Theoretical and experiments of mesoscopic statistical thermodynamics

Quan Hai-Tao; Dong Hui; Sun Chang-Pu

doi:10.7498/aps.72.20231608

Abstract

Does thermodynamics still hold true for mecroscopic small systems with only limited degrees of freedom? Do concepts such as temperature, entropy, work done, heat transfer, isothermal processes, and the Carnot cycle remain valid? Does the thermodynamic theory for small systems need modifying or supplementing compared with traditional thermodynamics applicable to macroscopic systems? Taking a single-particle system for example, we investigate the applicability of thermodynamic concepts and laws in small systems. We have found that thermodynamic laws still hold true in small systems at an ensemble-averaged level. After considering the information erasure of the Maxwell’s demon, the second law of thermodynamics is not violated. Additionally, ‘small systems’ bring some new features. Fluctuations in thermodynamic quantities become prominent. In any process far from equilibrium, the distribution functions of thermodynamic quantities satisfy certain rigorously established identities. These identities are known as fluctuation theorems. The second law of thermodynamics can be derived from them. Therefore, fluctuation theorems can be considered an upgradation to the second law of thermodynamics. They enable physicists to obtain equilibrium properties (e.g. free energy difference) by measuring physical quantities associated with non-equilibrium processes (e.g. work distributions). Furthermore, despite some distinct quantum features, the performance of quantum heat engine does not outperform that of classical heat engine. The introduction of motion equations into small system makes the relationship between thermodynamics and mechanics closer than before. Physicists can study energy dissipation in non-equilibrium process and optimize the power and efficiency of heat engine from the first principle. These findings enrich the content of thermodynamic theory and provide new ideas for establishing a general framework for non-equilibrium thermodynamics.

Keywords:

Authors and contacts

1.
School of Physics, Peking University, Beijing 100871, China
2.
Graduate School of China Academy of Engineering Physics, Beijing 100193, China

Corresponding author: Quan Hai-Tao, htquan@pku.edu.cn ; Sun Chang-Pu, suncp@gscaep.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12088101, 11825501, 12375028, U2330401).

FullText HTML

References

[1]	Scovil H E D, Schulz-DuBois E O 1959 Phys. Rev. Lett. 2 262 Google Scholar
[2]	Geusic J E, Schulz-DuBios E O, Scovil H E D 1967 Phys. Rev. 156 343 Google Scholar
[3]	Alicki R 1979 J. Phys. A: Math. Gen. 12 L103
[4]	Kosloff R 2013 Entropy 15 2100 Google Scholar
[5]	Kieu T D 2004 Phys. Rev. Lett. 93 140403 Google Scholar
[6]	Kieu T D 2006 Eur. Phys. J. D 39 115 Google Scholar
[7]	Scully M O 2003 Science 299 862 Google Scholar
[8]	Quan H T, Zhang P, Sun C P 2006 Phys. Rev. E 73 036122 Google Scholar
[9]	Quan H T, Liu Y xi, Sun C P, Nori F 2007 Phys. Rev. E 76 031105 Google Scholar
[10]	Binder F, Correa L A, Gogolin C, Anders J, Adesso G 2018 Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions (Cham: Springer International Publishing
[11]	Callen H B 1985 Thermodynamics and An Introduction to Thermostatistics (2nd Ed.) (New York: J. Wiley & Sons
[12]	Schrödinger E 1989 Statistical Thermodynamics (New York: Dover Publications
[13]	Reif F 2009 Fundamentals of Statistical and Thermal Physics (Waveland Press
[14]	Hill T L 2013 Thermodynamics of Small Systems (New York: Dover Publications, Inc
[15]	Allahverdyan A E, Serral Gracià R, Nieuwenhuizen Th M 2005 Phys. Rev. E 71 046106 Google Scholar
[16]	Leff H S, Rex A F 2003 Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Philadelphia: Institute of Physics
[17]	Szilard L 1929 Z. Physik 53 840 Google Scholar
[18]	Landauer R 1961 IBM J. Res. Dev. 5 183 Google Scholar
[19]	Bennett C H 1982 Int. J. Theor. Phys. 21 905 Google Scholar
[20]	Bérut A, Arakelyan A, Petrosyan A, Ciliberto S, Dillenschneider R, Lutz E 2012 Nature 483 187 Google Scholar
[21]	Parrondo J M R, Horowitz J M, Sagawa T 2015 Nat. Phys. 11 131 Google Scholar
[22]	Quan H T, Wang Y D, Liu Y xi, Sun C P, Nori F 2006 Phys. Rev. Lett. 97 180402 Google Scholar
[23]	孙昌璞, 全海涛 2013 物理 42 756 Sun C P, Quan H T 2013 Physics 42 756
[24]	Dong H, Xu D Z, Cai C Y, Sun C P 2011 Phys. Rev. E 83 061108 Google Scholar
[25]	Cai C Y, Dong H, Sun C P 2012 Phys. Rev. E 85 031114 Google Scholar
[26]	Mandal D, Quan H T, Jarzynski C 2013 Phys. Rev. Lett. 111 030602 Google Scholar
[27]	Sekimoto K 1997 J. Phys. Soc. Jpn. 66 1234 Google Scholar
[28]	Jarzynski C 1997 Phys. Rev. Lett. 78 2690 Google Scholar
[29]	Jarzynski C 2011 Annu. Rev. Condens. Matter Phys. 2 329 Google Scholar
[30]	Jarzynski C 1997 Phys. Rev. E 56 5018 Google Scholar
[31]	Sekimoto K 1998 Prog. Theor. Phys. Suppl. 130 17 Google Scholar
[32]	Seifert U 2012 Rep. Prog. Phys. 75 126001 Google Scholar
[33]	Oono Y, Paniconi M 1998 Prog. Theor. Phys. Suppl. 130 29 Google Scholar
[34]	Hatano T, Sasa S ichi 2001 Phys. Rev. Lett. 86 3463 Google Scholar
[35]	Sekimoto K 2010 Stochastic Energetics (Berlin Heidelberg: Springer
[36]	Jiang D Q, Qian M, Qian M P 2004 Mathematical Theory of Nonequilibrium Steady States: On the Frontier of Probability and Dynamical Systems (Berlin: Springer
[37]	Esposito M, van den Broeck C 2010 Phys. Rev. Lett. 104 090601 Google Scholar
[38]	Kim K H, Qian H 2004 Phys. Rev. Lett. 93 120602 Google Scholar
[39]	葛颢 2014 数学进展 43 161 Ge J 2014 Adv. Math. 43 161
[40]	Jarzynskia C 2008 Eur. Phys. J. B 64 331 Google Scholar
[41]	Liphardt J, Dumont S, Smith S B, Tinoco I, Bustamante C 2002 Science 296 1832 Google Scholar
[42]	Collin D, Ritort F, Jarzynski C, Smith S B, Tinoco I, Bustamante C 2005 Nature 437 231 Google Scholar
[43]	Crooks G E 1999 Phys. Rev. E 60 2721 Google Scholar
[44]	Hummer G, Szabo A 2001 Proc. Natl. Acad. Sci. USA 98 3658 Google Scholar
[45]	Jarzynski C 2000 J. Stat. Phys. 98 77 Google Scholar
[46]	Maragakis P, Spichty M, Karplus M 2008 J. Phys. Chem. B 112 6168 Google Scholar
[47]	Hoang T M, Pan R, Ahn J, Bang J, Quan H T, Li T 2018 Phys. Rev. Lett. 120 080602 Google Scholar
[48]	Jarzynski C, Wójcik D K 2004 Phys. Rev. Lett. 92 230602 Google Scholar
[49]	Chen J F, Quan H T 2023 Phys. Rev. E 107 024135 Google Scholar
[50]	Evans D J, Cohen E G D, Morriss G P 1993 Phys. Rev. Lett. 71 2401 Google Scholar
[51]	Evans D J, Searles D J 1994 Phys. Rev. E 50 1645
[52]	Gallavotti G, Cohen E G D 1995 Phys. Rev. Lett. 74 2694 Google Scholar
[53]	Gallavotti G, Cohen E G D 1995 J. Stat. Phys. 80 931 Google Scholar
[54]	Barato A C, Seifert U 2015 Phys. Rev. Lett. 114 158101 Google Scholar
[55]	Timpanaro A M, Guarnieri G, Goold J, Landi G T 2019 Phys. Rev. Lett. 123 090604 Google Scholar
[56]	Kurchan J 2000 A Quantum Fluctuation Theorem (arXiv:cond-mat/0007360
[57]	Tasaki H 2000 Jarzynski Relations for Quantum Systems and Some Applications (arXiv:cond-mat/0009244
[58]	Talkner P, Lutz E, Hänggi P 2007 Phys. Rev. E 75 050102 Google Scholar
[59]	Dorner R, Clark S R, Heaney L, Fazio R, Goold J, Vedral V 2013 Phys. Rev. Lett. 110 230601 Google Scholar
[60]	Mazzola L, De Chiara G, Paternostro M 2013 Phys. Rev. Lett. 110 230602 Google Scholar
[61]	Batalhão T B, Souza A M, Mazzola L, Auccaise R, Sarthour R S, Oliveira I S, Goold J, De Chiara G, Paternostro M, Serra R M 2014 Phys. Rev. Lett. 113 140601 Google Scholar
[62]	An S, Zhang J N, Um M, Lv D, Lu Y, Zhang J, Yin Z Q, Quan H T, Kim K 2015 Nat. Phys. 11 193 Google Scholar
[63]	Jarzynski C, Quan H T, Rahav S 2015 Phys. Rev. X 5 031038 Google Scholar
[64]	Funo K, Quan H T 2018 Phys. Rev. Lett. 121 040602 Google Scholar
[65]	Gong Z, Lan Y, Quan H T 2016 Phys. Rev. Lett. 117 180603 Google Scholar
[66]	Fei Z, Zhang J, Pan R, Qiu T, Quan H T 2019 Phys. Rev. A 99 052508 Google Scholar
[67]	Fei Z, Quan H T 2020 Phys. Rev. Lett. 124 240603 Google Scholar
[68]	Funo K, Quan H T 2018 Phys. Rev. E 98 012113 Google Scholar
[69]	Chen J F, Qiu T, Quan H T 2021 Entropy 23 1602 Google Scholar
[70]	Wang B, Zhang J, Quan H T 2018 Phys. Rev. E 97 052136 Google Scholar
[71]	Chen J, Dong H, Sun C P 2018 Phys. Rev. E 98 062119 Google Scholar
[72]	Tasaki H 1998 Phys. Rev. Lett. 80 1373 Google Scholar
[73]	Goldstein S, Lebowitz J L, Tumulka R, Zanghì N 2006 Phys. Rev. Lett. 96 050403 Google Scholar
[74]	Popescu S, Short A J, Winter A 2006 Nat. Phys. 2 754 Google Scholar
[75]	Dong H, Yang S, Liu X F, Sun C P 2007 Phys. Rev. A 76 044104 Google Scholar
[76]	Parikh M K, Wilczek F 2000 Phys. Rev. Lett. 85 5042 Google Scholar
[77]	Zhang B, Cai Q Y, You L, Zhan M S 2009 Phys. Lett. B 675 98 Google Scholar
[78]	Dong H, Cai Q Y, Liu X F, Sun C P 2014 Commun. Theor. Phys. 61 289 Google Scholar
[79]	Ma Y H, Cai Q Y, Dong H, Sun C P 2018 EPL 122 30001 Google Scholar
[80]	Srednicki M 1994 Phys. Rev. E 50 888 Google Scholar
[81]	D’Alessio L, Kafri Y, Polkovnikov A, Rigol M 2016 Adv. Phys. 65 239 Google Scholar
[82]	Yvon J 1955 Proceedings of the International Conference on the Peaceful Uses of Atomic Energy Geneva, Switzerland, August 8—20, 1955
[83]	Curzon F L, Ahlborn B 1975 Am. J. Phys. 43 22 Google Scholar
[84]	Andresen B, Salamon P, Berry R S 1984 Phys. Today 37 62 Google Scholar
[85]	Chen J C, Su S H, Su G Z 2023 Reflection and Exploration on Hot Issues in Thermodynamics and Statistical Physics (2nd Ed.) (Beijing: Science Press) [陈金灿, 苏山河, 苏国珍 2023 热力学与统计物理学热点问题思考与探索(第二版)(北京: 科学出版社) Chen J C, Su S H, Su G Z 2023 Reflection and Exploration on Hot Issues in Thermodynamics and Statistical Physics (2nd Ed.) (Beijing: Science Press)
[86]	Esposito M, Kawai R, Lindenberg K, Van den Broeck C 2010 Phys. Rev. Lett. 105 150603 Google Scholar
[87]	Ma Y H, Xu D, Dong H, Sun C P 2018 Phys. Rev. E 98 042112 Google Scholar
[88]	Ryabov A, Holubec V 2016 Phys. Rev. E 93 050101 Google Scholar
[89]	Long R, Liu W 2016 Phys. Rev. E 94 052114 Google Scholar
[90]	Chen J F, Sun C P, Dong H 2019 Phys. Rev. E 100 062140 Google Scholar
[91]	Chen J F, Sun C P, Dong H 2019 Phys. Rev. E 100 032144 Google Scholar
[92]	Ma Y H, Xu D, Dong H, Sun C P 2018 Phys. Rev. E 98 022133 Google Scholar
[93]	Cavina V, Mari A, Giovannetti V 2017 Phys. Rev. Lett. 119 050601 Google Scholar
[94]	Ma Y H, Zhai R X, Chen J, Sun C P, Dong H 2020 Phys. Rev. Lett. 125 210601 Google Scholar
[95]	Blickle V, Bechinger C 2012 Nat. Phys. 8 143 Google Scholar
[96]	Roßnagel J, Dawkins S T, Tolazzi K N, Abah O, Lutz E, Schmidt-Kaler F, Singer K 2016 Science 352 325 Google Scholar
[97]	Martínez I A, Roldán É, Dinis L, Petrov D, Parrondo J M R, Rica R A 2016 Nat. Phys. 12 67 Google Scholar
[98]	Zhai R X, Cui F M, Ma Y H, Sun C P, Dong H 2023 Phys. Rev. E 107 L042101 Google Scholar
[99]	Tu Z C 2021 Front. Phys. 16 33202 Google Scholar
[100]	Salamon P, Berry R S 1983 Phys. Rev. Lett. 51 1127 Google Scholar
[101]	Weinhold F 1975 J. Chem. Phys. 63 2479 Google Scholar
[102]	Ruppeiner G 1979 Phys. Rev. A 20 1608 Google Scholar
[103]	Crooks G E 2007 Phys. Rev. Lett. 99 100602 Google Scholar
[104]	Scandi M, Perarnau-Llobet M 2019 Quantum 3 197 Google Scholar
[105]	Abiuso P, Miller H J D, Perarnau-Llobet M, Scandi M 2020 Entropy 22 1076 Google Scholar
[106]	Chen J F, Sun C P, Dong H 2021 Phys. Rev. E 104 034117 Google Scholar
[107]	Li G, Chen J F, Sun C P, Dong H 2022 Phys. Rev. Lett. 128 230603 Google Scholar
[108]	Li G, Quan H T, Tu Z C 2017 Phys. Rev. E 96 012144 Google Scholar
[109]	Chen J F 2022 Phys. Rev. E 106 054108
[110]	Van Vu T, Hasegawa Y 2021 Phys. Rev. Lett. 126 010601 Google Scholar
[111]	Dago S, Pereda J, Barros N, Ciliberto S, Bellon L 2021 Phys. Rev. Lett. 126 170601 Google Scholar
[112]	Zhen Y Z, Egloff D, Modi K, Dahlsten O 2021 Phys. Rev. Lett. 127 190602 Google Scholar
[113]	Ma Y H, Chen J F, Sun C P, Dong H 2022 Phys. Rev. E 106 034112 Google Scholar
[114]	Tsirlin A M, Kazakov V, Zubov D V 2002 J. Phys. Chem. A 106 10926 Google Scholar
[115]	Chen J F, Zhai R X, Sun C P, Dong H 2023 PRX Energy 2 033003 Google Scholar

Cited By

图 1 统计热力学发展“新相图”

Figure 1. New diagram for the development of statistical thermodynamics.

DownLoad: Full-Size Img PowerPoint

图 2 一个基于离散能级系统的量子热力学循环

Figure 2. A quantum thermodynamic cycle based on a discrete system.

DownLoad: Full-Size Img PowerPoint

图 3 作为量子热机工作物质的多能级量子系统, 这里展示一个量子绝热过程

Figure 3. A quantum system with discrete energy levels as the working substance, here a quantum adiabatic process is illustrated.

DownLoad: Full-Size Img PowerPoint

图 4 一个基于二能级量子系统的量子Carnot热机循环, $ {T}_{{\mathrm{h}}} $和$ {T}_{{\mathrm{l}}} $分别代表高温和低温热库的温度, $ \varDelta $和$ {P}_{{\mathrm{e}}} $代表二能级系统的能级差和激发态上的布居数, $ P $和$ V $代表经典理想气体的压强和体积

Figure 4. A quantum Carnot cycle based on a two-level system, $ {T}_{{\mathrm{h}}} $ and $ {T}_{{\mathrm{l}}} $ denote the temperatures of two reservoirs, $ \varDelta $ and $ {P}_{{\mathrm{e}}} $ denote the level spacing and the probability in the excited state, $ P $ and $ V $ denote the pressure and the volume of the ideal gas.

DownLoad: Full-Size Img PowerPoint

图 5 Maxwell妖和Szilard单分子热机模型

Figure 5. Maxwell’s demon and Szilard single-molecule heat engine model.

DownLoad: Full-Size Img PowerPoint

图 6 Feynman棘轮, $ {T}_{{\mathrm{A}}} $和$ {T}_{{\mathrm{B}}} $代表两个热库的温度, 有关Feynman棘轮的介绍可参考文献[27]

Figure 6. Feynman’s ratchet. $ {T}_{{\mathrm{A}}} $ and $ {T}_{{\mathrm{B}}} $ denote the temperatures of two reservoirs. For an introduction to Feynman’s ratchet please refer to Ref. [27].

DownLoad: Full-Size Img PowerPoint

图 7 涨落定理家族, 这些涨落定理并不互相等价, 箭头代表“可以推导出”, 详情见文献[47]

Figure 7. Hierarchy of fluctuation theorems, these fluctuation theorems are not equivalent to each other. The arrows indicate “can lead to”. Details see Ref. [47].

DownLoad: Full-Size Img PowerPoint

图 8 基于两次投影定义的轨道功. 这里显示了一次实验的结果, 从第2个本征态跳到第4个瞬时本征态

Figure 8. Trajectory work based on two-point measurement. This figure illustrates a trajectory: from the 2^nd to the 4^th instantaneous eigenstate.

DownLoad: Full-Size Img PowerPoint

图 9 能壳变形图, 其中红色是无相互作用的能壳, 蓝色为有相互作用的修正

Figure 9. The distortion of the energy shell, the red area shows the case with no interaction, and the blue one shows the case with weak interaction

DownLoad: Full-Size Img PowerPoint

图 10 有限时间Carnot循环的功率效率约束关系　(a)基于二能级系统的有限时间量子Carnot循环; (b)量子Carnot循环中功率-效率和一般约束关系的对比, 其中图中的棕色虚线和灰色点线是由(25)式给出, 绿色三角代表最大功率的位置

Figure 10. The power-efficiency constraints of a finite-time Carnot cycle: (a) Finite-time quantum Carnot cycle based on a two-level system; (b) comparison between power-efficiency and general constraint relationships in a quantum Carnot cycle, where the brown dashed line and gray dotted line in the graph are given by Eq. (25), and the green triangles represent the positions of maximum power.

DownLoad: Full-Size Img PowerPoint

图 11 有限时间热力学$ 1/\tau $关系的实验验证　(a) 温度50 ℃下做功的$ 1/\tau $ 标度关系; (b) 标度关系的系数对操控方式的依赖关系$ L\left(t\right)={L}_{0}\propto {t}^{\alpha } $, 能量损耗最优操控是匀速控制$ \alpha =1 $

Figure 11. Experimental verification of the finite-time thermodynamic $ 1/\tau $ relationship: (a) The 1/τ scaling relationship for work done at a temperature of 50 ℃; (b) the dependence of the scaling relationship coefficient on the control method $ L\left(t\right)={L}_{0}\propto {t}^{\alpha }, $ with the energy-optimal control being uniform-speed control where $ \alpha =1 $.

DownLoad: Full-Size Img PowerPoint

图 12 有限时间Carnot循环　(a)有限时间Carnot循环的工作示意图; (b)循环中功率-效率约束关系; (c)最大功率效率对Carnot效率的依赖关系η_EMP = (0.524 ± 0.034)η_C

Figure 12. Finite-time Carnot cycle: (a) Schematic diagram of the finite-time Carnot cycle; (b) graph of the power-efficiency constraint relationship in the cycle; (c) dependency of maximum power efficiency on Carnot efficiency η_EMP = (0.524 ± 0.034)η_C.

DownLoad: Full-Size Img PowerPoint

图 13 能量最小消耗路径的几何描述

Figure 13. Riemann geometry for the minimum energy cost.

DownLoad: Full-Size Img PowerPoint

map

[1]	Scovil H E D, Schulz-DuBois E O 1959 Phys. Rev. Lett. 2 262 Google Scholar
[2]	Geusic J E, Schulz-DuBios E O, Scovil H E D 1967 Phys. Rev. 156 343 Google Scholar
[3]	Alicki R 1979 J. Phys. A: Math. Gen. 12 L103
[4]	Kosloff R 2013 Entropy 15 2100 Google Scholar
[5]	Kieu T D 2004 Phys. Rev. Lett. 93 140403 Google Scholar
[6]	Kieu T D 2006 Eur. Phys. J. D 39 115 Google Scholar
[7]	Scully M O 2003 Science 299 862 Google Scholar
[8]	Quan H T, Zhang P, Sun C P 2006 Phys. Rev. E 73 036122 Google Scholar
[9]	Quan H T, Liu Y xi, Sun C P, Nori F 2007 Phys. Rev. E 76 031105 Google Scholar
[10]	Binder F, Correa L A, Gogolin C, Anders J, Adesso G 2018 Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions (Cham: Springer International Publishing
[11]	Callen H B 1985 Thermodynamics and An Introduction to Thermostatistics (2nd Ed.) (New York: J. Wiley & Sons
[12]	Schrödinger E 1989 Statistical Thermodynamics (New York: Dover Publications
[13]	Reif F 2009 Fundamentals of Statistical and Thermal Physics (Waveland Press
[14]	Hill T L 2013 Thermodynamics of Small Systems (New York: Dover Publications, Inc
[15]	Allahverdyan A E, Serral Gracià R, Nieuwenhuizen Th M 2005 Phys. Rev. E 71 046106 Google Scholar
[16]	Leff H S, Rex A F 2003 Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Philadelphia: Institute of Physics
[17]	Szilard L 1929 Z. Physik 53 840 Google Scholar
[18]	Landauer R 1961 IBM J. Res. Dev. 5 183 Google Scholar
[19]	Bennett C H 1982 Int. J. Theor. Phys. 21 905 Google Scholar
[20]	Bérut A, Arakelyan A, Petrosyan A, Ciliberto S, Dillenschneider R, Lutz E 2012 Nature 483 187 Google Scholar
[21]	Parrondo J M R, Horowitz J M, Sagawa T 2015 Nat. Phys. 11 131 Google Scholar
[22]	Quan H T, Wang Y D, Liu Y xi, Sun C P, Nori F 2006 Phys. Rev. Lett. 97 180402 Google Scholar
[23]	孙昌璞, 全海涛 2013 物理 42 756 Sun C P, Quan H T 2013 Physics 42 756
[24]	Dong H, Xu D Z, Cai C Y, Sun C P 2011 Phys. Rev. E 83 061108 Google Scholar
[25]	Cai C Y, Dong H, Sun C P 2012 Phys. Rev. E 85 031114 Google Scholar
[26]	Mandal D, Quan H T, Jarzynski C 2013 Phys. Rev. Lett. 111 030602 Google Scholar
[27]	Sekimoto K 1997 J. Phys. Soc. Jpn. 66 1234 Google Scholar
[28]	Jarzynski C 1997 Phys. Rev. Lett. 78 2690 Google Scholar
[29]	Jarzynski C 2011 Annu. Rev. Condens. Matter Phys. 2 329 Google Scholar
[30]	Jarzynski C 1997 Phys. Rev. E 56 5018 Google Scholar
[31]	Sekimoto K 1998 Prog. Theor. Phys. Suppl. 130 17 Google Scholar
[32]	Seifert U 2012 Rep. Prog. Phys. 75 126001 Google Scholar
[33]	Oono Y, Paniconi M 1998 Prog. Theor. Phys. Suppl. 130 29 Google Scholar
[34]	Hatano T, Sasa S ichi 2001 Phys. Rev. Lett. 86 3463 Google Scholar
[35]	Sekimoto K 2010 Stochastic Energetics (Berlin Heidelberg: Springer
[36]	Jiang D Q, Qian M, Qian M P 2004 Mathematical Theory of Nonequilibrium Steady States: On the Frontier of Probability and Dynamical Systems (Berlin: Springer
[37]	Esposito M, van den Broeck C 2010 Phys. Rev. Lett. 104 090601 Google Scholar
[38]	Kim K H, Qian H 2004 Phys. Rev. Lett. 93 120602 Google Scholar
[39]	葛颢 2014 数学进展 43 161 Ge J 2014 Adv. Math. 43 161
[40]	Jarzynskia C 2008 Eur. Phys. J. B 64 331 Google Scholar
[41]	Liphardt J, Dumont S, Smith S B, Tinoco I, Bustamante C 2002 Science 296 1832 Google Scholar
[42]	Collin D, Ritort F, Jarzynski C, Smith S B, Tinoco I, Bustamante C 2005 Nature 437 231 Google Scholar
[43]	Crooks G E 1999 Phys. Rev. E 60 2721 Google Scholar
[44]	Hummer G, Szabo A 2001 Proc. Natl. Acad. Sci. USA 98 3658 Google Scholar
[45]	Jarzynski C 2000 J. Stat. Phys. 98 77 Google Scholar
[46]	Maragakis P, Spichty M, Karplus M 2008 J. Phys. Chem. B 112 6168 Google Scholar
[47]	Hoang T M, Pan R, Ahn J, Bang J, Quan H T, Li T 2018 Phys. Rev. Lett. 120 080602 Google Scholar
[48]	Jarzynski C, Wójcik D K 2004 Phys. Rev. Lett. 92 230602 Google Scholar
[49]	Chen J F, Quan H T 2023 Phys. Rev. E 107 024135 Google Scholar
[50]	Evans D J, Cohen E G D, Morriss G P 1993 Phys. Rev. Lett. 71 2401 Google Scholar
[51]	Evans D J, Searles D J 1994 Phys. Rev. E 50 1645
[52]	Gallavotti G, Cohen E G D 1995 Phys. Rev. Lett. 74 2694 Google Scholar
[53]	Gallavotti G, Cohen E G D 1995 J. Stat. Phys. 80 931 Google Scholar
[54]	Barato A C, Seifert U 2015 Phys. Rev. Lett. 114 158101 Google Scholar
[55]	Timpanaro A M, Guarnieri G, Goold J, Landi G T 2019 Phys. Rev. Lett. 123 090604 Google Scholar
[56]	Kurchan J 2000 A Quantum Fluctuation Theorem (arXiv:cond-mat/0007360
[57]	Tasaki H 2000 Jarzynski Relations for Quantum Systems and Some Applications (arXiv:cond-mat/0009244
[58]	Talkner P, Lutz E, Hänggi P 2007 Phys. Rev. E 75 050102 Google Scholar
[59]	Dorner R, Clark S R, Heaney L, Fazio R, Goold J, Vedral V 2013 Phys. Rev. Lett. 110 230601 Google Scholar
[60]	Mazzola L, De Chiara G, Paternostro M 2013 Phys. Rev. Lett. 110 230602 Google Scholar
[61]	Batalhão T B, Souza A M, Mazzola L, Auccaise R, Sarthour R S, Oliveira I S, Goold J, De Chiara G, Paternostro M, Serra R M 2014 Phys. Rev. Lett. 113 140601 Google Scholar
[62]	An S, Zhang J N, Um M, Lv D, Lu Y, Zhang J, Yin Z Q, Quan H T, Kim K 2015 Nat. Phys. 11 193 Google Scholar
[63]	Jarzynski C, Quan H T, Rahav S 2015 Phys. Rev. X 5 031038 Google Scholar
[64]	Funo K, Quan H T 2018 Phys. Rev. Lett. 121 040602 Google Scholar
[65]	Gong Z, Lan Y, Quan H T 2016 Phys. Rev. Lett. 117 180603 Google Scholar
[66]	Fei Z, Zhang J, Pan R, Qiu T, Quan H T 2019 Phys. Rev. A 99 052508 Google Scholar
[67]	Fei Z, Quan H T 2020 Phys. Rev. Lett. 124 240603 Google Scholar
[68]	Funo K, Quan H T 2018 Phys. Rev. E 98 012113 Google Scholar
[69]	Chen J F, Qiu T, Quan H T 2021 Entropy 23 1602 Google Scholar
[70]	Wang B, Zhang J, Quan H T 2018 Phys. Rev. E 97 052136 Google Scholar
[71]	Chen J, Dong H, Sun C P 2018 Phys. Rev. E 98 062119 Google Scholar
[72]	Tasaki H 1998 Phys. Rev. Lett. 80 1373 Google Scholar
[73]	Goldstein S, Lebowitz J L, Tumulka R, Zanghì N 2006 Phys. Rev. Lett. 96 050403 Google Scholar
[74]	Popescu S, Short A J, Winter A 2006 Nat. Phys. 2 754 Google Scholar
[75]	Dong H, Yang S, Liu X F, Sun C P 2007 Phys. Rev. A 76 044104 Google Scholar
[76]	Parikh M K, Wilczek F 2000 Phys. Rev. Lett. 85 5042 Google Scholar
[77]	Zhang B, Cai Q Y, You L, Zhan M S 2009 Phys. Lett. B 675 98 Google Scholar
[78]	Dong H, Cai Q Y, Liu X F, Sun C P 2014 Commun. Theor. Phys. 61 289 Google Scholar
[79]	Ma Y H, Cai Q Y, Dong H, Sun C P 2018 EPL 122 30001 Google Scholar
[80]	Srednicki M 1994 Phys. Rev. E 50 888 Google Scholar
[81]	D’Alessio L, Kafri Y, Polkovnikov A, Rigol M 2016 Adv. Phys. 65 239 Google Scholar
[82]	Yvon J 1955 Proceedings of the International Conference on the Peaceful Uses of Atomic Energy Geneva, Switzerland, August 8—20, 1955
[83]	Curzon F L, Ahlborn B 1975 Am. J. Phys. 43 22 Google Scholar
[84]	Andresen B, Salamon P, Berry R S 1984 Phys. Today 37 62 Google Scholar
[85]	Chen J C, Su S H, Su G Z 2023 Reflection and Exploration on Hot Issues in Thermodynamics and Statistical Physics (2nd Ed.) (Beijing: Science Press) [陈金灿, 苏山河, 苏国珍 2023 热力学与统计物理学热点问题思考与探索(第二版)(北京: 科学出版社) Chen J C, Su S H, Su G Z 2023 Reflection and Exploration on Hot Issues in Thermodynamics and Statistical Physics (2nd Ed.) (Beijing: Science Press)
[86]	Esposito M, Kawai R, Lindenberg K, Van den Broeck C 2010 Phys. Rev. Lett. 105 150603 Google Scholar
[87]	Ma Y H, Xu D, Dong H, Sun C P 2018 Phys. Rev. E 98 042112 Google Scholar
[88]	Ryabov A, Holubec V 2016 Phys. Rev. E 93 050101 Google Scholar
[89]	Long R, Liu W 2016 Phys. Rev. E 94 052114 Google Scholar
[90]	Chen J F, Sun C P, Dong H 2019 Phys. Rev. E 100 062140 Google Scholar
[91]	Chen J F, Sun C P, Dong H 2019 Phys. Rev. E 100 032144 Google Scholar
[92]	Ma Y H, Xu D, Dong H, Sun C P 2018 Phys. Rev. E 98 022133 Google Scholar
[93]	Cavina V, Mari A, Giovannetti V 2017 Phys. Rev. Lett. 119 050601 Google Scholar
[94]	Ma Y H, Zhai R X, Chen J, Sun C P, Dong H 2020 Phys. Rev. Lett. 125 210601 Google Scholar
[95]	Blickle V, Bechinger C 2012 Nat. Phys. 8 143 Google Scholar
[96]	Roßnagel J, Dawkins S T, Tolazzi K N, Abah O, Lutz E, Schmidt-Kaler F, Singer K 2016 Science 352 325 Google Scholar
[97]	Martínez I A, Roldán É, Dinis L, Petrov D, Parrondo J M R, Rica R A 2016 Nat. Phys. 12 67 Google Scholar
[98]	Zhai R X, Cui F M, Ma Y H, Sun C P, Dong H 2023 Phys. Rev. E 107 L042101 Google Scholar
[99]	Tu Z C 2021 Front. Phys. 16 33202 Google Scholar
[100]	Salamon P, Berry R S 1983 Phys. Rev. Lett. 51 1127 Google Scholar
[101]	Weinhold F 1975 J. Chem. Phys. 63 2479 Google Scholar
[102]	Ruppeiner G 1979 Phys. Rev. A 20 1608 Google Scholar
[103]	Crooks G E 2007 Phys. Rev. Lett. 99 100602 Google Scholar
[104]	Scandi M, Perarnau-Llobet M 2019 Quantum 3 197 Google Scholar
[105]	Abiuso P, Miller H J D, Perarnau-Llobet M, Scandi M 2020 Entropy 22 1076 Google Scholar
[106]	Chen J F, Sun C P, Dong H 2021 Phys. Rev. E 104 034117 Google Scholar
[107]	Li G, Chen J F, Sun C P, Dong H 2022 Phys. Rev. Lett. 128 230603 Google Scholar
[108]	Li G, Quan H T, Tu Z C 2017 Phys. Rev. E 96 012144 Google Scholar
[109]	Chen J F 2022 Phys. Rev. E 106 054108
[110]	Van Vu T, Hasegawa Y 2021 Phys. Rev. Lett. 126 010601 Google Scholar
[111]	Dago S, Pereda J, Barros N, Ciliberto S, Bellon L 2021 Phys. Rev. Lett. 126 170601 Google Scholar
[112]	Zhen Y Z, Egloff D, Modi K, Dahlsten O 2021 Phys. Rev. Lett. 127 190602 Google Scholar
[113]	Ma Y H, Chen J F, Sun C P, Dong H 2022 Phys. Rev. E 106 034112 Google Scholar
[114]	Tsirlin A M, Kazakov V, Zubov D V 2002 J. Phys. Chem. A 106 10926 Google Scholar
[115]	Chen J F, Zhai R X, Sun C P, Dong H 2023 PRX Energy 2 033003 Google Scholar

[1]	Guan Yun, Wang Bo-Bo. Thermodynamic functions of toroidal black holes. Acta Physica Sinica, 2022, 71(11): 110401. doi: 10.7498/aps.70.20212370
[2]	. Thermodynamic functions of toroidal black holes. Acta Physica Sinica, 2022, (): . doi: 10.7498/aps.71.20212370
[3]	Wang Zi, Ren Jie. Nonequilibrium thermal transport and thermodynamic geometry in periodically driven systems. Acta Physica Sinica, 2021, 70(23): 230503. doi: 10.7498/aps.70.20211723
[4]	Shen Jue, Liu Cheng-Zhou, Zhu Ning-Ning, Tong Yi-Nuo, Yan Chen-Cheng, Xue Ke-Lei. Thermodynamics and its quantum correction of non-commutative Schwarichild black hole. Acta Physica Sinica, 2019, 68(20): 200401. doi: 10.7498/aps.68.20191054
[5]	Jin Xiao, Wang Li-Min. Enthalpy relaxation studies of memory effect in various glass formers in the vicinity of glass transition. Acta Physica Sinica, 2017, 66(17): 176406. doi: 10.7498/aps.66.176406
[6]	Yuan Du-Qi. Thermodynamics of trapped finite unitary Fermi gas. Acta Physica Sinica, 2016, 65(18): 180302. doi: 10.7498/aps.65.180302
[7]	Zhang Tian-Bao, Yu Xuan-Ping, Chen A-Hai. Thermodynamic properties of one-dimensional Gaudin-Yang model at finite temperature. Acta Physica Sinica, 2015, 64(15): 156402. doi: 10.7498/aps.64.156402
[8]	Aili Mieralimujiang, Mamat Mamatrishat, Ghupur Yasenjan. Thermodynamic properties of harmonic oscillator system in noncommutative phase space. Acta Physica Sinica, 2015, 64(14): 140201. doi: 10.7498/aps.64.140201
[9]	Sun Qi-Cheng. Granular structure and the nonequilibrium thermodynamics. Acta Physica Sinica, 2015, 64(7): 076101. doi: 10.7498/aps.64.076101
[10]	Xia Shao-Jun, Chen Lin-Gen, Ge Yan-Lin, Sun Feng-Rui. Influence of heat leakage on entransy dissipation minimization of heat exchanger. Acta Physica Sinica, 2014, 63(2): 020505. doi: 10.7498/aps.63.020505
[11]	Xia Shao-Jun, Chen Lin-Gen, Ge Yan-Lin, Sun Feng-Rui. Entransy dissiaption minimization for isothermal throttling process. Acta Physica Sinica, 2013, 62(18): 180202. doi: 10.7498/aps.62.180202
[12]	Li Jun, Chen Lin-Gen, Ge Yan-Lin, Sun Feng-Rui. Progress in the study on finite time thermodynamic optimization for direct and reverse two-heat-reservoir thermodynamic cycles. Acta Physica Sinica, 2013, 62(13): 130501. doi: 10.7498/aps.62.130501
[13]	Lin Min, Zhang Mei-Li. Interaction of force and coupled system and stochastic energetic resonance. Acta Physica Sinica, 2011, 60(2): 020501. doi: 10.7498/aps.60.020501
[14]	Yang Hong-Wei, Zhong Wan-Xie, Hou Bi-Hui. The canonical transformation and symplectic description in mechanics, thermodynamics and electromagnetic waveguide. Acta Physica Sinica, 2010, 59(7): 4437-4441. doi: 10.7498/aps.59.4437
[15]	Zhang Ya-Nan, Yan Shi-Lei. Thermodynamic properties of random transverse mixed Ising spin system with cryst al field. Acta Physica Sinica, 2003, 52(11): 2890-2895. doi: 10.7498/aps.52.2890
[16]	Xiao Xing-Guo, Zhao Zheng. . Acta Physica Sinica, 1995, 44(5): 832-840. doi: 10.7498/aps.44.832
[17]	OU FA. . Acta Physica Sinica, 1995, 44(10): 1541-1550. doi: 10.7498/aps.44.1541
[18]	WANG WEI-HUA, BAI HAI-YANG, ZHANG YUN, CHEN HONG, WANG WEN-KUI. THERMODYNAMIC AND KINETIC ASPECTS OF SOLID STATE AMORPHIZATION REACTION IN Ni/a-Si MULTILAYER SYSTEM. Acta Physica Sinica, 1993, 42(9): 1499-1504. doi: 10.7498/aps.42.1499
[19]	. . Acta Physica Sinica, 1964, 20(9): 940-946. doi: 10.7498/aps.20.940
[20]	CHENG KAI-CHIA. A THERMODYNAMICAL STUDY OF INTERNAL FRICTION. Acta Physica Sinica, 1955, 11(2): 163-178. doi: 10.7498/aps.11.163

Catalog

Vol. 72, Issue 23 - 2023-12-05

Metrics

Abstract views: 5193
PDF Downloads: 264
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板