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).

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Cited By

图 1 统计热力学发展“新相图”

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

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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.

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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.

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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.

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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.

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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].

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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].

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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.

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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

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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.

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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 $.

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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.

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图 13 能量最小消耗路径的几何描述

Figure 13. Riemann geometry for the minimum energy cost.

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