) 2 J � = I � I V I + 1 J e = � � ( V I · F IJ )( R I − R J ) 2 I I � = J

insights from classical mechanics � � I ( R , V ) = �� I � I ( R , V ) = 1 I + 1 2 M I V 2 � v ( | R I − R J | )(1 + Γ IJ ) 2 J � = I � I V I + 1 J e = � � ( V I · F IJ )( R I − R J ) 2 I I � = J + 1 Γ IJ [ V I v ( | R I − R J | ) + ( V I · F IJ )( R I − R I )] � 2 I � = J

insights from classical mechanics � I V I + 1 J e = � � ( V I · F IJ )( R I − R J ) 2 I I � = J + 1 � Γ IJ [ V I v ( | R I − R J | ) + ( V I · F IJ )( R I − R I )] 2 I � = J

insights from classical mechanics � I V I + 1 J e = � � ( V I · F IJ )( R I − R J ) 2 I I � = J + 1 � Γ IJ [ V I v ( | R I − R J | ) + ( V I · F IJ )( R I − R I )] 2 I � = J  0     Γ IJ = 1 2 ( A IJ − A JI ) , A IJ ∼ U [0 , γ ]    sgn ( I − J ) × γ � J(t)J(0) �  0 . 1 0 . 2 t

insights from classical mechanics � I V I + 1 J e = � � ( V I · F IJ )( R I − R J ) 2 I I � = J + 1 � Γ IJ [ V I v ( | R I − R J | ) + ( V I · F IJ )( R I − R I )] 2 I � = J   0 0         Γ IJ = 1 1 2 ( A IJ − A JI ) , A IJ ∼ U [0 , γ ] Γ IJ = 2 ( A IJ − A JI ) , A IJ ∼ U [0 , γ ]      sgn ( I − J ) × γ  sgn ( I − J ) × γ � J(t)J(0) �   κ 0 . 1 0 . 2 0 . 2 0 . 4 t t

insights from classical mechanics � I V I + 1 J e = � � ( V I · F IJ )( R I − R J ) 2 I I � = J + 1 Γ IJ [ V I v ( | R I − R J | ) + ( V I · F IJ )( R I − R I )] � 2 I � = J

insights from classical mechanics � I V I + 1 J e = � � ( V I · F IJ )( R I − R J ) 2 I I � = J + 1 Γ IJ [ V I v ( | R I − R J | ) + ( V I · F IJ )( R I − R I )] � 2 I � = J P = d 1 ˙ � Γ IJ v ( | R I − R J | )( R I − R I ) dt 4 I � = J

insights from classical mechanics J ′ = J + ˙ P

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sha1_base64="BQ9WLjBu/xIPFrkWUAR+TlFh5gE=">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</latexit> insights from classical mechanics J ′ = J + ˙ P var D � � � t κ ∼ 1 2 t var D ( t ) D ( t ) = J ( t � ) dt � � � 0 � D J D � ( t ) = D ( t )+ P ( t ) − P (0) � � � � � � � � var D � ( t ) = var D ( t ) � + var ∆ P ( t ) + 2 cov D ( t ) · ∆ P ( t ) � � � � � � � � var D var D var P cov D P · � � � � � � � O ( t ) O (1)

insights from classical mechanics J ′ = J + ˙ P κ ′ = κ

gauge invariance E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ]

gauge invariance E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∂ 𝝯

gauge invariance E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] � e ( r ) d r E [ Ω ] = Ω

gauge invariance E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] � e ( r ) d r E [ Ω ] = Ω E ′ [ Ω ] = E [ Ω ] + O [ ∂ Ω ]

gauge invariance E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] � e ( r ) d r E [ Ω ] = Ω E ′ [ Ω ] = E [ Ω ] + O [ ∂ Ω ] e ′ ( r ) = e ( r ) − ∇ · p ( r )

gauge invariance E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] � e ( r ) d r E [ Ω ] = Ω E ′ [ Ω ] = E [ Ω ] + O [ ∂ Ω ] e ′ ( r ) = e ( r ) − ∇ · p ( r ) e ′ ( r , t ) = −∇ · j ( r , t ) + ˙ p ( r , t ) � � ˙

gauge invariance E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] � e ( r ) d r E [ Ω ] = Ω E ′ [ Ω ] = E [ Ω ] + O [ ∂ Ω ] e ′ ( r ) = e ( r ) − ∇ · p ( r ) e ′ ( r , t ) = −∇ · j ( r , t ) + ˙ p ( r , t ) � � ˙ j ′ ( r , t ) = j ( r , t ) + ˙ p ( r , t )

gauge invariance E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] � e ( r ) d r E [ Ω ] = Ω E ′ [ Ω ] = E [ Ω ] + O [ ∂ Ω ] e ′ ( r ) = e ( r ) − ∇ · p ( r ) e ′ ( r , t ) = −∇ · j ( r , t ) + ˙ p ( r , t ) � � ˙ j ′ ( r , t ) = j ( r , t ) + ˙ p ( r , t ) J ′ ( t ) = J ( t ) + ˙ P ( t )

gauge invariance any two energy densiFes that differ E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ by the divergence of a (bounded) ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] vector field are physically equivalent � e ( r ) d r E [ Ω ] = Ω E ′ [ Ω ] = E [ Ω ] + O [ ∂ Ω ] e ′ ( r ) = e ( r ) − ∇ · p ( r ) e ′ ( r , t ) = −∇ · j ( r , t ) + ˙ p ( r , t ) � � ˙ j ′ ( r , t ) = j ( r , t ) + ˙ p ( r , t ) J ′ ( t ) = J ( t ) + ˙ P ( t )

gauge invariance any two energy densiFes that differ E E E E [ Ω 1 ∪ Ω 2 ] = E [ Ω 1 ] + E [ Ω 2 ] + � [ ∂ Ω ] ∪ by the divergence of a (bounded) ∂ 𝝯 ? = E [ Ω 1 ] + E [ Ω 2 ] vector field are physically equivalent � e ( r ) d r E [ Ω ] = Ω E ′ [ Ω ] = E [ Ω ] + O [ ∂ Ω ] the corresponding energy fluxes differ e ′ ( r ) = e ( r ) − ∇ · p ( r ) e ′ ( r , t ) = −∇ · j ( r , t ) + ˙ p ( r , t ) � � ˙ by a total Fme derivaFve, and the j ′ ( r , t ) = j ( r , t ) + ˙ p ( r , t ) heat transport coefficients coincide J ′ ( t ) = J ( t ) + ˙ P ( t )

density-funcFonal theory I + e 2 Z I Z J E DF T = 1 � � M I V 2 R IJ 2 2 I I � = J � v − 1 � � � 2 E H + � XC ( r ) − µ XC ( r ) � ( r ) d r � + v

the DFT energy density I + e 2 Z I Z J E DF T = 1 � � M I V 2 R IJ 2 2 I I � = J � v − 1 � � � 2 E H + � XC ( r ) − µ XC ( r ) � ( r ) d r � + v e DF T ( r ) = e 0 ( r ) + e KS ( r ) + e H ( r ) + e XC ( r ) � � � r r R r Re r r r r r r r r r

the DFT energy density I + e 2 Z I Z J E DF T = 1 � � M I V 2 R IJ 2 2 I I � = J � v − 1 � � � 2 E H + � XC ( r ) − µ XC ( r ) � ( r ) d r � + v e DF T ( r ) = e 0 ( r ) + e KS ( r ) + e H ( r ) + e XC ( r ) r r r r r � 1 � � � � � e 0 ( r ) = r � ( r − R I ) r R 2 M I V 2 I + w I I � ˆ � e KS ( r ) = Re r Re v ( r ) r H KS � v ( r ) r � � ∗ v e H ( r ) = − 1 r 2 � ( r ) v H ( r ) r r e XC ( r ) = ( � XC ( r ) − v XC ( r )) � ( r ) r r r r

the DFT energy current � J DF T = r ˙ e DF T ( r , t ) d r = J KS + J H + J � 0 + J 0 + J XC

the DFT energy current � J DF T = r ˙ e DF T ( r , t ) d r = J KS + J H + J � 0 + J 0 + J XC � � � � � v | r ˆ J KS = � v | r | � v � H KS | ˙ � v � + � v � ˙ v J H = 1 � v H ( r ) � v H ( r ) d r ˙ 4 � � J � � � v | ( r � R I ) ( V I · � I ˆ 0 = v 0 ) | � v � v,I � � � � J 0 = V I e 0 ( R I � R L ) ( V L · � L w I ) I + I L � = I � (LDA) 0 J XC = � ( r ) ˙ � ( r ) �� GGA ( r ) d r (GGA) � �

the DFT energy current � J DF T = r ˙ e DF T ( r , t ) d r = J KS + J H + J � 0 + J 0 + J XC � � � � � � � � v | r ˆ J J KS = r � v | r | � v � r H KS | ˙ � v � + � v � ˙ v J H = 1 � � J v H ( r ) � v H ( r ) d r r r r ˙ 4 � � J J � � � v | ( r � R I ) ( V I · � I ˆ r R V • | ˙ ϕ v � and ϕ v � orthogonal to the ˆ 0 = H KS | ˙ v 0 ) | � v � occupied-state manifold v,I � � � � J J 0 = V I e 0 V ( R I � R L ) ( V L · � L w I ) R R V I + • ˆ P c r | ϕ v � computed from standard DFPT I L � = I � (LDA) (LDA) 0 J XC = J � ( r ) ˙ r � ( r ) �� GGA ( r ) d r r r r (GGA) (GGA) � �

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

a benchmark 108 “LDA Ar” atoms 1 @bp density, T= 250 K a � J ( t ) · J (0) � C ✏ 100 ps CP trajectory 100 ps classical FF trajectory 1 ns classical FF trajectory 100 b � t κ κ 1 � J ( t � ) · J (0) � d t � 3 V k B T 2 0 t"[ps] 1"

a benchmark 108 “LDA Ar” atoms 1 @bp density, T= 250 K a � J ( t ) · J (0) � C ✏ 100 ps CP trajectory 100 ps classical FF trajectory 1 ns classical FF trajectory 100 b same behavior at T=400 K � t κ κ 1 � J ( t � ) · J (0) � d t � 3 V k B T 2 0 t"[ps] 1"

liquid (heavy) water 64 molecules, T=385 K expt density @ac � t 1 � J ( t � ) · J (0) � d t � 3 V k B T 2 0 1 2 t[ps]

liquid (heavy) water 64 molecules, T=385 K expt density @ac � t 1 � J ( t � ) · J (0) � d t � 3 V k B T 2 0 1 2 t[ps] � ∞ � J ( t ) · J (0) � e i ω t d t S ( ω ) = −∞ ω

liquid (heavy) water 64 molecules, T=385 K expt density @ac � t 1 � J ( t � ) · J (0) � d t � 3 V k B T 2 0 1 2 Einstein’s relaFon t[ps] � t � t �� 2 � t � 1 J ( t � ) d t � � J ( t � ) · J (0) � d t � � � ≈ � � 6 V k B T 2 3 V k B T 2 � � 0 0

liquid (heavy) water 64 molecules, T=385 K expt density @ac � t 1 � J ( t � ) · J (0) � d t � 3 V k B T 2 0 1 2 t[ps] � t �� 2 � � 1 J ( t � ) d t � � � � � 6 V k B T 2 � � 0 1 2 t[ps]

liquid (heavy) water 64 molecules, T=385 K expt density @ac � t 1 � J ( t � ) · J (0) � d t � 3 V k B T 2 0 κ DFT = 0 . 74 ± 0 . 12 W /( mK ) 1 2 κ expt = 0 . 61 ( light@AC ) t[ps] κ DFT = 0 . 60 ( heavy@AC ) � t �� 2 � � 1 J ( t � ) d t � � � � � 6 V k B T 2 � � 0 1 2 t[ps]

hurdles towards an ab iniFo Green-Kubo theory week ending P H Y S I C A L R E V I E W L E T T E R S PRL 104, 208501 (2010) 21 MAY 2010 Thermal Conductivity of Periclase (MgO) from First Principles Stephen Stackhouse, Lars Stixrude, and Bijaya B. Karki 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.

hurdles towards an ab iniFo Green-Kubo theory week ending P H Y S I C A L R E V I E W L E T T E R S PRL 104, 208501 (2010) 21 MAY 2010 Thermal Conductivity of Periclase (MgO) from First Principles Stephen Stackhouse, Lars Stixrude, and Bijaya B. Karki 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. week ending P H Y S I C A L R E V I E W L E T T E R S PRL 118, 175901 (2017) 28 APRIL 2017 Ab Initio Green-Kubo Approach for the Thermal Conductivity of Solids 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

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