非平衡系统

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此词条由Jie翻译。

模板:Cleanup rewrite模板:Thermodynamics

Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of variables (non-equilibrium state variables) that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Non-equilibrium thermodynamics is concerned with transport processes and with the rates of chemical reactions. It relies on what may be thought of as more or less nearness to thermodynamic equilibrium.

Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of variables (non-equilibrium state variables) that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Non-equilibrium thermodynamics is concerned with transport processes and with the rates of chemical reactions. It relies on what may be thought of as more or less nearness to thermodynamic equilibrium.

非平衡热力学Non-equilibrium thermodynamics是热力学的一个分支，它处理的物理系统并不处于 热力学平衡状态Thermodynamic equilibrium ，不过可以通过变量（非平衡状态变量）来进行描述，相对于对处于热力学平衡状态下的指定系统进行变量外推。非平衡热力学与系统某一性质在物系内部的运输过程和化学反应速率有关。它仍然或多或少地取决于热力学平衡。


Almost all systems found in nature are not in thermodynamic equilibrium, for they are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems and to chemical reactions. Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of non-equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost.[1]

Almost all systems found in nature are not in thermodynamic equilibrium, for they are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems and to chemical reactions. Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of non-equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost.

自然界中，几乎所有系统都不处于热力学平衡状态。因为它们时刻在变化，或者因某些外界因素触发而产生变化。它们会断断续续地受到其他系统的物质和能量通量的影响，反之亦然。同时它们还会不间断的进行化学反应。但是部分系统及其热力学反应过程在某种有效的意义上，是接近于热力学平衡的。因此，允许就目前所知的非平衡热力学理论对系统进行准确性描述。然而，仍然有许多自然系统和其热力学反应过程远远超出了非平衡热力学方法的描述能力范围，由于非变分动力学的存在，自由能的概念并未被考虑。


The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. One fundamental difference between equilibrium thermodynamics and non-equilibrium thermodynamics lies in the behaviour of inhomogeneous systems, which require for their study knowledge of rates of reaction which are not considered in equilibrium thermodynamics of homogeneous systems. This is discussed below. Another fundamental and very important difference is the difficulty or impossibility, in general, in defining entropy at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium; it can be done, to useful approximation, only in carefully chosen special cases, namely those that are throughout in local thermodynamic equilibrium.[2][3]

The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. One fundamental difference between equilibrium thermodynamics and non-equilibrium thermodynamics lies in the behaviour of inhomogeneous systems, which require for their study knowledge of rates of reaction which are not considered in equilibrium thermodynamics of homogeneous systems. This is discussed below. Another fundamental and very important difference is the difficulty or impossibility, in general, in defining entropy at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium; it can be done, to useful approximation, only in carefully chosen special cases, namely those that are throughout in local thermodynamic equilibrium.

非平衡系统的热力学研究比平衡热力学需要懂更多的专业概念知识。非平衡热力学与平衡热力学之间的一个根本区别在于其非均相系统的性质，这就要求研究者们对关于反应速率的相关知识有一定的掌握，而均相系统的平衡热力学中并未考虑这一点。这将在下面章节进行讨论。另一个根本且非常重要的区别是，对于不是处于热力学平衡状态的系统，想在宏观上来定义 熵Entropy的瞬间非常困难，或者说几乎是不可能的。只有在精心挑选的特殊情况下，即局部处于热力学平衡状态的情况下，才能做到有效地近似。



Scope 范围

Difference between equilibrium and non-equilibrium thermodynamics 平衡与非平衡热力学之间的差异

A profound difference separates equilibrium from non-equilibrium thermodynamics. Equilibrium thermodynamics ignores the time-courses of physical processes. In contrast, non-equilibrium thermodynamics attempts to describe their time-courses in continuous detail.

A profound difference separates equilibrium from non-equilibrium thermodynamics. Equilibrium thermodynamics ignores the time-courses of physical processes. In contrast, non-equilibrium thermodynamics attempts to describe their time-courses in continuous detail.

巨大的差异将平衡与非平衡热力学区分开来。平衡热力学忽略了物理过程的时程分析。相反，非平衡热力学则试图通过连续的细节描述来进行时程分析。


Equilibrium thermodynamics restricts its considerations to processes that have initial and final states of thermodynamic equilibrium; the time-courses of processes are deliberately ignored. Consequently, equilibrium thermodynamics allows processes that pass through states far from thermodynamic equilibrium, that cannot be described even by the variables admitted for non-equilibrium thermodynamics,[4] such as time rates of change of temperature and pressure.[5] For example, in equilibrium thermodynamics, a process is allowed to include even a violent explosion that cannot be described by non-equilibrium thermodynamics.[4] Equilibrium thermodynamics does, however, for theoretical development, use the idealized concept of the "quasi-static process". A quasi-static process is a conceptual (timeless and physically impossible) smooth mathematical passage along a continuous path of states of thermodynamic equilibrium.[6] It is an exercise in differential geometry rather than a process that could occur in actuality.

Equilibrium thermodynamics restricts its considerations to processes that have initial and final states of thermodynamic equilibrium; the time-courses of processes are deliberately ignored. Consequently, equilibrium thermodynamics allows processes that pass through states far from thermodynamic equilibrium, that cannot be described even by the variables admitted for non-equilibrium thermodynamics, such as time rates of change of temperature and pressure. For example, in equilibrium thermodynamics, a process is allowed to include even a violent explosion that cannot be described by non-equilibrium thermodynamics. It is an exercise in differential geometry rather than a process that could occur in actuality.

平衡热力学在分析过程中仅将其考虑因素限制在具有热力学平衡的初始状态和最终状态上；而其时程分析被故意忽略。因此，对于反应过程中处于远非平衡状态下的系统，平衡热力学都选择放过不进行分析，而实际上，即使通过非平衡热力学所允许的变量（例如温度和压力的时间变化率）也无法对该过程进行描述。例如，在平衡热力学中，甚至可以包含一个猛烈的爆炸过程，该过程无法用非平衡热力学来描述。但是，为了进行理论发展地研究，平衡热力学确实使用了“准静态过程”的理想概念。准静态过程指的是沿着热力学平衡状态的连续路径，进行的概念性（不受时间影响且物理上不可能）平滑数学分析过程。这整个反应过程存在于微分几何中，而不是实际可能发生的情况下。


Non-equilibrium thermodynamics, on the other hand, attempting to describe continuous time-courses, needs its state variables to have a very close connection with those of equilibrium thermodynamics.[7] This profoundly restricts the scope of non-equilibrium thermodynamics, and places heavy demands on its conceptual framework.

Non-equilibrium thermodynamics, on the other hand, attempting to describe continuous time-courses, needs its state variables to have a very close connection with those of equilibrium thermodynamics. This profoundly restricts the scope of non-equilibrium thermodynamics, and places heavy demands on its conceptual framework.

另一方面，非平衡热力学试图描述连续的时间过程，需要其状态变量与平衡热力学的 状态变量State variables保持非常紧密的联系。这极大地限制了非平衡热力学的范围，并对它的概念框架提出了很高的要求。


Non-equilibrium state variables 非平衡状态变量

The suitable relationship that defines non-equilibrium thermodynamic state variables is as follows. On occasions when the system happens to be in states that are sufficiently close to thermodynamic equilibrium, non-equilibrium state variables are such that they can be measured locally with sufficient accuracy by the same techniques as are used to measure thermodynamic state variables, or by corresponding time and space derivatives, including fluxes of matter and energy. In general, non-equilibrium thermodynamic systems are spatially and temporally non-uniform, but their non-uniformity still has a sufficient degree of smoothness to support the existence of suitable time and space derivatives of non-equilibrium state variables. Because of the spatial non-uniformity, non-equilibrium state variables that correspond to extensive thermodynamic state variables have to be defined as spatial densities of the corresponding extensive equilibrium state variables. On occasions when the system is sufficiently close to thermodynamic equilibrium, intensive non-equilibrium state variables, for example temperature and pressure, correspond closely with equilibrium state variables. It is necessary that measuring probes be small enough, and rapidly enough responding, to capture relevant non-uniformity. Further, the non-equilibrium state variables are required to be mathematically functionally related to one another in ways that suitably resemble corresponding relations between equilibrium thermodynamic state variables.[8] In reality, these requirements are very demanding, and it may be difficult or practically, or even theoretically, impossible to satisfy them. This is part of why non-equilibrium thermodynamics is a work in progress.

The suitable relationship that defines non-equilibrium thermodynamic state variables is as follows. On occasions when the system happens to be in states that are sufficiently close to thermodynamic equilibrium, non-equilibrium state variables are such that they can be measured locally with sufficient accuracy by the same techniques as are used to measure thermodynamic state variables, or by corresponding time and space derivatives, including fluxes of matter and energy. In general, non-equilibrium thermodynamic systems are spatially and temporally non-uniform, but their non-uniformity still has a sufficient degree of smoothness to support the existence of suitable time and space derivatives of non-equilibrium state variables. Because of the spatial non-uniformity, non-equilibrium state variables that correspond to extensive thermodynamic state variables have to be defined as spatial densities of the corresponding extensive equilibrium state variables. On occasions when the system is sufficiently close to thermodynamic equilibrium, intensive non-equilibrium state variables, for example temperature and pressure, correspond closely with equilibrium state variables. It is necessary that measuring probes be small enough, and rapidly enough responding, to capture relevant non-uniformity. Further, the non-equilibrium state variables are required to be mathematically functionally related to one another in ways that suitably resemble corresponding relations between equilibrium thermodynamic state variables. Onsager 1931, also), time rate of entropy production (Onsager 1931), dissipative structure, but they are hardly touched on in the present article.

定义非平衡热力学状态变量的关系如下：在系统恰好处于很接近热力学平衡状态的情况下，非平衡状态变量可以通过与测量热力学状态变量相同的技术，或通过相应的时空导数，包括物质和能量通量，以足够的精度在本地进行测量。通常，非平衡热力学系统在空间和时间上都是非均匀的，但是它们的非均匀性仍然具有足够的平滑度，以保证非平衡状态变量的时空导数适当存在。另外由于空间的不均匀性，必须将非平衡状态变量（对应于广义热力学状态变量）定义为相应的广义平衡状态变量的空间密度。在系统足够接近热力学平衡的情况下，密集的非平衡状态变量（例如温度和压力）与平衡状态变量紧密对应。测量时探头必须足够小，并且响应速度要足够快，以捕获相关的不均匀性。此外，要求非平衡状态变量在数学上彼此函数相关，其方式应类似于平衡热力学状态变量之间的对应关系。实际上，这些要求非常苛刻，可能很难实现，或实际上，甚至在理论上无法满足它们。这就是非平衡热力学的研究一直处在探索中的部分原因。

Overview 概述

Non-equilibrium thermodynamics is a work in progress, not an established edifice. This article is an attempt to sketch some approaches to it and some concepts important for it.

非平衡热力学目前仍然处在探索中，距离理论成熟仍需要一定时间。本文试图勾勒出一些方法和一些重要的概念。


Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873,[9] Onsager 1931,[10] also[8][11]), time rate of entropy production (Onsager 1931),[10] thermodynamic fields,[12][13][14] dissipative structure,[15] and non-linear dynamical structure.[11]

对于非平衡热力学，特别重要的一些概念包括：能量耗散的时间速率（Rayleigh 1873，Onsager 1931），熵产生的时间速率（Onsager 1931），热力学场， 耗散结构Dissipative structure和非线性动力学结构。


One problem of interest is the thermodynamic study of non-equilibrium steady states, in which entropy production and some flows are non-zero, but there is no time variation of physical variables.

感兴趣的问题之一是对 非平衡稳态Non-equilibrium steady states的热力学研究，其中包括熵产生，和某些非零流量，不过这些物理变量不具有时间变化性。


One initial approach to non-equilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'.[3] There are other approaches to non-equilibrium thermodynamics, for example extended irreversible thermodynamics,[3][16] and generalized thermodynamics,[17] but they are hardly touched on in the present article.

有一种分析非平衡热力学的初始方法被称为“ 经典不可逆热力学Classical irreversible thermodynamics”。同时还有其他方法例如： 扩展的不可逆热力学Extended irreversible thermodynamics和 广义热力学Generalized thermodynamics，但在本文中并没有涉及。


Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions 实验室条件下物质的准无辐射非平衡热力学

According to Wildt[18] (see also Essex[19][20][21]), current versions of non-equilibrium thermodynamics ignore radiant heat; they can do so because they refer to laboratory quantities of matter under laboratory conditions with temperatures well below those of stars. At laboratory temperatures, in laboratory quantities of matter, thermal radiation is weak and can be practically nearly ignored. But, for example, atmospheric physics is concerned with large amounts of matter, occupying cubic kilometers, that, taken as a whole, are not within the range of laboratory quantities; then thermal radiation cannot be ignored.

怀尔德Wild（也参见Essex）认为，当前版本的非平衡热力学忽略了辐射热；之所以可以这么做，是因为它们的研究对象是实验室条件下，温度远低于星体温度的实验室物质量。在实验室温度下，基于实验室物质量，其热辐射非常微弱，几乎可以忽略不计。但是，例如大气物理学中涉及的大量物质，他们占有立方公里的空间，总体上讲，不属于实验室数量范围内；那么其热辐射就不能忽略。


Local equilibrium thermodynamics 局部平衡热力学

The terms 'classical irreversible thermodynamics'[3] and 'local equilibrium thermodynamics' are sometimes used to refer to a version of non-equilibrium thermodynamics that demands certain simplifying assumptions, as follows. The assumptions have the effect of making each very small volume element of the system effectively homogeneous, or well-mixed, or without an effective spatial structure, and without kinetic energy of bulk flow or of diffusive flux. Even within the thought-frame of classical irreversible thermodynamics, care[11] is needed in choosing the independent variables[22] for systems. In some writings, it is assumed that the intensive variables of equilibrium thermodynamics are sufficient as the independent variables for the task (such variables are considered to have no 'memory', and do not show hysteresis); in particular, local flow intensive variables are not admitted as independent variables; local flows are considered as dependent on quasi-static local intensive variables.

术语“经典不可逆热力学”和“局部平衡热力学”有时用于指代那些需要简化假设的非平衡热力学。这些假设的作用是使系统中每个非常小体积的元素能有效地均质化，或混合充分，或无有效的空间结构，同时也没有大流量或扩散通量的动能。即使在经典不可逆热力学的思想框架内，在选择系统的自变量时也需要谨慎。在某些著作中，假设平衡热力学的密集变量足以作为研究的自变量（此类变量被认为没有“内存”，并且不显示迟滞）；不允许将局部流量密集型变量视为自变量；本地流量被认为依赖于准静态局部集约变量。


Also it is assumed that the local entropy density is the same function of the other local intensive variables as in equilibrium; this is called the local thermodynamic equilibrium assumption[8][11][15][16][23][24][25][26] (see also Keizer (1987)[27]). Radiation is ignored because it is transfer of energy between regions, which can be remote from one another. In the classical irreversible thermodynamic approach, there is allowed very small spatial variation, from very small volume element to adjacent very small volume element, but it is assumed that the global entropy of the system can be found by simple spatial integration of the local entropy density; this means that spatial structure cannot contribute as it properly should to the global entropy assessment for the system. This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density. All of these are very stringent demands. Consequently, this approach can deal with only a very limited range of phenomena. This approach is nevertheless valuable because it can deal well with some macroscopically observable phenomena.模板:Examples

同时还假设局部熵密度与热力学平衡中的其他局部强度变量的函数相同。这称为局部热力学平衡假设（另请参见Keizer（1987）。需要注意的是辐射被忽略了，因为它是区域之间的能量转移，而这些区域可能彼此远离。在经典的不可逆热力学方法中，允许非常小的空间变化，从很小的体积元素到相邻的很小的体积元素，但是是基于假设可以将局部熵密度进行简单的空间积分来找到系统的全局熵的。这意味着空间结构无法适当地为系统全局熵的评估做出贡献。这种方法假设了空间和时间的连续性，甚至假设了局部定义的强度变量（例如温度和内部能量密度）的可微性。所有这些都是非常严格的要求。因此，这种方法只能处理非常有限的现象。不过这种方法很有价值，因为它可以很好地处理一些宏观上可观察到的现象。


In other writings, local flow variables are considered; these might be considered as classical by analogy with the time-invariant long-term time-averages of flows produced by endlessly repeated cyclic processes; examples with flows are in the thermoelectric phenomena known as the Seebeck and the Peltier effects, considered by Kelvin in the nineteenth century and by Lars Onsager in the twentieth.[23][28] These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation.

在其他研究著作中，还考虑了局部流动变量。其经典思想是将局部热量认为是通过不断循环产生的长效定常时均的流量，其相关例子如 热电现象Thermoelectric phenomena，即 塞贝克效应Seebeck effect和 珀尔帖效应Peltier effect，由开尔文Kelvin在19世纪和拉尔斯·昂萨格Lars Onsager在20世纪提出。这些效应发生在金属链接处，这些链接最初被有效地视为二维表面，没有空间体积，也没有空间变化。


Local equilibrium thermodynamics with materials with "memory" 具有“记忆”材料的局部平衡热力学

A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their constitutive equations depend not only on present values but also on past values of local equilibrium variables. Thus time comes into the picture more deeply than for time-dependent local equilibrium thermodynamics with memoryless materials, but fluxes are not independent variables of state.[29]

局部平衡热力学的进一步扩展是允许材料具有“记忆”，因此它们的 本构方程Constitutive equations不仅取决于局部平衡变量的当前值，而且还取决于过去的值。因此，与无记忆材料的依时性局部平衡热力学相比，时间对图像的影响更深，不过通量不是状态的独立变量。


Extended irreversible thermodynamics 扩展的不可逆热力学

Extended irreversible thermodynamics is a branch of non-equilibrium thermodynamics that goes outside the restriction to the local equilibrium hypothesis. The space of state variables is enlarged by including the fluxes of mass, momentum and energy and eventually higher order fluxes.

扩展的不可逆热力学是非平衡热力学的一个分支，它超出了局部平衡假设的限制。通过包括质量，动量和能量通量以及最终的高阶通量来扩大状态变量的空间。


The formalism is well-suited for describing high-frequency processes and small-length scales materials. 该形式体系非常适合描述高频过程和小尺度材料。

Basic concepts 基本概念

There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary Couette flow, a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. Laser action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed. Damping of acoustic perturbations or shock waves are non-stationary non-equilibrium processes. Driven complex fluids, turbulent systems and glasses are other examples of non-equilibrium systems.

静态非平衡系统有许多案例，其中一些非常简单，例如，将一个系统限制在两个温度不同的恒温器之间，或者是常规 库埃特流体Couette flow运动模型的两个平板之间（封闭状态），该平板互相沿反方向运动，而平板壁上需要定义非平衡条件。另外 激光作用Laser action也是一个非平衡过程，但它依赖于从局部热力学平衡出发，因此超出了经典不可逆热力学的范围。在此，两个分子自由度（分子激光，振动和旋转的分子运动）之间维持着明显的温差，要求在一个很小的空间区域中存在两组“温度”组成部分，其中不包括局部热力学平衡，因为后者仅需要一个“温度”。另外声扰动或冲击波阻尼过程是非静态非平衡的过程。而驱动的 复杂流体Complex fluids，湍流系统和玻璃是非平衡系统的其他案例。


The mechanics of macroscopic systems depends on a number of extensive quantities. It should be stressed that all systems are permanently interacting with their surroundings, thereby causing unavoidable fluctuations of extensive quantities. Equilibrium conditions of thermodynamic systems are related to the maximum property of the entropy. If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful. The system's properties are then most conveniently described using the thermodynamic potential Helmholtz free energy (A = U - TS), a Legendre transformation of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the Gibbs free energy (G = U + PV - TS), where the system's properties are determined both by the temperature and by the pressure.

宏观系统的力学取决于广延量。这里需要强调的是，所有的系统都在与周围环境永久性地相互作用，从而导致不可避免的大量波动。热力学系统的平衡条件与熵的极限性质有关。如果允许波动的唯一广延量是其内部能量，而所有其他能量都严格保持恒定，则系统温度是可测量且有意义的。那么使用热力学势 亥姆霍兹自由能Helmholtz free energy（A = U-TS）（能量的 勒让德变换Legendre transformation）可以最方便地描述系统的属性。如果在能量波动后，系统的宏观尺寸（体积）能同样保持波动，则我们可以使用 吉布斯自由能（G = U + PV-TS），其中系统的特性既取决于温度又取决于压力。


Non-equilibrium systems are much more complex and they may undergo fluctuations of more extensive quantities. The boundary conditions impose on them particular intensive variables, like temperature gradients or distorted collective motions (shear motions, vortices, etc.), often called thermodynamic forces. If free energies are very useful in equilibrium thermodynamics, it must be stressed that there is no general law defining stationary non-equilibrium properties of the energy as is the second law of thermodynamics for the entropy in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential.

非平衡系统要复杂得多，并且可能会发生更大范围的波动。边界条件将特殊的强度变量强加给系统，例如温度梯度或扭曲的集体运动（剪切运动，涡旋等），通常称为热力学力。如果自由能在平衡热力学中非常有用，则必须要强调的是，没有任何定律能像热力学第二定律去定义平衡热力学中的熵那样，去定义能量的静态非平衡属性。这就是为什么在这种情况下，应考虑使用更广义的勒让德变换。这是扩展的马休势Massieu potential。


By definition, the entropy (S) is a function of the collection of extensive quantities [math]\displaystyle{ E_i }[/math]. Each extensive quantity has a conjugate intensive variable [math]\displaystyle{ I_i }[/math] (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:

根据定义，熵（S）是广延量集合的函数[math]\displaystyle{ E_i }[/math]。每个广延量都有一个共轭强化变量[math]\displaystyle{ I_i }[/math]（通过与该链接中给出的定义进行比较，在此使用了强化变量的受限定义）：


[math]\displaystyle{ I_i = \frac{\partial{S}}{\partial{E_i}}. }[/math]


We then define the extended Massieu function as follows: 然后，我们定义扩展的 马休函数Massieu function，如下所示：


[math]\displaystyle{ \ k_{\rm B} M = S - \sum_i( I_i E_i), }[/math]


where [math]\displaystyle{ \ k_{\rm B} }[/math] is Boltzmann's constant, whence 其中[math]\displaystyle{ \ k_{\rm B} }[/math]是 玻尔兹曼常数Boltzmann's constant，因此


[math]\displaystyle{ \ k_{\rm B} \, dM = \sum_i (E_i \, dI_i). }[/math]


The independent variables are the intensities.

其自变量是强度。


Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the thermodynamic forces driving fluxes of extensive properties through the system.

强度是全局值，对整个系统有效。当边界设定强加给系统不同的局部条件（例如，温度差异）时，将存在代表平均值的密集变量，还有一些代表梯度或更高矩的变量。后者是通过系统驱动广延特性通量的热动力。


It may be shown that the Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not.

可以证明，不管是否建立在平衡状态下，勒让德变换都在扩展的马休函数的最小状态下改变熵（在平衡时有效）的最大状态。

Stationary states, fluctuations, and stability 平稳状态，波动性和稳定性

In thermodynamics one is often interested in a stationary state of a process, allowing that the stationary state include the occurrence of unpredictable and experimentally unreproducible fluctuations in the state of the system. The fluctuations are due to the system's internal sub-processes and to exchange of matter or energy with the system's surroundings that create the constraints that define the process.

在热力学中，人们通常对过程的静态感兴趣，因为它涵盖了系统状态下发生的不可预测的和实验上不可再现的波动。波动是由于系统内部的子过程以及与系统周围的物质或能量交换引起的，从而形成了定义过程的约束。


If the stationary state of the process is stable, then the unreproducible fluctuations involve local transient decreases of entropy. The reproducible response of the system is then to increase the entropy back to its maximum by irreversible processes: the fluctuation cannot be reproduced with a significant level of probability. Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323). [30] The stable stationary state has a local maximum of entropy and is locally the most reproducible state of the system. There are theorems about the irreversible dissipation of fluctuations. Here 'local' means local with respect to the abstract space of thermodynamic coordinates of state of the system.

如果静态反应过程是稳定的，则不可重现的波动会涉及到熵的局部瞬时减小过程。系统的可重现响应是通过不可逆过程将熵增加回最大值：即不能以很大的概率再现波动。除了接近临界点外，关于稳态的波动很小（Kondepudi和Prigogine 1998，第323页）。稳定的静止状态具有局部的熵最大值，并且该系统最可能出现局部性重现响应状态。关于波动的不可逆耗散存在一些与之相关的定理。这里的“局部”是指相对于系统状态下热力学坐标的抽象空间而言的。


If the stationary state is unstable, then any fluctuation will almost surely trigger the virtually explosive departure of the system from the unstable stationary state. This can be accompanied by increased export of entropy.

如果静态反应过程是不稳定的，那么任何波动都很大概率会触发系统从不稳定的静止状态下产生爆炸，并伴随着熵输出的增加。

Local thermodynamic equilibrium 局部热力学平衡

The scope of present-day non-equilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in non-equilibrium thermodynamics of matter is that they deal with what is known as local thermodynamic equilibrium.

当今非平衡热力学的范围并不涵盖所有物理过程。在物质的非平衡热力学中有许多研究有效性的条件是：他们与所谓的局部热力学平衡相关。


Ponderable matter 物质的可估量性

Local thermodynamic equilibrium of matter[8][15][24][25][26] (see also Keizer (1987)[27] means that conceptually, for study and analysis, the system can be spatially and temporally divided into 'cells' or 'micro-phases' of small (infinitesimal) size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation. These conditions are unfulfilled, for example, in very rarefied gases, in which molecular collisions are infrequent; and in the boundary layers of a star, where radiation is passing energy to space; and for interacting fermions at very low temperature, where dissipative processes become ineffective. When these 'cells' are defined, one admits that matter and energy may pass freely between contiguous 'cells', slowly enough to leave the 'cells' in their respective individual local thermodynamic equilibria with respect to intensive variables.

从概念上讲，为了进行研究和分析，物质的局部热力学平衡（另请参见Keizer（1987）可以假设系统在空间和时间上划分为小尺寸（无穷小）的“细胞”或“微相”，那么其中物质的经典热力学平衡条件就能很好地满足。但是仍然存在某些条件无法得到满足，例如在极稀有的气体中，很少会发生分子碰撞；在恒星的边界层，辐射将能量传递到太空；以及在很低的温度下，与费米子的相互作用（其耗散过程变得无效）。当定义了这些“单元”时，人们承认物质和能量可以在相邻的“单元”之间自由地通过，其速度足以使“单元”（相对于强度变量）保持各自的局部热力学平衡。


One can think here of two 'relaxation times' separated by order of magnitude.[31] The longer relaxation time is of the order of magnitude of times taken for the macroscopic dynamical structure of the system to change. The shorter is of the order of magnitude of times taken for a single 'cell' to reach local thermodynamic equilibrium. If these two relaxation times are not well separated, then the classical non-equilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning[31] and other approaches have to be proposed, see for instance Extended irreversible thermodynamics. For example, in the atmosphere, the speed of sound is much greater than the wind speed; this favours the idea of local thermodynamic equilibrium of matter for atmospheric heat transfer studies at altitudes below about 60 km where sound propagates, but not above 100 km, where, because of the paucity of intermolecular collisions, sound does not propagate.

在这里，人们可以想到两个“弛豫时间”之间的数量级分隔。较长的弛豫时间约为系统宏观动力学结构发生变化所需的时间量级。较短的是独立“单元”达到局部热力学平衡所需的时间量级。如果这两个驰豫时间没有很好地分开，那么局部热力学平衡的经典非平衡热力学概念就失去了意义，那么必须提出其他方法，例如扩展的不可逆热力学。例如，在大气中，音速远大于风速；这有利于物质的局部热力学平衡的想法，对于在低于60 km的高空进行大气传热研究，声音可以在其中传播，但需要限制在100 km以内，因为分子间碰撞发生的很少，因此声音无法传播。

Milne's definition in terms of radiative equilibrium 米尔恩Milne在辐射平衡系统方面的定义

Edward A. Milne, thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the thermal radiation of the matter in each small local 'cell'.[32] He defined 'local thermodynamic equilibrium' in a 'cell' by requiring that it macroscopically absorb and spontaneously emit radiation as if it were in radiative equilibrium in a cavity at the temperature of the matter of the 'cell'. Then it strictly obeys Kirchhoff's law of equality of radiative emissivity and absorptivity, with a black body source function. The key to local thermodynamic equilibrium here is that the rate of collisions of ponderable matter particles such as molecules should far exceed the rates of creation and annihilation of photons.

爱德华·米尔恩Edward A. Milne在研究恒星时，根据每个局部“小单元”中物质的热辐射来定义“局部热力学平衡”。他通过设定“吸收并自发辐射（宏观意义上）”这一基本要求，定义研究对象处在“细胞”物质温度的空腔中，类似辐射平衡状态一样。然后，它严格遵守关于辐射发射率和吸收率相等的基尔霍夫定律Kirchhoff's law，以及黑体源函数。这里达到局部热力学平衡的关键在于重要物质颗粒的碰撞速率，例如分子应远远超过光子的产生和湮灭的速率。

Entropy in evolving systems 进化系统中的熵

It is pointed out by W.T. Grandy Jr,[33][34][35][36] that entropy, though it may be defined for a non-equilibrium system is—when strictly considered—only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.

WT Grandy Jr小W·T·格兰迪指出，尽管熵可能是为非平衡系统定义的，但严格来说，它只是一个宏观量，是指整个系统，不是动态变量，通常不充当描述局部物理力的局部势能。但是，在特殊情况下，人们可以隐喻地认为热变量的行为就像局部物理力一样。构成经典不可逆热力学的近似想法是建立在这种隐喻思维之上的。


This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics,[37][38][39][40] which evolved completely independently of statistical mechanics and maximum-entropy principles.

这种观点与连续热力学中熵的概念和用法有很多共同点，而后者完全独立于统计力学和最大熵原理而发展。


Entropy in non-equilibrium 非平衡状态的熵

To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables [math]\displaystyle{ x_1, x_2, ..., x_n }[/math] that are used to fix the equilibrium state, as was described above, a set of variables [math]\displaystyle{ \xi_1, \xi_2,\ldots }[/math] that are called internal variables have been introduced. The equilibrium state is considered to be stable and the main property of the internal variables, as measures of non-equilibrium of the system, is their trending to disappear; the local law of disappearing can be written as relaxation equation for each internal variable

为了描述热力学系统与平衡之间的偏差，除了如上所述的用于固定平衡状态的本构变量[math]\displaystyle{ x_1, x_2, ..., x_n }[/math]外，还引入了一组称为内部变量的变量[math]\displaystyle{ \xi_1, \xi_2,\ldots }[/math]。平衡状态被认为是稳定的，内部变量的主要性质（作为系统的非平衡度量）趋于消失；消失的局部定律可以写成每个内部变量的弛豫方程


[math]\displaystyle{ \frac{d\xi_i}{dt} = - \frac{1}{\tau_i} \, \left(\xi_i - \xi_i^{(0)} \right),\quad i =1,\,2,\ldots , }[/math]

 

 

 

 

(1)


where [math]\displaystyle{ \tau_i= \tau_i(T, x_1, x_2, \ldots, x_n) }[/math] is a relaxation time of a corresponding variables. It is convenient to consider the initial value [math]\displaystyle{ \xi_i^0 }[/math] are equal to zero. The above equation is valid for small deviations from equilibrium; The dynamics of internal variables in general case is considered by Pokrovskii.[41]

其中[math]\displaystyle{ \tau_i= \tau_i(T, x_1, x_2, \ldots, x_n) }[/math]是相应变量的弛豫时间。出于方便考虑初始值[math]\displaystyle{ \xi_i^0 }[/math]等于零。上面的方程对于偏离平衡态的小偏差是有效的；通常情况下，内部变量的动力学可以按照Pokrovskii的方法考虑。


Entropy of the system in non-equilibrium is a function of the total set of variables 系统在非平衡状态下的熵相当于变量总数的函数


[math]\displaystyle{ S=S(T, x_1, x_2, , x_n; \xi_1, \xi_2, \ldots) }[/math]

 

 

 

 

(1)


The essential contribution to the thermodynamics of the non-equilibrium systems was brought by Prigogine, when he and his collaborators investigated the systems of chemically reacting substances. The stationary states of such systems exists due to exchange both particles and energy with the environment. In section 8 of the third chapter of his book,[42] Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature [math]\displaystyle{ T }[/math] . The increment of entropy [math]\displaystyle{ S }[/math] can be calculated according to the formula

普利高因Prigogine在他和他的合作者研究化学反应物质的系统时，对非平衡系统的热力学做出了重要贡献。由于与环境交换粒子和能量，因此是存在这种稳态的系统。在他的书的第三章第8节中，Prigogine指定了在给定体积和恒定温度[math]\displaystyle{ T }[/math]下所考虑的系统熵变化的三个贡献。可以根据以下公式计算熵的增量S


[math]\displaystyle{ T\,dS = \Delta Q - \sum_{j} \, \Xi_{j} \,\Delta \xi_j + \sum_{\alpha =1}^k\, \mu_\alpha \, \Delta N_\alpha. }[/math]

 

 

 

 

(1)


The first term on the right hand side of the equation presents a stream of thermal energy into the system; the last term—a stream of energy into the system coming with the stream of particles of substances [math]\displaystyle{ \Delta N_\alpha }[/math] that can be positive or negative, [math]\displaystyle{ \mu_\alpha }[/math] is chemical potential of substance [math]\displaystyle{ \alpha }[/math]. The middle term in (1) depicts energy dissipation (entropy production) due to the relaxation of internal variables [math]\displaystyle{ \xi_j }[/math]. In the case of chemically reacting substances, which was investigated by Prigogine, the internal variables appear to be measures of incompleteness of chemical reactions, that is measures of how much the considered system with chemical reactions is out of equilibrium. The theory can be generalised,[43][41] to consider any deviation from the equilibrium state as an internal variable, so that we consider the set of internal variables [math]\displaystyle{ \xi_j }[/math] in equation (1) to consist of the quantities defining not only degrees of completeness of all chemical reactions occurring in the system, but also the structure of the system, gradients of temperature, difference of concentrations of substances and so on.

等式右边的第一项表示进入系统的热能流。最后一项是进入系统的能量流与可能为正或负的物质粒子流[math]\displaystyle{ \Delta N_\alpha }[/math]一起出现，[math]\displaystyle{ \mu_\alpha }[/math]为物质[math]\displaystyle{ \alpha }[/math]的 化学势Chemical potential。（1）中的中间项描述了由于内部变量[math]\displaystyle{ \xi_j }[/math]的松弛而导致的 能量耗散Energy dissipation（熵产生）。就化学反应物质而言（由Prigogine调查），内部变量似乎是化学反应不完整的量度，即所考虑的具有化学反应的系统失衡程度。该理论可以推广为，与平衡状态的任何偏差视为内部变量，因此我们认为方程式（1）中的内部变量[math]\displaystyle{ \xi_j }[/math]不仅由定义系统中发生的所有化学反应的完成度量组成，而且还有系统的结构，温度梯度，物质浓度的差异等等组成。

Flows and forces

The fundamental relation of classical equilibrium thermodynamics [44]


[math]\displaystyle{ dS=\frac{1}{T}dU+\frac{p}{T}dV-\sum_{i=1}^s\frac{\mu_i}{T}dN_i }[/math]


expresses the change in entropy [math]\displaystyle{ dS }[/math] of a system as a function of the intensive quantities temperature [math]\displaystyle{ T }[/math], pressure [math]\displaystyle{ p }[/math] and [math]\displaystyle{ i^{th} }[/math] chemical potential [math]\displaystyle{ \mu_i }[/math] and of the differentials of the extensive quantities energy [math]\displaystyle{ U }[/math], volume [math]\displaystyle{ V }[/math] and [math]\displaystyle{ i^{th} }[/math] particle number [math]\displaystyle{ N_i }[/math].


Following Onsager (1931,I),[10] let us extend our considerations to thermodynamically non-equilibrium systems. As a basis, we need locally defined versions of the extensive macroscopic quantities [math]\displaystyle{ U }[/math], [math]\displaystyle{ V }[/math] and [math]\displaystyle{ N_i }[/math] and of the intensive macroscopic quantities [math]\displaystyle{ T }[/math], [math]\displaystyle{ p }[/math] and [math]\displaystyle{ \mu_i }[/math].


For classical non-equilibrium studies, we will consider some new locally defined intensive macroscopic variables. We can, under suitable conditions, derive these new variables by locally defining the gradients and flux densities of the basic locally defined macroscopic quantities.


Such locally defined gradients of intensive macroscopic variables are called 'thermodynamic forces'. They 'drive' flux densities, perhaps misleadingly often called 'fluxes', which are dual to the forces. These quantities are defined in the article on Onsager reciprocal relations.


Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities ([math]\displaystyle{ J_i }[/math]) may be coupled. The article on Onsager reciprocal relations considers the stable near-steady thermodynamically non-equilibrium regime, which has dynamics linear in the forces and flux densities.


In stationary conditions, such forces and associated flux densities are by definition time invariant, as also are the system's locally defined entropy and rate of entropy production. Notably, according to Ilya Prigogine and others, when an open system is in conditions that allow it to reach a stable stationary thermodynamically non-equilibrium state, it organizes itself so as to minimize total entropy production defined locally. This is considered further below.


One wants to take the analysis to the further stage of describing the behaviour of surface and volume integrals of non-stationary local quantities; these integrals are macroscopic fluxes and production rates. In general the dynamics of these integrals are not adequately described by linear equations, though in special cases they can be so described.


Onsager reciprocal relations

Main article: Onsager reciprocal relations

Following Section III of Rayleigh (1873),[9] Onsager (1931, I)[10] showed that in the regime where both the flows ([math]\displaystyle{ J_i }[/math]) are small and the thermodynamic forces ([math]\displaystyle{ F_i }[/math]) vary slowly, the rate of creation of entropy [math]\displaystyle{ (\sigma) }[/math] is linearly related to the flows:


[math]\displaystyle{ \sigma = \sum_i J_i\frac{\partial F_i}{\partial x_i} }[/math]


and the flows are related to the gradient of the forces, parametrized by a matrix of coefficients conventionally denoted [math]\displaystyle{ L }[/math]:


[math]\displaystyle{ J_i = \sum_{j} L_{ij} \frac{\partial F_j}{\partial x_j} }[/math]


from which it follows that:


[math]\displaystyle{ \sigma = \sum_{i,j} L_{ij} \frac{\partial F_i}{\partial x_i}\frac{\partial F_j}{\partial x_j} }[/math]


The second law of thermodynamics requires that the matrix [math]\displaystyle{ L }[/math] be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix [math]\displaystyle{ L }[/math] is symmetric. This fact is called the Onsager reciprocal relations.


The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.[41]


Speculated extremal principles for non-equilibrium processes

Main article: Extremal principles in non-equilibrium thermodynamics


Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999)[45] concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008)[2] is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in Onsager's 1931[10] origination of this subject. Other writers have also felt that prospects for general global extremal principles are clouded. Such writers include Glansdorff and Prigogine (1971), Lebon, Jou and Casas-Vásquez (2008), and Šilhavý (1997).

There is good experimental evidence that heat convection does not obey extremal principles for time rate of entropy production.[46] Theoretical analysis shows that chemical reactions do not obey extremal principles for the second differential of time rate of entropy production.[47] The development of a general extremal principle seems infeasible in the current state of knowledge.


Applications

Non-equilibrium thermodynamics has been successfully applied to describe biological processes such as protein folding/unfolding and transport through membranes.[48][49]

It is also used to give a description of the dynamics of nanoparticles, which can be out of equilibrium in systems where catalysis and electrochemical conversion is involved.[50]

Also, ideas from non-equilibrium thermodynamics and the informatic theory of entropy have been adapted to describe general economic systems.[51]

[52]


See also

Category:Concepts in physics

分类: 物理概念

Category:Branches of thermodynamics

分类: 热力学的分支


This page was moved from wikipedia:en:Non-equilibrium thermodynamics. Its edit history can be viewed at 非平衡系统/edithistory

  1. ↑ Bodenschatz, Eberhard; Cannell, David S.; de Bruyn, John R.; Ecke, Robert; Hu, Yu-Chou; Lerman, Kristina; Ahlers, Guenter (December 1992). "Experiments on three systems with non-variational aspects". Physica D: Nonlinear Phenomena. 61 (1–4): 77–93. doi:10.1016/0167-2789(92)90150-L.
  2. ↑ 2.0 2.1 Grandy, W.T., Jr (2008).
  3. ↑ 3.0 3.1 3.2 3.3 Lebon, G., Jou, D., Casas-Vázquez, J. (2008). Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers, Springer-Verlag, Berlin, e-
    .
  4. ↑ 4.0 4.1 Lieb, E.H., Yngvason, J. (1999), p. 5.
  5. ↑ Gyarmati, I. (1967/1970), pp. 8–12.
  6. ↑ Callen, H.B. (1960/1985), § 4–2.
  7. ↑ Glansdorff, P., Prigogine, I. (1971), Ch. II,§ 2.
  8. ↑ 8.0 8.1 8.2 8.3 Gyarmati, I. (1967/1970).
  9. ↑ 9.0 9.1 Strutt, J. W. (1871). "Some General Theorems relating to Vibrations". Proceedings of the London Mathematical Society. s1-4: 357–368. doi:10.1112/plms/s1-4.1.357.
  10. ↑ 10.0 10.1 10.2 10.3 10.4 Onsager, L. (1931). "Reciprocal relations in irreversible processes, I". Physical Review. 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405.
  11. ↑ 11.0 11.1 11.2 11.3 Lavenda, B.H. (1978). Thermodynamics of Irreversible Processes, Macmillan, London,
    .
  12. ↑ Gyarmati, I. (1967/1970), pages 4-14.
  13. ↑ Ziegler, H., (1983). An Introduction to Thermomechanics, North-Holland, Amsterdam,
    .
  14. ↑ Balescu, R. (1975). Equilibrium and Non-equilibrium Statistical Mechanics, Wiley-Interscience, New York,
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  15. ↑ 15.0 15.1 15.2 Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley-Interscience, London, 1971,
    .
  16. ↑ 16.0 16.1 Jou, D., Casas-Vázquez, J., Lebon, G. (1993). Extended Irreversible Thermodynamics, Springer, Berlin,
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  17. ↑ Eu, B.C. (2002).
  18. ↑ Wildt, R. (1972). "Thermodynamics of the gray atmosphere. IV. Entropy transfer and production". Astrophysical Journal. 174: 69–77. Bibcode:1972ApJ...174...69W. doi:10.1086/151469.
  19. ↑ Essex, C. (1984a). "Radiation and the irreversible thermodynamics of climate". Journal of the Atmospheric Sciences. 41 (12): 1985–1991. Bibcode:1984JAtS...41.1985E. doi:10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2..
  20. ↑ Essex, C. (1984b). "Minimum entropy production in the steady state and radiative transfer". Astrophysical Journal. 285: 279–293. Bibcode:1984ApJ...285..279E. doi:10.1086/162504.
  21. ↑ Essex, C. (1984c). "Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes". Planetary and Space Science. 32 (8): 1035–1043. Bibcode:1984P&SS...32.1035E. doi:10.1016/0032-0633(84)90060-6.
  22. ↑ Prigogine, I., Defay, R. (1950/1954). Chemical Thermodynamics, Longmans, Green & Co, London, page 1.
  23. ↑ 23.0 23.1 De Groot, S.R., Mazur, P. (1962). Non-equilibrium Thermodynamics, North-Holland, Amsterdam.
  24. ↑ 24.0 24.1 Balescu, R. (1975). Equilibrium and Non-equilibrium Statistical Mechanics, John Wiley & Sons, New York,
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  25. ↑ 25.0 25.1 Mihalas, D., Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press, New York
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  26. ↑ 26.0 26.1 Schloegl, F. (1989). Probability and Heat: Fundamentals of Thermostatistics, Freidr. Vieweg & Sohn, Braunschweig,
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  27. ↑ 27.0 27.1 Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York,
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  28. ↑ Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester UK,
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  29. ↑ Coleman, B.D.; Noll, W. (1963). "The thermodynamics of elastic materials with heat conduction and viscosity". Arch. Ration. Mach. Analysis. 13 (1): 167–178. Bibcode:1963ArRMA..13..167C. doi:10.1007/bf01262690. S2CID 189793830.
  30. ↑ Kondepudi, D., Prigogine, I, (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures, Wiley, Chichester, 1998,
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  31. ↑ 31.0 31.1 Zubarev D. N.,(1974). Nonequilibrium Statistical Thermodynamics, translated from the Russian by P.J. Shepherd, New York, Consultants Bureau.
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  32. ↑ Milne, E.A. (1928). "The effect of collisions on monochromatic radiative equilibrium". Monthly Notices of the Royal Astronomical Society. 88 (6): 493–502. Bibcode:1928MNRAS..88..493M. doi:10.1093/mnras/88.6.493.
  33. ↑ Grandy, W.T., Jr. (2004). "Time Evolution in Macroscopic Systems. I. Equations of Motion". Foundations of Physics. 34 (1): 1. arXiv:cond-mat/0303290. Bibcode:2004FoPh...34....1G. doi:10.1023/B:FOOP.0000012007.06843.ed.
  34. ↑ Grandy, W.T., Jr. (2004). "Time Evolution in Macroscopic Systems. II. The Entropy". Foundations of Physics. 34 (1): 21. arXiv:cond-mat/0303291. Bibcode:2004FoPh...34...21G. doi:10.1023/B:FOOP.0000012008.36856.c1. S2CID 18573684.
  35. ↑ Grandy, W. T., Jr (2004). "Time Evolution in Macroscopic Systems. III: Selected Applications". Foundations of Physics. 34 (5): 771. Bibcode:2004FoPh...34..771G. doi:10.1023/B:FOOP.0000022187.45866.81. S2CID 119406182.
  36. ↑ Grandy 2004 see also [1].
  37. ↑ Truesdell, Clifford (1984). Rational Thermodynamics (2 ed.). Springer. 
  38. ↑ Maugin, Gérard A. (2002). Continuum Thermomechanics. Kluwer. 
  39. ↑ Gurtin, Morton E. (2010). The Mechanics and Thermodynamics of Continua. Cambridge University Press. 
  40. ↑ Amendola, Giovambattista (2012). Thermodynamics of Materials with Memory: Theory and Applications. Springer. 
  41. ↑ 41.0 41.1 41.2 Pokrovskii V.N. (2013) A derivation of the main relations of non-equilibrium thermodynamics. Hindawi Publishing Corporation: ISRN Thermodynamics, vol. 2013, article ID 906136, 9 p. https://dx.doi.org/10.1155/2013/906136.
  42. ↑ Prigogine, I. (1955/1961/1967). Introduction to Thermodynamics of Irreversible Processes. 3rd edition, Wiley Interscience, New York.
  43. ↑ Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.
  44. ↑ W. Greiner, L. Neise, and H. Stöcker (1997), Thermodynamics and Statistical Mechanics (Classical Theoretical Physics) ,Springer-Verlag, New York, P85, 91, 101,108,116,
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  45. ↑ Nicolis, C. (1999). "Entropy production and dynamical complexity in a low-order atmospheric model". Quarterly Journal of the Royal Meteorological Society. 125 (557): 1859–1878. Bibcode:1999QJRMS.125.1859N. doi:10.1002/qj.49712555718.
  46. ↑ Attard, P. (2012). "Optimising Principle for Non-Equilibrium Phase Transitions and Pattern Formation with Results for Heat Convection". arXiv:1208.5105 [cond-mat.stat-mech]. {{cite arxiv}}: Cite has empty unknown parameter: |journal= (help)
  47. ↑ Keizer, J.; Fox, R. (January 1974). "Qualms Regarding the Range of Validity of the Glansdorff-Prigogine Criterion for Stability of Non-Equilibrium States". PNAS. 71: 192–196. doi:10.1073/pnas.71.1.192. PMID 16592132.
  48. ↑ Kimizuka, Hideo; Kaibara, Kozue (September 1975). "Nonequilibrium thermodynamics of ion transport through membranes". Journal of Colloid and Interface Science. 52 (3): 516–525. doi:10.1016/0021-9797(75)90276-3.
  49. ↑ Baranowski, B. (April 1991). "Non-equilibrium thermodynamics as applied to membrane transport". Journal of Membrane Science. 57 (2–3): 119–159. doi:10.1016/S0376-7388(00)80675-4.
  50. ↑ Bazant, Martin Z. (22 March 2013). "Theory of Chemical Kinetics and Charge Transfer based on Nonequilibrium Thermodynamics". Accounts of Chemical Research. 46 (5): 1144–1160. arXiv:1208.1587. doi:10.1021/ar300145c. PMID 23520980. S2CID 10827167.
  51. ↑ Pokrovskii, Vladimir (2011). Econodynamics. The Theory of Social Production.. https://www.springer.com/physics/complexity/book/978-94-007-2095-4: Springer, Dordrecht-Heidelberg-London-New York.. 
  52. ↑ Chen, Jing (2015). The Unity of Science and Economics: A New Foundation of Economic Theory. https://www.springer.com/us/book/9781493934645: Springer. 
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