介观统计热力学理论与实验

全海涛; 董辉; 孙昌璞

doi:10.7498/aps.72.20231608

摘要

对于只有有限自由度的介观小系统, 传统的热力学定律是否成立? 温度、熵、做功、传热、等温过程、Carnot循环这些概念还是否有效? 是否需要对原来适用于宏观系统的传统热力学理论进行修改或扩充、以适用于这样的小系统? 在过去近20年里, 我们深入研究了在介观小系统和量子系统中热力学基本概念的推广(例如什么是量子等温过程)以及基本热力学定律的适用性问题. 研究表明, 在系综平均意义上热力学定律仍然适用于小系统; 考虑了Maxwell妖的信息擦除功耗后, 热力学第二定律不会被违反; “小系统”的统计热力学具有一些新的特性, 由于系统和环境之间的耦合不可忽略, 有限系统的平衡态分布偏离正则系综, 这可以描述诸如黑洞等小系统的辐射关联及其信息丢失现象; 在任意远离平衡的情况下, 热力学量的涨落变得十分显著, 并且热力学量的分布函数满足一些严格成立的恒等式. 这些恒等式定义了所谓的涨落定理, 由此通过测量非平衡过程的物理量(如功分布)可以获得平衡过程的物理量相对值(如自由能差等). 此外, 尽管量子属性和信息论的考虑为统计热力学带来一些有别于经典和传统的特性, 有助于理解Gibbs佯谬和Maxwell妖等基本问题, 但需要指出的是, 量子热机和信息辅助热机的效率并没有超越经典热机. 随着在小系统中引入运动方程, 热力学和力学之间的联系变得更加紧密, 能够从第一性原理出发研究非平衡过程的能量耗散和热机的功率、效率优化及其最优调控微分几何化等问题. 在对具体热循环过程熵产生问题的研究中, 对得到的功率-效率约束关系进行了系统性的实验检验.

关键词:

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:

作者及机构信息

1.
北京大学物理学院, 北京　100871

2.
中国工程物理研究院研究生院, 北京　100193

通信作者: 全海涛, htquan@pku.edu.cn ; 孙昌璞, suncp@gscaep.ac.cn

基金项目: 国家自然科学基金(批准号: 12088101, 11825501, 12375028, U2330401)资助的课题.

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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施引文献

图 1 统计热力学发展“新相图”

Fig. 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

Fig. 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

Fig. 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

Fig. 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

Fig. 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]

Fig. 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]

Fig. 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

Fig. 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

Fig. 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

Fig. 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 $

Fig. 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

Fig. 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

Fig. 13. Riemann geometry for the minimum energy cost.

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map

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