THERMODYNAMIC UNCERTAINTY RELATIONS AND THEIR CONNECTION WITH FLUCTUATION THEOREMS
Gabriel T. Landi
Instituto de Física da Universidade de São Paulo
Grhafite IFUSP
September 2019 www.fmt.if.usp.br/~gtlandi
Summary
I. Entropy production.
II. Fluctuation theorems (FTs) III. Thermodynamic uncertainty
relations (TURs).
IV. TURs from FTs.
V. Applications to quantum heat engines.
André M. Timpanaro, Giacomo Guarnieri, John Goold, GTL,
“Thermodynamic uncertainty relations from exchange fluctuation theorems”.
Phys. Rev. Lett. 123, 090604 (2019).
arXiv 1904.07574
The 2nd law
Entropy production
If the process is irreversible, we get Clausius’ inequality instead:
S Q T
<latexit sha1_base64="xOZ3oXXfPqHhBZu00y3KmVcjETI=">AAACKHicdVDLSgMxFM3UV62vqitxEyyCqzIjou2uoAuXLfYFnVIy6Z02NJMZk4xQhuLXuHGhv+JOuvUrXJo+BFv1QOBwzr039x4v4kxp2x5bqZXVtfWN9GZma3tndy+7f1BXYSwp1GjIQ9n0iALOBNQ00xyakQQSeBwa3uB64jceQCoWiqoeRtAOSE8wn1GijdTJHrk3wDXBd9jtwT12fUloUhkl1VEnm7PzRdspXjr4N3Hy9hQ5NEe5k/10uyGNAxCacqJUy7Ej3U6I1IxyGGXcWEFE6ID0oGWoIAGodjI9YYRPjdLFfijNExpP1Z8dCQmUGgaeqQyI7qtlbyL+6XnBws+JYBQmRy7to/1CO2EiijUIOlvHjznWIZ6khrtMAtV8aAihkpmLMO0TM0abbDMmqu888P+kfp537LxTuciVCvPQ0ugYnaAz5KArVEK3qIxqiKJH9IRe0Kv1bL1Z79Z4Vpqy5j2HaAHWxxeCaabk</latexit><latexit sha1_base64="xOZ3oXXfPqHhBZu00y3KmVcjETI=">AAACKHicdVDLSgMxFM3UV62vqitxEyyCqzIjou2uoAuXLfYFnVIy6Z02NJMZk4xQhuLXuHGhv+JOuvUrXJo+BFv1QOBwzr039x4v4kxp2x5bqZXVtfWN9GZma3tndy+7f1BXYSwp1GjIQ9n0iALOBNQ00xyakQQSeBwa3uB64jceQCoWiqoeRtAOSE8wn1GijdTJHrk3wDXBd9jtwT12fUloUhkl1VEnm7PzRdspXjr4N3Hy9hQ5NEe5k/10uyGNAxCacqJUy7Ej3U6I1IxyGGXcWEFE6ID0oGWoIAGodjI9YYRPjdLFfijNExpP1Z8dCQmUGgaeqQyI7qtlbyL+6XnBws+JYBQmRy7to/1CO2EiijUIOlvHjznWIZ6khrtMAtV8aAihkpmLMO0TM0abbDMmqu888P+kfp537LxTuciVCvPQ0ugYnaAz5KArVEK3qIxqiKJH9IRe0Kv1bL1Z79Z4Vpqy5j2HaAHWxxeCaabk</latexit><latexit sha1_base64="xOZ3oXXfPqHhBZu00y3KmVcjETI=">AAACKHicdVDLSgMxFM3UV62vqitxEyyCqzIjou2uoAuXLfYFnVIy6Z02NJMZk4xQhuLXuHGhv+JOuvUrXJo+BFv1QOBwzr039x4v4kxp2x5bqZXVtfWN9GZma3tndy+7f1BXYSwp1GjIQ9n0iALOBNQ00xyakQQSeBwa3uB64jceQCoWiqoeRtAOSE8wn1GijdTJHrk3wDXBd9jtwT12fUloUhkl1VEnm7PzRdspXjr4N3Hy9hQ5NEe5k/10uyGNAxCacqJUy7Ej3U6I1IxyGGXcWEFE6ID0oGWoIAGodjI9YYRPjdLFfijNExpP1Z8dCQmUGgaeqQyI7qtlbyL+6XnBws+JYBQmRy7to/1CO2EiijUIOlvHjznWIZ6khrtMAtV8aAihkpmLMO0TM0abbDMmqu888P+kfp537LxTuciVCvPQ0ugYnaAz5KArVEK3qIxqiKJH9IRe0Kv1bL1Z79Z4Vpqy5j2HaAHWxxeCaabk</latexit><latexit sha1_base64="xOZ3oXXfPqHhBZu00y3KmVcjETI=">AAACKHicdVDLSgMxFM3UV62vqitxEyyCqzIjou2uoAuXLfYFnVIy6Z02NJMZk4xQhuLXuHGhv+JOuvUrXJo+BFv1QOBwzr039x4v4kxp2x5bqZXVtfWN9GZma3tndy+7f1BXYSwp1GjIQ9n0iALOBNQ00xyakQQSeBwa3uB64jceQCoWiqoeRtAOSE8wn1GijdTJHrk3wDXBd9jtwT12fUloUhkl1VEnm7PzRdspXjr4N3Hy9hQ5NEe5k/10uyGNAxCacqJUy7Ej3U6I1IxyGGXcWEFE6ID0oGWoIAGodjI9YYRPjdLFfijNExpP1Z8dCQmUGgaeqQyI7qtlbyL+6XnBws+JYBQmRy7to/1CO2EiijUIOlvHjznWIZ6khrtMAtV8aAihkpmLMO0TM0abbDMmqu888P+kfp537LxTuciVCvPQ0ugYnaAz5KArVEK3qIxqiKJH9IRe0Kv1bL1Z79Z4Vpqy5j2HaAHWxxeCaabk</latexit>
We may be write this as
S = ⌃ + Q
T , ⌃ 0
<latexit sha1_base64="VZyYzO7Gxde1ztRzP1Piyso2Dt0=">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</latexit><latexit sha1_base64="VZyYzO7Gxde1ztRzP1Piyso2Dt0=">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</latexit><latexit sha1_base64="VZyYzO7Gxde1ztRzP1Piyso2Dt0=">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</latexit><latexit sha1_base64="VZyYzO7Gxde1ztRzP1Piyso2Dt0=">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</latexit>
In thermodynamics we learn that when a process is done reversibly
S = Q T
<latexit sha1_base64="vxIDTSum3vYnY9lYGXAb50h6HQo=">AAACJnicdVDLSgMxFM34rPVVdeHCTbAIrsqMiLYLoaALly32BW0pmfROG5rJDElGKMN8jRsX+ivuRNz5GS7NtBVs1QOBwzn33tx73JAzpW373VpaXlldW89sZDe3tnd2c3v7DRVEkkKdBjyQLZco4ExAXTPNoRVKIL7LoemOrlO/eQ9SsUDU9DiErk8GgnmMEm2kXu6wcwNcE3yHrzDueJLQuJrEtaSXy9uFku2ULhz8mzgFe4I8mqHSy312+gGNfBCacqJU27FD3Y2J1IxySLKdSEFI6IgMoG2oID6objw5IMEnRuljL5DmCY0n6s+OmPhKjX3XVPpED9Wil4p/eq4/93MsGIX0yIV9tFfsxkyEkQZBp+t4Ecc6wGlmuM8kUM3HhhAqmbkI0yExY7RJNmui+s4D/08aZwXHLjjV83y5OAstg47QMTpFDrpEZXSLKqiOKErQA3pCz9aj9WK9Wm/T0iVr1nOA5mB9fAETO6WU</latexit><latexit sha1_base64="vxIDTSum3vYnY9lYGXAb50h6HQo=">AAACJnicdVDLSgMxFM34rPVVdeHCTbAIrsqMiLYLoaALly32BW0pmfROG5rJDElGKMN8jRsX+ivuRNz5GS7NtBVs1QOBwzn33tx73JAzpW373VpaXlldW89sZDe3tnd2c3v7DRVEkkKdBjyQLZco4ExAXTPNoRVKIL7LoemOrlO/eQ9SsUDU9DiErk8GgnmMEm2kXu6wcwNcE3yHrzDueJLQuJrEtaSXy9uFku2ULhz8mzgFe4I8mqHSy312+gGNfBCacqJU27FD3Y2J1IxySLKdSEFI6IgMoG2oID6objw5IMEnRuljL5DmCY0n6s+OmPhKjX3XVPpED9Wil4p/eq4/93MsGIX0yIV9tFfsxkyEkQZBp+t4Ecc6wGlmuM8kUM3HhhAqmbkI0yExY7RJNmui+s4D/08aZwXHLjjV83y5OAstg47QMTpFDrpEZXSLKqiOKErQA3pCz9aj9WK9Wm/T0iVr1nOA5mB9fAETO6WU</latexit><latexit sha1_base64="vxIDTSum3vYnY9lYGXAb50h6HQo=">AAACJnicdVDLSgMxFM34rPVVdeHCTbAIrsqMiLYLoaALly32BW0pmfROG5rJDElGKMN8jRsX+ivuRNz5GS7NtBVs1QOBwzn33tx73JAzpW373VpaXlldW89sZDe3tnd2c3v7DRVEkkKdBjyQLZco4ExAXTPNoRVKIL7LoemOrlO/eQ9SsUDU9DiErk8GgnmMEm2kXu6wcwNcE3yHrzDueJLQuJrEtaSXy9uFku2ULhz8mzgFe4I8mqHSy312+gGNfBCacqJU27FD3Y2J1IxySLKdSEFI6IgMoG2oID6objw5IMEnRuljL5DmCY0n6s+OmPhKjX3XVPpED9Wil4p/eq4/93MsGIX0yIV9tFfsxkyEkQZBp+t4Ecc6wGlmuM8kUM3HhhAqmbkI0yExY7RJNmui+s4D/08aZwXHLjjV83y5OAstg47QMTpFDrpEZXSLKqiOKErQA3pCz9aj9WK9Wm/T0iVr1nOA5mB9fAETO6WU</latexit><latexit sha1_base64="vxIDTSum3vYnY9lYGXAb50h6HQo=">AAACJnicdVDLSgMxFM34rPVVdeHCTbAIrsqMiLYLoaALly32BW0pmfROG5rJDElGKMN8jRsX+ivuRNz5GS7NtBVs1QOBwzn33tx73JAzpW373VpaXlldW89sZDe3tnd2c3v7DRVEkkKdBjyQLZco4ExAXTPNoRVKIL7LoemOrlO/eQ9SsUDU9DiErk8GgnmMEm2kXu6wcwNcE3yHrzDueJLQuJrEtaSXy9uFku2ULhz8mzgFe4I8mqHSy312+gGNfBCacqJU27FD3Y2J1IxySLKdSEFI6IgMoG2oID6objw5IMEnRuljL5DmCY0n6s+OmPhKjX3XVPpED9Wil4p/eq4/93MsGIX0yIV9tFfsxkyEkQZBp+t4Ecc6wGlmuM8kUM3HhhAqmbkI0yExY7RJNmui+s4D/08aZwXHLjjV83y5OAstg47QMTpFDrpEZXSLKqiOKErQA3pCz9aj9WK9Wm/T0iVr1nOA5mB9fAETO6WU</latexit>
The quantity 𝚺 is called the Entropy Production.
2nd law:
𝚺 ≥ 0
Why entropy production matters
1st and 2nd laws for a system coupled to two baths:
Entropy production is therefore the reason the efficiency is smaller than Carnot:
⌘ = ⌘ C T c Q ˙ h
⌃ ˙
<latexit sha1_base64="wRtJSPFjFdWn9Ym0oZZ5G2h6KO8=">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</latexit><latexit sha1_base64="wRtJSPFjFdWn9Ym0oZZ5G2h6KO8=">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</latexit><latexit sha1_base64="wRtJSPFjFdWn9Ym0oZZ5G2h6KO8=">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</latexit><latexit sha1_base64="wRtJSPFjFdWn9Ym0oZZ5G2h6KO8=">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</latexit>
The efficiency of the engine may then be written as
⌘ = W ˙ Q ˙ h
<latexit sha1_base64="anyJAjrQuIiZ6P4ALnsGa7gR31s=">AAACMXicbVA9SwNBEN2L3/ErammzJAg2hjsRTCMEbCwVjAkkIext5pIle3vH7pwQjuv9NTYW+lfSia1/wNJNcoVJfLDweG9mdub5sRQGXXfiFNbWNza3tneKu3v7B4elo+MnEyWaQ4NHMtItnxmQQkEDBUpoxRpY6Eto+qPbqd98Bm1EpB5xHEM3ZAMlAsEZWqlXKncAGb2hF7QTaMbTTj/CtJllc/KQ9YZZr1Rxq+4MdJV4OamQHPe90o9t5kkICrlkxrQ9N8ZuyjQKLiErdhIDMeMjNoC2pYqFYLrp7JaMnlmlT4NI26eQztS/HSkLjRmHvq0MGQ7NsjcV//X8cOHnVAkO05OX9sGg1k2FihMExefrBImkGNFpfLQvNHCUY0sY18JeRPmQ2TFoQy7aqLzlYFbJ02XVc6vew1WlXstD2yanpEzOiUeuSZ3ckXvSIJy8kFfyTj6cN2fifDpf89KCk/eckAU4378IX6q+</latexit><latexit sha1_base64="anyJAjrQuIiZ6P4ALnsGa7gR31s=">AAACMXicbVA9SwNBEN2L3/ErammzJAg2hjsRTCMEbCwVjAkkIext5pIle3vH7pwQjuv9NTYW+lfSia1/wNJNcoVJfLDweG9mdub5sRQGXXfiFNbWNza3tneKu3v7B4elo+MnEyWaQ4NHMtItnxmQQkEDBUpoxRpY6Eto+qPbqd98Bm1EpB5xHEM3ZAMlAsEZWqlXKncAGb2hF7QTaMbTTj/CtJllc/KQ9YZZr1Rxq+4MdJV4OamQHPe90o9t5kkICrlkxrQ9N8ZuyjQKLiErdhIDMeMjNoC2pYqFYLrp7JaMnlmlT4NI26eQztS/HSkLjRmHvq0MGQ7NsjcV//X8cOHnVAkO05OX9sGg1k2FihMExefrBImkGNFpfLQvNHCUY0sY18JeRPmQ2TFoQy7aqLzlYFbJ02XVc6vew1WlXstD2yanpEzOiUeuSZ3ckXvSIJy8kFfyTj6cN2fifDpf89KCk/eckAU4378IX6q+</latexit><latexit sha1_base64="anyJAjrQuIiZ6P4ALnsGa7gR31s=">AAACMXicbVA9SwNBEN2L3/ErammzJAg2hjsRTCMEbCwVjAkkIext5pIle3vH7pwQjuv9NTYW+lfSia1/wNJNcoVJfLDweG9mdub5sRQGXXfiFNbWNza3tneKu3v7B4elo+MnEyWaQ4NHMtItnxmQQkEDBUpoxRpY6Eto+qPbqd98Bm1EpB5xHEM3ZAMlAsEZWqlXKncAGb2hF7QTaMbTTj/CtJllc/KQ9YZZr1Rxq+4MdJV4OamQHPe90o9t5kkICrlkxrQ9N8ZuyjQKLiErdhIDMeMjNoC2pYqFYLrp7JaMnlmlT4NI26eQztS/HSkLjRmHvq0MGQ7NsjcV//X8cOHnVAkO05OX9sGg1k2FihMExefrBImkGNFpfLQvNHCUY0sY18JeRPmQ2TFoQy7aqLzlYFbJ02XVc6vew1WlXstD2yanpEzOiUeuSZ3ckXvSIJy8kFfyTj6cN2fifDpf89KCk/eckAU4378IX6q+</latexit><latexit sha1_base64="anyJAjrQuIiZ6P4ALnsGa7gR31s=">AAACMXicbVA9SwNBEN2L3/ErammzJAg2hjsRTCMEbCwVjAkkIext5pIle3vH7pwQjuv9NTYW+lfSia1/wNJNcoVJfLDweG9mdub5sRQGXXfiFNbWNza3tneKu3v7B4elo+MnEyWaQ4NHMtItnxmQQkEDBUpoxRpY6Eto+qPbqd98Bm1EpB5xHEM3ZAMlAsEZWqlXKncAGb2hF7QTaMbTTj/CtJllc/KQ9YZZr1Rxq+4MdJV4OamQHPe90o9t5kkICrlkxrQ9N8ZuyjQKLiErdhIDMeMjNoC2pYqFYLrp7JaMnlmlT4NI26eQztS/HSkLjRmHvq0MGQ7NsjcV//X8cOHnVAkO05OX9sGg1k2FihMExefrBImkGNFpfLQvNHCUY0sY18JeRPmQ2TFoQy7aqLzlYFbJ02XVc6vew1WlXstD2yanpEzOiUeuSZ3ckXvSIJy8kFfyTj6cN2fifDpf89KCk/eckAU4378IX6q+</latexit>
= 1 + Q ˙ c Q ˙ h
<latexit sha1_base64="N+sS3/SFxfFLuk3LRJa6erdeEKk=">AAACMHicbVDLSgMxFM34rPU16tJNtAiCUGZEsBuh4MZlC/YB7TBk0kwbmmSGJCOUYdZ+jRsX+iu6Erd+gUsz7SC29UDgcM69N/eeIGZUacd5t1ZW19Y3Nktb5e2d3b19++CwraJEYtLCEYtkN0CKMCpIS1PNSDeWBPGAkU4wvs39zgORikbiXk9i4nE0FDSkGGkj+fbJDXThBeyHEuG0P4h02sx8nP3SUebbFafqTAGXiVuQCijQ8O1v04wTToTGDCnVc51YeymSmmJGsnI/USRGeIyGpGeoQJwoL52eksEzowxgGEnzhIZT9W9HirhSEx6YSo70SC16ufivF/C5n1NBMcmPXthHhzUvpSJONBF4tk6YMKgjmKcHB1QSrNnEEIQlNRdBPEJmjDYZl01U7mIwy6R9WXWdqtu8qtRrRWglcAxOwTlwwTWogzvQAC2AwSN4Ai/g1Xq23qwP63NWumIVPUdgDtbXD6/fqgk=</latexit><latexit sha1_base64="N+sS3/SFxfFLuk3LRJa6erdeEKk=">AAACMHicbVDLSgMxFM34rPU16tJNtAiCUGZEsBuh4MZlC/YB7TBk0kwbmmSGJCOUYdZ+jRsX+iu6Erd+gUsz7SC29UDgcM69N/eeIGZUacd5t1ZW19Y3Nktb5e2d3b19++CwraJEYtLCEYtkN0CKMCpIS1PNSDeWBPGAkU4wvs39zgORikbiXk9i4nE0FDSkGGkj+fbJDXThBeyHEuG0P4h02sx8nP3SUebbFafqTAGXiVuQCijQ8O1v04wTToTGDCnVc51YeymSmmJGsnI/USRGeIyGpGeoQJwoL52eksEzowxgGEnzhIZT9W9HirhSEx6YSo70SC16ufivF/C5n1NBMcmPXthHhzUvpSJONBF4tk6YMKgjmKcHB1QSrNnEEIQlNRdBPEJmjDYZl01U7mIwy6R9WXWdqtu8qtRrRWglcAxOwTlwwTWogzvQAC2AwSN4Ai/g1Xq23qwP63NWumIVPUdgDtbXD6/fqgk=</latexit><latexit sha1_base64="N+sS3/SFxfFLuk3LRJa6erdeEKk=">AAACMHicbVDLSgMxFM34rPU16tJNtAiCUGZEsBuh4MZlC/YB7TBk0kwbmmSGJCOUYdZ+jRsX+iu6Erd+gUsz7SC29UDgcM69N/eeIGZUacd5t1ZW19Y3Nktb5e2d3b19++CwraJEYtLCEYtkN0CKMCpIS1PNSDeWBPGAkU4wvs39zgORikbiXk9i4nE0FDSkGGkj+fbJDXThBeyHEuG0P4h02sx8nP3SUebbFafqTAGXiVuQCijQ8O1v04wTToTGDCnVc51YeymSmmJGsnI/USRGeIyGpGeoQJwoL52eksEzowxgGEnzhIZT9W9HirhSEx6YSo70SC16ufivF/C5n1NBMcmPXthHhzUvpSJONBF4tk6YMKgjmKcHB1QSrNnEEIQlNRdBPEJmjDYZl01U7mIwy6R9WXWdqtu8qtRrRWglcAxOwTlwwTWogzvQAC2AwSN4Ai/g1Xq23qwP63NWumIVPUdgDtbXD6/fqgk=</latexit><latexit sha1_base64="N+sS3/SFxfFLuk3LRJa6erdeEKk=">AAACMHicbVDLSgMxFM34rPU16tJNtAiCUGZEsBuh4MZlC/YB7TBk0kwbmmSGJCOUYdZ+jRsX+iu6Erd+gUsz7SC29UDgcM69N/eeIGZUacd5t1ZW19Y3Nktb5e2d3b19++CwraJEYtLCEYtkN0CKMCpIS1PNSDeWBPGAkU4wvs39zgORikbiXk9i4nE0FDSkGGkj+fbJDXThBeyHEuG0P4h02sx8nP3SUebbFafqTAGXiVuQCijQ8O1v04wTToTGDCnVc51YeymSmmJGsnI/USRGeIyGpGeoQJwoL52eksEzowxgGEnzhIZT9W9HirhSEx6YSo70SC16ufivF/C5n1NBMcmPXthHhzUvpSJONBF4tk6YMKgjmKcHB1QSrNnEEIQlNRdBPEJmjDYZl01U7mIwy6R9WXWdqtu8qtRrRWglcAxOwTlwwTWogzvQAC2AwSN4Ai/g1Xq23qwP63NWumIVPUdgDtbXD6/fqgk=</latexit>
= 1 T c T h
T c Q ˙ h
⌃ ˙
<latexit sha1_base64="zoyiwjpqIjMkxGdxPybmWeaw8J0=">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</latexit><latexit sha1_base64="zoyiwjpqIjMkxGdxPybmWeaw8J0=">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</latexit><latexit sha1_base64="zoyiwjpqIjMkxGdxPybmWeaw8J0=">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</latexit><latexit sha1_base64="zoyiwjpqIjMkxGdxPybmWeaw8J0=">AAACSXicbVDLSgMxFM20Pmp9jbp0EyyCG8uMCBZBKLhx2WJf0JaSSTNtaJIZkoxQhvkXv8aNC936Ge6KKzPTWdjWAwmHc+5N7j1eyKjSjvNlFYpb2zu7pb3y/sHh0bF9ctpRQSQxaeOABbLnIUUYFaStqWakF0qCuMdI15s9pn73hUhFA9HS85AMOZoI6lOMtJFG9v0DdOE1HPgS4bg1wom5psmqMhgHOm4mqZ7RwTOdcJSM7IpTdTLATeLmpAJyNEb2wrTjiBOhMUNK9V0n1MMYSU0xI0l5ECkSIjxDE9I3VCBO1DDOdkzgpVHG0A+kOULDTP3bESOu1Jx7ppIjPVXrXir+63l85edYUEzS3dfm0X5tGFMRRpoIvBzHjxjUAUxjhWMqCdZsbgjCkpqNIJ4i84w24ZdNVO56MJukc1N1narbvK3Ua3loJXAOLsAVcMEdqIMn0ABtgMEreAMf4NN6t76thfWzLC1Yec8ZWEGh+AvLJLL8</latexit>
dU
dt = ˙ Q h + ˙ Q c + ˙ W
<latexit sha1_base64="m9rGqCqlCF53zpwWHPIZvcaTkfI=">AAACQHicbZDNSgMxEMezftb6VfXoJVgEQSi7ItiLUPDisQW3LbRlyWazbWg2uySzQln2OXwaLx70GXwDb+JNPJl+KLZ1IOHHf2aSmb+fCK7Btl+tldW19Y3NwlZxe2d3b790cNjUcaooc2ksYtX2iWaCS+YCB8HaiWIk8gVr+cObcb51z5TmsbyDUcJ6EelLHnJKwEheyemGitAswG5uLsjxNe4GMWSN3Bvg81+mP9zKvVLZrtiTwMvgzKCMZlH3Sp+mk6YRk0AF0brj2An0MqKAU8HyYjfVLCF0SPqsY1CSiOleNlktx6dGCXAYK3Mk4In6tyMjkdajyDeVEYGBXsyNxX9zfjT3cyY5ZWMrFuaBsNrLuExSYJJOxwlTgSHGYzdxwBWjIEYGCFXcbITpgJhnwHheNFY5i8YsQ/Oi4tgVp3FZrlVnphXQMTpBZ8hBV6iGblEduYiiB/SIntGL9WS9We/Wx7R0xZr1HKG5sL6+AX7Hr8Y=</latexit><latexit sha1_base64="m9rGqCqlCF53zpwWHPIZvcaTkfI=">AAACQHicbZDNSgMxEMezftb6VfXoJVgEQSi7ItiLUPDisQW3LbRlyWazbWg2uySzQln2OXwaLx70GXwDb+JNPJl+KLZ1IOHHf2aSmb+fCK7Btl+tldW19Y3NwlZxe2d3b790cNjUcaooc2ksYtX2iWaCS+YCB8HaiWIk8gVr+cObcb51z5TmsbyDUcJ6EelLHnJKwEheyemGitAswG5uLsjxNe4GMWSN3Bvg81+mP9zKvVLZrtiTwMvgzKCMZlH3Sp+mk6YRk0AF0brj2An0MqKAU8HyYjfVLCF0SPqsY1CSiOleNlktx6dGCXAYK3Mk4In6tyMjkdajyDeVEYGBXsyNxX9zfjT3cyY5ZWMrFuaBsNrLuExSYJJOxwlTgSHGYzdxwBWjIEYGCFXcbITpgJhnwHheNFY5i8YsQ/Oi4tgVp3FZrlVnphXQMTpBZ8hBV6iGblEduYiiB/SIntGL9WS9We/Wx7R0xZr1HKG5sL6+AX7Hr8Y=</latexit><latexit sha1_base64="m9rGqCqlCF53zpwWHPIZvcaTkfI=">AAACQHicbZDNSgMxEMezftb6VfXoJVgEQSi7ItiLUPDisQW3LbRlyWazbWg2uySzQln2OXwaLx70GXwDb+JNPJl+KLZ1IOHHf2aSmb+fCK7Btl+tldW19Y3NwlZxe2d3b790cNjUcaooc2ksYtX2iWaCS+YCB8HaiWIk8gVr+cObcb51z5TmsbyDUcJ6EelLHnJKwEheyemGitAswG5uLsjxNe4GMWSN3Bvg81+mP9zKvVLZrtiTwMvgzKCMZlH3Sp+mk6YRk0AF0brj2An0MqKAU8HyYjfVLCF0SPqsY1CSiOleNlktx6dGCXAYK3Mk4In6tyMjkdajyDeVEYGBXsyNxX9zfjT3cyY5ZWMrFuaBsNrLuExSYJJOxwlTgSHGYzdxwBWjIEYGCFXcbITpgJhnwHheNFY5i8YsQ/Oi4tgVp3FZrlVnphXQMTpBZ8hBV6iGblEduYiiB/SIntGL9WS9We/Wx7R0xZr1HKG5sL6+AX7Hr8Y=</latexit><latexit sha1_base64="m9rGqCqlCF53zpwWHPIZvcaTkfI=">AAACQHicbZDNSgMxEMezftb6VfXoJVgEQSi7ItiLUPDisQW3LbRlyWazbWg2uySzQln2OXwaLx70GXwDb+JNPJl+KLZ1IOHHf2aSmb+fCK7Btl+tldW19Y3NwlZxe2d3b790cNjUcaooc2ksYtX2iWaCS+YCB8HaiWIk8gVr+cObcb51z5TmsbyDUcJ6EelLHnJKwEheyemGitAswG5uLsjxNe4GMWSN3Bvg81+mP9zKvVLZrtiTwMvgzKCMZlH3Sp+mk6YRk0AF0brj2An0MqKAU8HyYjfVLCF0SPqsY1CSiOleNlktx6dGCXAYK3Mk4In6tyMjkdajyDeVEYGBXsyNxX9zfjT3cyY5ZWMrFuaBsNrLuExSYJJOxwlTgSHGYzdxwBWjIEYGCFXcbITpgJhnwHheNFY5i8YsQ/Oi4tgVp3FZrlVnphXQMTpBZ8hBV6iGblEduYiiB/SIntGL9WS9We/Wx7R0xZr1HKG5sL6+AX7Hr8Y=</latexit>
dS
dt = ˙ ⌃ + Q ˙ h
T h + Q ˙ c T c
<latexit sha1_base64="ZwYufrxioWr2JgaB+SqvKllUaUE=">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</latexit><latexit sha1_base64="ZwYufrxioWr2JgaB+SqvKllUaUE=">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</latexit><latexit sha1_base64="ZwYufrxioWr2JgaB+SqvKllUaUE=">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</latexit><latexit sha1_base64="ZwYufrxioWr2JgaB+SqvKllUaUE=">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</latexit>
= 0
<latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit><latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit><latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit><latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit>
= 0
<latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit><latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit><latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit><latexit sha1_base64="2hr/yq/WZcT4XnLqM/HCrzR22Zc=">AAACEHicbVDLSgMxFL1TX7W+qi7dBIvgqsyIYDdCwY3LivYB7VAy6Z02NJMZkoxQhn6CGxf6K+7ErX/gn7g0bWdhqwcCh3PuTU5OkAiujet+OYW19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0f+2SfrniVt05yF/i5aQCORr98ndvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQu5ZKGqH2s3nUKTmzyoCEsbJHGjJXf29kNNJ6EgV2MqJmpFe9mfivF0RLL2eSMwwVZSt5TFjzMy6T1KBkizhhKoiJyawdMuAKmRETSyhT3P6IsBG11xjbYclW5a0W85e0LqqeW/XuLiv1Wl5aEU7gFM7Bgyuowy00oAkMhvAEL/DqPDtvzrvzsRgtOPnOMSzB+fwBrK2c9w==</latexit>
Carnot’s statement of the 2nd law
“The efficiency of a quasi-static or reversible Carnot
cycle depends only on the temperatures of the two heat
reservoirs, and is the same, whatever the working
substance. A Carnot engine operated in this way is the
most efficient possible heat engine using those two
temperatures.”
Flow of heat
But if there is no work involved,
⌃ ˙ = Q ˙ h T h
Q ˙ c
T c 0
<latexit sha1_base64="dDIFiznvngOyq9en/doD4FAihDQ=">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</latexit><latexit sha1_base64="dDIFiznvngOyq9en/doD4FAihDQ=">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</latexit><latexit sha1_base64="dDIFiznvngOyq9en/doD4FAihDQ=">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</latexit><latexit sha1_base64="dDIFiznvngOyq9en/doD4FAihDQ=">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</latexit>
The 2nd law reads
Q
<latexit sha1_base64="0YKtE+SJn+mYskpqbr2CBLdLSeE=">AAACJnicbVDLSgMxFM3UV62vURcu3ASL4MYyI4LdCAU3LluwD2iHkknvtKGZzJBkhDLM17hxob/iTsSdn+HStB3Btl4InJxzT3Lv8WPOlHacT6uwtr6xuVXcLu3s7u0f2IdHLRUlkkKTRjySHZ8o4ExAUzPNoRNLIKHPoe2P76Z6+xGkYpF40JMYvJAMBQsYJdpQffukN4h02sj6FN/iS/x7G/XtslNxZoVXgZuDMsqr3re/jZcmIQhNOVGq6zqx9lIiNaMcslIvURATOiZD6BooSAjKS2cLZPjcMAMcRNIcofGM/etISajUJPRNZ0j0SC1rU/JfzQ8Xfk4FoxBIQpfm0UHVS5mIEw2CzscJEo51hKeZ4QGTQDWfGECoZGYjTEfEPKNNsiUTlbsczCpoXVVcp+I2rsu1ah5aEZ2iM3SBXHSDauge1VETUZShJ/SCXq1n6816tz7mrQUr9xyjhbK+fgAXb6WO</latexit><latexit sha1_base64="0YKtE+SJn+mYskpqbr2CBLdLSeE=">AAACJnicbVDLSgMxFM3UV62vURcu3ASL4MYyI4LdCAU3LluwD2iHkknvtKGZzJBkhDLM17hxob/iTsSdn+HStB3Btl4InJxzT3Lv8WPOlHacT6uwtr6xuVXcLu3s7u0f2IdHLRUlkkKTRjySHZ8o4ExAUzPNoRNLIKHPoe2P76Z6+xGkYpF40JMYvJAMBQsYJdpQffukN4h02sj6FN/iS/x7G/XtslNxZoVXgZuDMsqr3re/jZcmIQhNOVGq6zqx9lIiNaMcslIvURATOiZD6BooSAjKS2cLZPjcMAMcRNIcofGM/etISajUJPRNZ0j0SC1rU/JfzQ8Xfk4FoxBIQpfm0UHVS5mIEw2CzscJEo51hKeZ4QGTQDWfGECoZGYjTEfEPKNNsiUTlbsczCpoXVVcp+I2rsu1ah5aEZ2iM3SBXHSDauge1VETUZShJ/SCXq1n6816tz7mrQUr9xyjhbK+fgAXb6WO</latexit><latexit sha1_base64="0YKtE+SJn+mYskpqbr2CBLdLSeE=">AAACJnicbVDLSgMxFM3UV62vURcu3ASL4MYyI4LdCAU3LluwD2iHkknvtKGZzJBkhDLM17hxob/iTsSdn+HStB3Btl4InJxzT3Lv8WPOlHacT6uwtr6xuVXcLu3s7u0f2IdHLRUlkkKTRjySHZ8o4ExAUzPNoRNLIKHPoe2P76Z6+xGkYpF40JMYvJAMBQsYJdpQffukN4h02sj6FN/iS/x7G/XtslNxZoVXgZuDMsqr3re/jZcmIQhNOVGq6zqx9lIiNaMcslIvURATOiZD6BooSAjKS2cLZPjcMAMcRNIcofGM/etISajUJPRNZ0j0SC1rU/JfzQ8Xfk4FoxBIQpfm0UHVS5mIEw2CzscJEo51hKeZ4QGTQDWfGECoZGYjTEfEPKNNsiUTlbsczCpoXVVcp+I2rsu1ah5aEZ2iM3SBXHSDauge1VETUZShJ/SCXq1n6816tz7mrQUr9xyjhbK+fgAXb6WO</latexit><latexit sha1_base64="0YKtE+SJn+mYskpqbr2CBLdLSeE=">AAACJnicbVDLSgMxFM3UV62vURcu3ASL4MYyI4LdCAU3LluwD2iHkknvtKGZzJBkhDLM17hxob/iTsSdn+HStB3Btl4InJxzT3Lv8WPOlHacT6uwtr6xuVXcLu3s7u0f2IdHLRUlkkKTRjySHZ8o4ExAUzPNoRNLIKHPoe2P76Z6+xGkYpF40JMYvJAMBQsYJdpQffukN4h02sj6FN/iS/x7G/XtslNxZoVXgZuDMsqr3re/jZcmIQhNOVGq6zqx9lIiNaMcslIvURATOiZD6BooSAjKS2cLZPjcMAMcRNIcofGM/etISajUJPRNZ0j0SC1rU/JfzQ8Xfk4FoxBIQpfm0UHVS5mIEw2CzscJEo51hKeZ4QGTQDWfGECoZGYjTEfEPKNNsiUTlbsczCpoXVVcp+I2rsu1ah5aEZ2iM3SBXHSDauge1VETUZShJ/SCXq1n6816tz7mrQUr9xyjhbK+fgAXb6WO</latexit>˙ c = Q ˙ h
) ⌃ ˙ =
✓ 1 T c
1 T h
◆
Q ˙ h 0
<latexit sha1_base64="VIpikKv/0e+UCRi3Mr8kfAEZdDw=">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</latexit><latexit sha1_base64="VIpikKv/0e+UCRi3Mr8kfAEZdDw=">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</latexit><latexit sha1_base64="VIpikKv/0e+UCRi3Mr8kfAEZdDw=">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</latexit><latexit sha1_base64="VIpikKv/0e+UCRi3Mr8kfAEZdDw=">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</latexit>
Heat flows from hot to cold.
Clausius’ statement of the 2nd law
“Heat can never pass from a colder to a warmer body
without some other change, connected therewith,
occurring at the same time.”
Work from a single bath
Finally, suppose there is only one bath present:
W
<latexit sha1_base64="fwMsM9emVGF5XkVhVh+egF4UyQM=">AAACJHicbVDLSgMxFM34rPU1Kq7cBIvgxjIjgt0IBTcuW7APaIeSSe+0oZnMkGSEMszHuHGhv+JOXLjxO1yaTmdhWw8EDufck9wcP+ZMacf5stbWNza3tks75d29/YND++i4raJEUmjRiEey6xMFnAloaaY5dGMJJPQ5dPzJ/czvPIFULBKPehqDF5KRYAGjRBtpYJ/2h5FOOxm+w1c4581sMB7YFafq5MCrxC1IBRVoDOwfk6VJCEJTTpTquU6svZRIzSiHrNxPFMSETsgIeoYKEoLy0nz9DF8YZYiDSJojNM7Vv4mUhEpNQ99MhkSP1bI3E//1/HDh5VQwCoEkdGkfHdS8lIk40SDofJ0g4VhHeNYYHjIJVPOpIYRKZn6E6ZiYa7TptWyqcpeLWSXt66rrVN3mTaVeK0oroTN0ji6Ri25RHT2gBmohilL0jF7Rm/VivVsf1ud8dM0qMidoAdb3L4NcpL4=</latexit><latexit sha1_base64="fwMsM9emVGF5XkVhVh+egF4UyQM=">AAACJHicbVDLSgMxFM34rPU1Kq7cBIvgxjIjgt0IBTcuW7APaIeSSe+0oZnMkGSEMszHuHGhv+JOXLjxO1yaTmdhWw8EDufck9wcP+ZMacf5stbWNza3tks75d29/YND++i4raJEUmjRiEey6xMFnAloaaY5dGMJJPQ5dPzJ/czvPIFULBKPehqDF5KRYAGjRBtpYJ/2h5FOOxm+w1c4581sMB7YFafq5MCrxC1IBRVoDOwfk6VJCEJTTpTquU6svZRIzSiHrNxPFMSETsgIeoYKEoLy0nz9DF8YZYiDSJojNM7Vv4mUhEpNQ99MhkSP1bI3E//1/HDh5VQwCoEkdGkfHdS8lIk40SDofJ0g4VhHeNYYHjIJVPOpIYRKZn6E6ZiYa7TptWyqcpeLWSXt66rrVN3mTaVeK0oroTN0ji6Ri25RHT2gBmohilL0jF7Rm/VivVsf1ud8dM0qMidoAdb3L4NcpL4=</latexit><latexit sha1_base64="fwMsM9emVGF5XkVhVh+egF4UyQM=">AAACJHicbVDLSgMxFM34rPU1Kq7cBIvgxjIjgt0IBTcuW7APaIeSSe+0oZnMkGSEMszHuHGhv+JOXLjxO1yaTmdhWw8EDufck9wcP+ZMacf5stbWNza3tks75d29/YND++i4raJEUmjRiEey6xMFnAloaaY5dGMJJPQ5dPzJ/czvPIFULBKPehqDF5KRYAGjRBtpYJ/2h5FOOxm+w1c4581sMB7YFafq5MCrxC1IBRVoDOwfk6VJCEJTTpTquU6svZRIzSiHrNxPFMSETsgIeoYKEoLy0nz9DF8YZYiDSJojNM7Vv4mUhEpNQ99MhkSP1bI3E//1/HDh5VQwCoEkdGkfHdS8lIk40SDofJ0g4VhHeNYYHjIJVPOpIYRKZn6E6ZiYa7TptWyqcpeLWSXt66rrVN3mTaVeK0oroTN0ji6Ri25RHT2gBmohilL0jF7Rm/VivVsf1ud8dM0qMidoAdb3L4NcpL4=</latexit><latexit sha1_base64="fwMsM9emVGF5XkVhVh+egF4UyQM=">AAACJHicbVDLSgMxFM34rPU1Kq7cBIvgxjIjgt0IBTcuW7APaIeSSe+0oZnMkGSEMszHuHGhv+JOXLjxO1yaTmdhWw8EDufck9wcP+ZMacf5stbWNza3tks75d29/YND++i4raJEUmjRiEey6xMFnAloaaY5dGMJJPQ5dPzJ/czvPIFULBKPehqDF5KRYAGjRBtpYJ/2h5FOOxm+w1c4581sMB7YFafq5MCrxC1IBRVoDOwfk6VJCEJTTpTquU6svZRIzSiHrNxPFMSETsgIeoYKEoLy0nz9DF8YZYiDSJojNM7Vv4mUhEpNQ99MhkSP1bI3E//1/HDh5VQwCoEkdGkfHdS8lIk40SDofJ0g4VhHeNYYHjIJVPOpIYRKZn6E6ZiYa7TptWyqcpeLWSXt66rrVN3mTaVeK0oroTN0ji6Ri25RHT2gBmohilL0jF7Rm/VivVsf1ud8dM0qMidoAdb3L4NcpL4=</latexit>˙ = Q ˙ h
⌃ ˙ = Q ˙ h
T h = W ˙
T h 0
<latexit sha1_base64="24qCmoLLVuWTTXy8PgHzvTHxfXo=">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</latexit><latexit sha1_base64="24qCmoLLVuWTTXy8PgHzvTHxfXo=">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</latexit><latexit sha1_base64="24qCmoLLVuWTTXy8PgHzvTHxfXo=">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</latexit><latexit sha1_base64="24qCmoLLVuWTTXy8PgHzvTHxfXo=">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</latexit>
Positive work (in my definition) means an external agent is doing work on
the system.
Kelvin-Planck statement of the 2nd law
“It is impossible to devise a cyclically operating device,
the sole effect of which is to absorb energy in the form of
heat from a single thermal reservoir and to deliver an
equivalent amount of work.”
Fluctuation theorems
Collin, et. al., Nature, 437 (2005)
Fluctuation theorems
Fluctuation theorems describe the stochastic behavior of the entropy production:
The most famous one is the Crooks fluctuation theorem:
P F (W )
P B ( W ) = e (W F )
<latexit sha1_base64="gqBZY7dIOvPO5eoqFyJJZjzMFWo=">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</latexit><latexit sha1_base64="gqBZY7dIOvPO5eoqFyJJZjzMFWo=">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</latexit><latexit sha1_base64="gqBZY7dIOvPO5eoqFyJJZjzMFWo=">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</latexit><latexit sha1_base64="gqBZY7dIOvPO5eoqFyJJZjzMFWo=">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</latexit>
P F (⌃)
P B (⌃) = e ⌃
<latexit sha1_base64="HmbQ8LlclIaKYRpbf+DjRG+7AYc=">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</latexit><latexit sha1_base64="HmbQ8LlclIaKYRpbf+DjRG+7AYc=">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</latexit><latexit sha1_base64="HmbQ8LlclIaKYRpbf+DjRG+7AYc=">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</latexit><latexit sha1_base64="HmbQ8LlclIaKYRpbf+DjRG+7AYc=">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</latexit>
Collin, et. al., Nature, 437 (2005)
Batalhão, et. al., Phys. Rev. Lett. 113 (2014).
Exchange fluctuation theorem
Jarzynski and Wójcik Phys. Rev. Lett. 92, 230602 (2004)
The heat exchange between two bodies satisfies a
stronger fluctuation theorem
P (Q)
P ( Q) = e Q
<latexit sha1_base64="zoLgeJdKnAq9rSJ07krU8ztcpOY=">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</latexit><latexit sha1_base64="zoLgeJdKnAq9rSJ07krU8ztcpOY=">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</latexit><latexit sha1_base64="zoLgeJdKnAq9rSJ07krU8ztcpOY=">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</latexit><latexit sha1_base64="zoLgeJdKnAq9rSJ07krU8ztcpOY=">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</latexit>
Here it’s the same P(Q) that appears on the numerator and denominator.
����
����
����
J. R. Gomes-Solano Phys. Rev. Lett. 106, 200602 (2011)
Thermodynamic Uncertainty
Relations
Thermodynamic Uncertainty Relations (TURs)
A. C. Barato, U. Seifert, “Thermodynamic Uncertainty Relation for Biomolecular Processes”, Physical Review Letters, 114, 158101 (2015)
var( Q ) hQi 2
2 h ⌃ i
<latexit sha1_base64="OBZ99n17aysDZ4Uep+UtHElhdSE=">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</latexit><latexit sha1_base64="OBZ99n17aysDZ4Uep+UtHElhdSE=">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</latexit><latexit sha1_base64="OBZ99n17aysDZ4Uep+UtHElhdSE=">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</latexit><latexit sha1_base64="OBZ99n17aysDZ4Uep+UtHElhdSE=">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</latexit>
• Proved for classical Markov processes.
• Physical origins are rather obscure.
• Regimes of validity?
• Quantum effects?
����
Implications for mesoscopic engines
In an autonomous engine the output power is The TUR in this case then reads
W
<latexit sha1_base64="C3nPJ6nxox17PMw82SBxutgHk7c=">AAACFHicbVBNSwMxFHxbv2r9qnr0EiyCp7Irgj0WvHisYLtCu5Rs+rYNzWaXJCuUpT/Ciwf9K97Eq3f/iUfTdg+2dSAwzLxJXiZMBdfGdb+d0sbm1vZOebeyt39weFQ9PunoJFMM2ywRiXoMqUbBJbYNNwIfU4U0DgX64fh25vtPqDRP5IOZpBjEdCh5xBk1VvJ7g8Tk/rRfrbl1dw6yTryC1KBAq1/9sUmWxSgNE1TrruemJsipMpwJnFZ6mcaUsjEdYtdSSWPUQT5fd0ourDIgUaLskYbM1b+JnMZaT+LQTsbUjPSqNxP/9cJ46eVccoaRomxlHxM1gpzLNDMo2WKdKBPEJGTWEBlwhcyIiSWUKW5/RNiI2muM7bFiq/JWi1knnau659a9++tas1GUVoYzOIdL8OAGmnAHLWgDgzE8wyu8OS/Ou/PhfC5GS06ROYUlOF+/RSWfhA==</latexit><latexit sha1_base64="C3nPJ6nxox17PMw82SBxutgHk7c=">AAACFHicbVBNSwMxFHxbv2r9qnr0EiyCp7Irgj0WvHisYLtCu5Rs+rYNzWaXJCuUpT/Ciwf9K97Eq3f/iUfTdg+2dSAwzLxJXiZMBdfGdb+d0sbm1vZOebeyt39weFQ9PunoJFMM2ywRiXoMqUbBJbYNNwIfU4U0DgX64fh25vtPqDRP5IOZpBjEdCh5xBk1VvJ7g8Tk/rRfrbl1dw6yTryC1KBAq1/9sUmWxSgNE1TrruemJsipMpwJnFZ6mcaUsjEdYtdSSWPUQT5fd0ourDIgUaLskYbM1b+JnMZaT+LQTsbUjPSqNxP/9cJ46eVccoaRomxlHxM1gpzLNDMo2WKdKBPEJGTWEBlwhcyIiSWUKW5/RNiI2muM7bFiq/JWi1knnau659a9++tas1GUVoYzOIdL8OAGmnAHLWgDgzE8wyu8OS/Ou/PhfC5GS06ROYUlOF+/RSWfhA==</latexit><latexit sha1_base64="C3nPJ6nxox17PMw82SBxutgHk7c=">AAACFHicbVBNSwMxFHxbv2r9qnr0EiyCp7Irgj0WvHisYLtCu5Rs+rYNzWaXJCuUpT/Ciwf9K97Eq3f/iUfTdg+2dSAwzLxJXiZMBdfGdb+d0sbm1vZOebeyt39weFQ9PunoJFMM2ywRiXoMqUbBJbYNNwIfU4U0DgX64fh25vtPqDRP5IOZpBjEdCh5xBk1VvJ7g8Tk/rRfrbl1dw6yTryC1KBAq1/9sUmWxSgNE1TrruemJsipMpwJnFZ6mcaUsjEdYtdSSWPUQT5fd0ourDIgUaLskYbM1b+JnMZaT+LQTsbUjPSqNxP/9cJ46eVccoaRomxlHxM1gpzLNDMo2WKdKBPEJGTWEBlwhcyIiSWUKW5/RNiI2muM7bFiq/JWi1knnau659a9++tas1GUVoYzOIdL8OAGmnAHLWgDgzE8wyu8OS/Ou/PhfC5GS06ROYUlOF+/RSWfhA==</latexit><latexit sha1_base64="C3nPJ6nxox17PMw82SBxutgHk7c=">AAACFHicbVBNSwMxFHxbv2r9qnr0EiyCp7Irgj0WvHisYLtCu5Rs+rYNzWaXJCuUpT/Ciwf9K97Eq3f/iUfTdg+2dSAwzLxJXiZMBdfGdb+d0sbm1vZOebeyt39weFQ9PunoJFMM2ywRiXoMqUbBJbYNNwIfU4U0DgX64fh25vtPqDRP5IOZpBjEdCh5xBk1VvJ7g8Tk/rRfrbl1dw6yTryC1KBAq1/9sUmWxSgNE1TrruemJsipMpwJnFZ6mcaUsjEdYtdSSWPUQT5fd0ourDIgUaLskYbM1b+JnMZaT+LQTsbUjPSqNxP/9cJ46eVccoaRomxlHxM1gpzLNDMo2WKdKBPEJGTWEBlwhcyIiSWUKW5/RNiI2muM7bFiq/JWi1knnau659a9++tas1GUVoYzOIdL8OAGmnAHLWgDgzE8wyu8OS/Ou/PhfC5GS06ROYUlOF+/RSWfhA==</latexit>˙
var( ˙ W ) h W ˙ i 2
2 h ⌃ ˙ i
<latexit sha1_base64="Eml2r0k7hKdLmd458P8MNQ+BO5o=">AAACbnicbVFNaxsxEJW3X4mbtm4LuZQSEVNwL2bXFJJjoJceU1rHgaxrZuXZjYik3UqzIUbsz+uP6G/opYcWeqxsbyBxMiB4evNmNPOUVUo6iuOfnejBw0ePn2xtd5/uPHv+ovfy1YkraytwLEpV2tMMHCppcEySFJ5WFkFnCifZxcdlfnKJ1snSfKVFhVMNhZG5FECBmvVmaW5B+JTwivwl2GaQzkvyk+Z941MFplDIWya1q+u3UcPTAr/zdeVoQ5d+kYWGa3Ez6/XjYbwKfhckLeizNo5nvX+hjag1GhIKnDtL4oqmHixJEfp109phBeICCjwL0IBGN/UrIxr+LjBznpc2HEN8xd6s8KCdW+gsKDXQudvMLcl7c5m+9bI3UuBy/Y15KD+cemmqmtCI9Th5rTiVfOk9n0uLgtQiABBWho24OIfQhsIPdYNVyaYxd8HJaJjEw+Tzh/7RYWvaFnvD9tmAJeyAHbFP7JiNmWA/2C/2h/3t/I52o7fR3loaddqa1+xWRIP/jEXBqQ==</latexit><latexit sha1_base64="Eml2r0k7hKdLmd458P8MNQ+BO5o=">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</latexit><latexit sha1_base64="Eml2r0k7hKdLmd458P8MNQ+BO5o=">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</latexit><latexit sha1_base64="Eml2r0k7hKdLmd458P8MNQ+BO5o=">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</latexit>
From our previously derived result:
⌘ = ⌘ C T c Q ˙ h
⌃ ˙ ! h ⌃ ˙ i = h Q ˙ h i
T c (⌘ C ⌘)
<latexit sha1_base64="tOw/zBbFreRwDmMSsG3+7W1YoWI=">AAACpXicbVFNj9MwEHXC11K+Chy5WFshFgFVgpDYy0or7YUL0la03ZU2VTRxJ621thPsCVIV5W/w3/gnHDjgpEHQLiNZfn7zPDN+zkolHUXRjyC8dfvO3XsH9wcPHj56/GT49NncFZUVOBOFKuxlBg6VNDgjSQovS4ugM4UX2fVZm7/4htbJwkxpU+JCw8rIXAogT6XD7wkS8BPebukZf8eT3IKop6lo6mRZUD1p0nXDO5h8kSsN/vC1giVPqPiDFJiVwh1RYrfcSV9wRzPpSm4VTdeLH/0dwIPX6XAUjaMu+E0Q92DE+jhPhz99ZVFpNCQUOHcVRyUtarAkhW8ySCqHJYhrWOGVhwY0ukXd+dfwl55Z8rywfhniHfvvjRq0cxudeaUGWrv9XEv+N5fpnc61kQJbO/bmofx4UUtTVoRGbMfJK8W9we2X8aW0KEhtPABhpX8RF2vwZch/7MBbFe8bcxPM34/jaBxPPoxOj3vTDtgLdsiOWMw+slP2iZ2zGRPsV3AYvAnehq/Cz+E0nG+lYdDfec52Ikx/A8i40Qk=</latexit><latexit sha1_base64="tOw/zBbFreRwDmMSsG3+7W1YoWI=">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</latexit><latexit sha1_base64="tOw/zBbFreRwDmMSsG3+7W1YoWI=">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</latexit><latexit sha1_base64="tOw/zBbFreRwDmMSsG3+7W1YoWI=">AAACpXicbVFNj9MwEHXC11K+Chy5WFshFgFVgpDYy0or7YUL0la03ZU2VTRxJ621thPsCVIV5W/w3/gnHDjgpEHQLiNZfn7zPDN+zkolHUXRjyC8dfvO3XsH9wcPHj56/GT49NncFZUVOBOFKuxlBg6VNDgjSQovS4ugM4UX2fVZm7/4htbJwkxpU+JCw8rIXAogT6XD7wkS8BPebukZf8eT3IKop6lo6mRZUD1p0nXDO5h8kSsN/vC1giVPqPiDFJiVwh1RYrfcSV9wRzPpSm4VTdeLH/0dwIPX6XAUjaMu+E0Q92DE+jhPhz99ZVFpNCQUOHcVRyUtarAkhW8ySCqHJYhrWOGVhwY0ukXd+dfwl55Z8rywfhniHfvvjRq0cxudeaUGWrv9XEv+N5fpnc61kQJbO/bmofx4UUtTVoRGbMfJK8W9we2X8aW0KEhtPABhpX8RF2vwZch/7MBbFe8bcxPM34/jaBxPPoxOj3vTDtgLdsiOWMw+slP2iZ2zGRPsV3AYvAnehq/Cz+E0nG+lYdDfec52Ikx/A8i40Qk=</latexit>
Thus:
var( ˙ W ) h W ˙ i 2
2T c h Q ˙ h i
1
⌘ C ⌘
<latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">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</latexit><latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">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</latexit><latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">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</latexit><latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">AAACh3icbVFNT9tAEF27QCF8pe2xl1WjSnAg2FHVcqTi0iNIhCDhYI0342TFeu3ujhHRyj+0l/6OHrsOPpTQkVZ6++bNd1YpaSmKfgXhm43NrbfbO73dvf2Dw/679ze2rI3AsShVaW4zsKikxjFJUnhbGYQiUzjJHi5a/+QRjZWlvqZlhdMC5lrmUgB5Ku0/JbkB4RLCJ3KPYJqjZFaSmzTHjUsU6LlC3jGJWX3vRw1P5viTP0eOrlOxJr1q0gXv1E0ni70GCdILfsJb0KT9QTSMVsZfg7gDA9bZZdr/45OLukBNQoG1d3FU0dSBISl8nV5SW6xAPMAc7zzUUKCdutWGGv7ZMzOel8Y/TXzF/hvhoLB2WWReWQAt7LqvJf/ry4oXlZ2WAtuB1/qh/GzqpK5qQi2e28lrxank7VH4TBoUpJYegDDST8TFAnwa8qfr+VXF64t5DW5GwzgaxldfBudn3dK22Uf2iR2xmH1j5+wHu2RjJtjvYCPYDw7CnfA0/Bp22jDoYj6wFxZ+/wuzAccc</latexit>
P. Pietzonka and U. Seifert, Phys. Rev. Lett., 120, 190602 (2017)
Finally, we note that . Whence ⌘ = h W ˙ i
h Q ˙ h i
<latexit sha1_base64="Xk/453P2pRUT2HlMZXbrGZ6E5Fs=">AAACUHicbVFNSwMxEJ2tX7V+VT16CRbBi2VXBL0IihePCtYKbinZdLYNZrNLMiuUpb/HX+PFg+I/6U3TWsVWB0Je3ptMZl6iTElLvv/ulebmFxaXysuVldW19Y3q5tatTXMjsCFSlZq7iFtUUmODJCm8ywzyJFLYjB4uRnrzEY2Vqb6hfoathHe1jKXg5Kh29TxE4uyUHYSx4aIIFdddhSzspFQ0Byw04/NgWrgetHs/Urta8+v+ONhfEExADSZx1a4OXRGRJ6hJKG7tfeBn1Cq4ISlcvUqYW8y4eOBdvHdQ8wRtqxiPOmB7jumwODVuaWJj9veNgifW9pPIZSacenZWG5H/alEy9XKhpcCRJTP9UHzSKqTOckItvtqJc8UoZSN3WUcaFKT6DnBhpJuIiR53Zcj9QcVZFcwa8xfcHtYDvx5cH9XOTiamlWEHdmEfAjiGM7iEK2iAgCd4hld48168ofdR8r5Sv3fYhqkoVT4BD3a1ng==</latexit><latexit sha1_base64="Xk/453P2pRUT2HlMZXbrGZ6E5Fs=">AAACUHicbVFNSwMxEJ2tX7V+VT16CRbBi2VXBL0IihePCtYKbinZdLYNZrNLMiuUpb/HX+PFg+I/6U3TWsVWB0Je3ptMZl6iTElLvv/ulebmFxaXysuVldW19Y3q5tatTXMjsCFSlZq7iFtUUmODJCm8ywzyJFLYjB4uRnrzEY2Vqb6hfoathHe1jKXg5Kh29TxE4uyUHYSx4aIIFdddhSzspFQ0Byw04/NgWrgetHs/Urta8+v+ONhfEExADSZx1a4OXRGRJ6hJKG7tfeBn1Cq4ISlcvUqYW8y4eOBdvHdQ8wRtqxiPOmB7jumwODVuaWJj9veNgifW9pPIZSacenZWG5H/alEy9XKhpcCRJTP9UHzSKqTOckItvtqJc8UoZSN3WUcaFKT6DnBhpJuIiR53Zcj9QcVZFcwa8xfcHtYDvx5cH9XOTiamlWEHdmEfAjiGM7iEK2iAgCd4hld48168ofdR8r5Sv3fYhqkoVT4BD3a1ng==</latexit><latexit sha1_base64="Xk/453P2pRUT2HlMZXbrGZ6E5Fs=">AAACUHicbVFNSwMxEJ2tX7V+VT16CRbBi2VXBL0IihePCtYKbinZdLYNZrNLMiuUpb/HX+PFg+I/6U3TWsVWB0Je3ptMZl6iTElLvv/ulebmFxaXysuVldW19Y3q5tatTXMjsCFSlZq7iFtUUmODJCm8ywzyJFLYjB4uRnrzEY2Vqb6hfoathHe1jKXg5Kh29TxE4uyUHYSx4aIIFdddhSzspFQ0Byw04/NgWrgetHs/Urta8+v+ONhfEExADSZx1a4OXRGRJ6hJKG7tfeBn1Cq4ISlcvUqYW8y4eOBdvHdQ8wRtqxiPOmB7jumwODVuaWJj9veNgifW9pPIZSacenZWG5H/alEy9XKhpcCRJTP9UHzSKqTOckItvtqJc8UoZSN3WUcaFKT6DnBhpJuIiR53Zcj9QcVZFcwa8xfcHtYDvx5cH9XOTiamlWEHdmEfAjiGM7iEK2iAgCd4hld48168ofdR8r5Sv3fYhqkoVT4BD3a1ng==</latexit><latexit sha1_base64="Xk/453P2pRUT2HlMZXbrGZ6E5Fs=">AAACUHicbVFNSwMxEJ2tX7V+VT16CRbBi2VXBL0IihePCtYKbinZdLYNZrNLMiuUpb/HX+PFg+I/6U3TWsVWB0Je3ptMZl6iTElLvv/ulebmFxaXysuVldW19Y3q5tatTXMjsCFSlZq7iFtUUmODJCm8ywzyJFLYjB4uRnrzEY2Vqb6hfoathHe1jKXg5Kh29TxE4uyUHYSx4aIIFdddhSzspFQ0Byw04/NgWrgetHs/Urta8+v+ONhfEExADSZx1a4OXRGRJ6hJKG7tfeBn1Cq4ISlcvUqYW8y4eOBdvHdQ8wRtqxiPOmB7jumwODVuaWJj9veNgifW9pPIZSacenZWG5H/alEy9XKhpcCRJTP9UHzSKqTOckItvtqJc8UoZSN3WUcaFKT6DnBhpJuIiR53Zcj9QcVZFcwa8xfcHtYDvx5cH9XOTiamlWEHdmEfAjiGM7iEK2iAgCd4hld48168ofdR8r5Sv3fYhqkoVT4BD3a1ng==</latexit>
var( ˙ W ) 2T c |h W ˙ i| ⌘
⌘ C ⌘
<latexit sha1_base64="3BjkFBOAxcCqfbaaJeY0eilCFA8=">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</latexit><latexit sha1_base64="3BjkFBOAxcCqfbaaJeY0eilCFA8=">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</latexit><latexit sha1_base64="3BjkFBOAxcCqfbaaJeY0eilCFA8=">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</latexit><latexit sha1_base64="3BjkFBOAxcCqfbaaJeY0eilCFA8=">AAACYnicbVFNT9wwEPWmX7D9WsqRHkZdVaKHrhJUqRyRuPRIVZZFIqto4p0sFo4T7AnqKuSH9af0xIVD+w96rBNyKEtHsvz83szY85yWWjkOw5+D4NHjJ0+fbWwOn794+er1aOvNiSsqK2kqC13Y0xQdaWVoyoo1nZaWME81zdKLw1afXZF1qjDHvCppnuPSqExJZE8lo28x03eur9A2u/Gi4HrWfIB4SZewB8eJhOtYo1lqgl6MbXe8hjizKOuYGKHptuQQPkILmmQ0DidhF/AQRD0Yiz6OktEf311WORmWGp07i8KS5zVaVlJTM4wrRyXKC1zSmYcGc3Lzuhu+gfeeWUBWWL8MQ8f+W1Fj7twqT31mjnzu1rWW/K+W5vduro2S1A699h7O9ue1MmXFZOTdc7JKAxfQ+g0LZUmyXnmA0io/Echz9G3Y/8rQWxWtG/MQnOxNonASff00PtjvTdsQO+Kd2BWR+CwOxBdxJKZCih/iRvwSvwe3wTDYCrbvUoNBX7Mt7kXw9i+RYrps</latexit>
If you wish to operate the engine close to Carnot efficiency, you pay the price that the fluctuations may become very large.
To reduce fluctuations, the engine should be operated irreversibly!
Thus:
var( ˙ W ) h W ˙ i 2
2T c h Q ˙ h i
1
⌘ C ⌘
<latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">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</latexit><latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">AAACh3icbVFNT9tAEF27QCF8pe2xl1WjSnAg2FHVcqTi0iNIhCDhYI0342TFeu3ujhHRyj+0l/6OHrsOPpTQkVZ6++bNd1YpaSmKfgXhm43NrbfbO73dvf2Dw/679ze2rI3AsShVaW4zsKikxjFJUnhbGYQiUzjJHi5a/+QRjZWlvqZlhdMC5lrmUgB5Ku0/JbkB4RLCJ3KPYJqjZFaSmzTHjUsU6LlC3jGJWX3vRw1P5viTP0eOrlOxJr1q0gXv1E0ni70GCdILfsJb0KT9QTSMVsZfg7gDA9bZZdr/45OLukBNQoG1d3FU0dSBISl8nV5SW6xAPMAc7zzUUKCdutWGGv7ZMzOel8Y/TXzF/hvhoLB2WWReWQAt7LqvJf/ry4oXlZ2WAtuB1/qh/GzqpK5qQi2e28lrxank7VH4TBoUpJYegDDST8TFAnwa8qfr+VXF64t5DW5GwzgaxldfBudn3dK22Uf2iR2xmH1j5+wHu2RjJtjvYCPYDw7CnfA0/Bp22jDoYj6wFxZ+/wuzAccc</latexit><latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">AAACh3icbVFNT9tAEF27QCF8pe2xl1WjSnAg2FHVcqTi0iNIhCDhYI0342TFeu3ujhHRyj+0l/6OHrsOPpTQkVZ6++bNd1YpaSmKfgXhm43NrbfbO73dvf2Dw/679ze2rI3AsShVaW4zsKikxjFJUnhbGYQiUzjJHi5a/+QRjZWlvqZlhdMC5lrmUgB5Ku0/JbkB4RLCJ3KPYJqjZFaSmzTHjUsU6LlC3jGJWX3vRw1P5viTP0eOrlOxJr1q0gXv1E0ni70GCdILfsJb0KT9QTSMVsZfg7gDA9bZZdr/45OLukBNQoG1d3FU0dSBISl8nV5SW6xAPMAc7zzUUKCdutWGGv7ZMzOel8Y/TXzF/hvhoLB2WWReWQAt7LqvJf/ry4oXlZ2WAtuB1/qh/GzqpK5qQi2e28lrxank7VH4TBoUpJYegDDST8TFAnwa8qfr+VXF64t5DW5GwzgaxldfBudn3dK22Uf2iR2xmH1j5+wHu2RjJtjvYCPYDw7CnfA0/Bp22jDoYj6wFxZ+/wuzAccc</latexit><latexit sha1_base64="flTTNasU07mi0g2WidmkAV8LtWs=">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</latexit>