非平衡系统

此词条由Jie翻译。

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.

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.

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

Difference between equilibrium and non-equilibrium thermodynamics

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.

平衡与非平衡态热力学之间有着深刻的区别。平衡态热力学忽略了物理过程的时间过程。相比之下，非平衡态热力学试图不断详细地描述他们的时间过程。

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. 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. 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.

平衡态热力学将它的考虑局限于具有热力学平衡初始和终止状态的过程，过程的时间过程被故意忽略。因此，平衡态热力学允许过程通过远离热力学平衡的状态，这些过程甚至不能用非平衡态热力学所允许的变量来描述，比如温度和压力的时间变化率。例如，在平衡态热力学中，一个过程甚至可以包括一个非平衡态热力学无法描述的剧烈爆炸。这是一个微分几何的练习，而不是一个可能在现实中发生的过程。

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.

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，非平衡态热力学，试图描述连续的时间过程，需要它的状态变量与平衡态热力学的状态变量有非常密切的联系。这深刻地限制了非平衡态热力学的范围，并对其概念框架安全部门提出了沉重的要求。

Non-equilibrium state variables

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. 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合适关系如下。当系统碰巧处于足够接近热力学平衡的状态时，非平衡态变量可以通过与测量热力学状态变量相同的技术，或者通过相应的时间和空间导数，包括物质和能量的流动，足够精确地在局部测量。一般来说，非平衡态热力学系统在空间和时间上都是不均匀的，但是它们的不均匀性仍然具有足够的光滑度来支持存在合适的非平衡态变量的时间和空间导数。由于空间非均匀性，对应于广义热力学状态变量的非平衡状态变量必须定义为相应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，与平衡状态变量密切对应。为了获得相应的非均匀性，测量探头必须足够小，响应速度也必须足够快。此外，非平衡状态变量需要在数学上相互之间以适当类似于平衡热力学状态变量之间对应关系的方式进行功能联系。昂萨格1931)、产生熵时间速率(昂萨格1931)、耗散结构，但在本文中几乎没有涉及。

Overview

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.

According to Wildt (see also Essex), 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.

根据 Wildt 的说法，当前版本的非平衡态热力学星云忽略了辐射热; 它们之所以能做到这一点，是因为它们指的是实验室条件下的物质数量，而实验室条件下的物质温度远低于恒星的温度。在实验室温度下，在实验室数量的物质中，热辐射很弱，几乎可以忽略不计。但是，例如，大气物理学关注的是占据立方公里的大量物质，作为一个整体，不在实验室数量的范围内; 那么热辐射就不能被忽视。

Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873, Onsager 1931, also), time rate of entropy production (Onsager 1931), thermodynamic fields, dissipative structure, and non-linear dynamical structure.

The terms 'classical irreversible thermodynamics' 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.

系统的经典不可逆热力学术语。在一些著作中，假定平衡热力学的密集变量充分作为任务的独立变量(这些变量被认为没有”记忆” ，不显示滞后) ; 特别是，局部流密集变量不被承认为独立变量; 局部流被认为是依赖于准静态的局部密集变量。

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.

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 (see also Keizer (1987)). 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.

同时假设局部熵密度与平衡状态下其他局部密集型变量的函数相同，这被称为局部热力学平衡假设(参见 Keizer (1987))。辐射之所以被忽略，是因为它是能量在区域之间的转移，而区域之间可以相互远离。在经典的不可逆热力学方法中，允许有非常小的空间变化，从非常小的体积元到相邻的非常小的体积元，但是假定系统的总体熵可以通过简单的局部熵密度的空间积分得到，这意味着空间结构不能对系统的总体熵评价作出贡献，因为它应该对系统的总体熵评价作出贡献。这种方法假设空间和时间的连续性，甚至可微的局部定义的强度变量，如温度和内部能量密度。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范围的现象。然而，这种方法是有价值的，因为它可以很好地处理一些宏观上可观察到的现象。

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

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. These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation.

在其他著作中，考虑了局部流动变量; 这些可以被认为是经典的，类比于由无休止的重复循环过程产生的流动的时不变的长期时间平均值; 有关流动的例子是被称为 Seebeck 和 Peltier 效应的热电现象，开尔文在十九世纪和拉斯昂萨格尔在二十世纪考虑。这些效应发生在金属连接处，最初被有效地处理为二维表面，没有空间体积，也没有空间变化。

Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions

According to Wildt (see also Essex), 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.

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.

局域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不仅依赖于现值，而且依赖于局域平衡变量的过去值。因此，对于无记忆材料，时间比依赖时间的局域平衡热力学更为深入，但是通量并不是状态的独立变量。

Local equilibrium thermodynamics

The terms 'classical irreversible thermodynamics' 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 is needed in choosing the independent variables 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.

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.

扩展不可逆热力学是非平衡态热力学热力学的一个分支，它超越了局部平衡假说的限制。状态变量的空间通过包括质量、动量和能量的通量以及最终的高阶通量来扩大。

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 (see also Keizer (1987)). 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

The formalism is well-suited for describing high-frequency processes and small-length scales materials.

形式主义非常适合于描述高频过程和小尺度材料。

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. These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation.

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.

有许多固定的非平衡系统的例子，其中一些非常简单，例如在不同温度下被限制在两个恒温器之间的系统，或普通的 Couette 流动，两个平板壁之间的流体沿相反方向运动，并定义了壁上的非平衡条件。激光作用也是一个非平衡过程，但它依赖于脱离局部热力学平衡，因此超出了经典不可逆热力学的范围; 在这里，两个分子自由度(分子激光、振动和转动分子运动)之间保持了很大的温差，在一个很小的空间区域需要两个分量的‘温度’ ，排除了局部热力学平衡，这就要求只需要一个温度。声扰动或激波的阻尼是非平稳的非平衡过程。被驱动的复杂流体、湍流系统和玻璃是非平衡系统的其他例子。

Local equilibrium thermodynamics with materials with "memory"

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.

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.

宏观系统的力学依赖于大量的数据。应当强调的是，所有系统都与其周围环境永久地相互作用，从而造成大量不可避免的波动。热力学系统的平衡条件与熵的最大性质有关。如果唯一允许大范围波动的量是内能，而其它量都严格保持恒定，那么系统的温度就是可测量的，也是有意义的。系统的性质可以用热动力位能亥姆霍兹自由能(a = u-TS)来描述，它是能量的勒壤得转换。如果除了能量的波动，系统的宏观尺寸(体积)仍然保持波动状态，我们使用吉布斯自由能(g = u + PV-TS) ，其中系统的性质既由温度决定，也由压力决定。

Extended irreversible thermodynamics

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，它们可能经历更广泛的量的波动。边界条件强加给它们特殊的强度变量，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。如果自由能在平衡态热力学中非常有用，那么必须强调的是，在平衡态热力学中，定义能量的稳态非平衡性质的一般定律，并不像平衡态热力学中熵的热力学第二定律定律那样。这就是为什么在这种情况下，应该考虑一个更广泛的勒壤得转换。这就是延伸的马西欧势。

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.

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)是大量集合的函数。每个扩展量都有一个共轭密集变量 i _ i </math > (通过与本链接中给出的定义进行比较，这里使用了密集变量的有限定义) ，因此:

The formalism is well-suited for describing high-frequency processes and small-length scales materials.

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

[数学] i = frac { partial { s }{ e _ i }

Basic concepts

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.

We then define the extended Massieu function as follows:

然后我们将扩展的 Massieu 函数定义如下:

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.

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

[ math > k { rm b } m = s-sum _ i (i _ i e _ i) ,

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.

where [math]\displaystyle{ \ k_{\rm B} }[/math] is Boltzmann's constant, whence

波尔兹曼常数是从哪里来的

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:

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

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

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

The independent variables are the intensities.

自变量是强度。

We then define the extended Massieu function as follows:

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，有效的系统作为一个整体。当边界对系统施加不同的局部条件时，例如:。温度差) ，有密集的变量代表平均值和其他代表梯度或更高的矩。后者是热力学力量，驱动广泛性质的通量通过系统。

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

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，无论是否处于平衡状态，勒壤得转换改变了定态扩展 Massieu 函数的最小条件下熵的最大条件(平衡时有效)。

where [math]\displaystyle{ \ k_{\rm B} }[/math] is Boltzmann's constant, whence

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

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，人们经常对一个过程的定态感兴趣，允许系统状态的定态包括不可预测和实验上不可重复的涨落的发生。波动是由于系统的内部子过程和物质或能量与系统周围环境的交换所造成的，这些环境制约了这一过程。

The independent variables are the intensities.

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). 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.

如果这个过程的定态是稳定的，那么不可复制的涨落就包含了局部瞬时的熵减。系统的可重复响应是通过不可逆过程将熵增加到最大值: 涨落不能以显著的概率水平再现。稳定定态的波动极小，除了临界点附近(Kondepudi 和 Prigogine 1998,323页)。稳定的定态有一个局部最大熵，是系统局部最可复制的状态。关于涨落的不可逆耗散有几个定理。这里“局部”是指相对于系统状态热力学坐标的抽象空间的局部。

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.

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.

如果定态是不稳定的，那么任何波动几乎肯定会触发系统几乎爆炸性地偏离不稳定的定态。这可能伴随着熵输出的增加。

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.

Stationary states, fluctuations, and stability

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.

当今非平衡态热力学的范围并不包括所有的物理过程。在物质的非平衡态热力学几年，许多研究有效的一个条件是，他们处理的是所谓的局部热力学平衡。

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.

Local thermodynamic equilibrium of matter 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 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.

物质的局部热力学平衡较长的弛豫时间是系统宏观动力学结构发生变化所需时间的数量级。较短的一个数量级是一个单细胞到达本地热力学平衡所需的时间。如果这两个弛豫时间没有很好地分开，那么局部热力学平衡的经典的非平衡热力学概念就失去了它的意义。他定义了“细胞中的局部热力学平衡” ，要求它在宏观上吸收和自发地发射辐射，就好像它在一个与“细胞”物质温度相当的辐射平衡中一样。然后严格遵守基尔霍夫辐射发射率和吸收率相等的定律，使用一个黑体源函数。这里局域热力学平衡的关键在于，可重物质粒子的碰撞速率，比如分子，应该远远超过光子的产生和湮灭速率。

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). 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.

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.

It is pointed out by W.T. Grandy Jr, 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.

这是 w.t. 指出的。Grandy Jr 认为，熵虽然可以被定义为非平衡系统，但是在严格考虑时，它只是一个指向整个系统的宏观量，而不是一个动力学变量，一般不作为描述局部物理力的局部势。然而，在特殊情况下，人们可以隐喻地认为，热变量表现得像局部物理力量。构成经典不可逆热力学的近似是建立在这种隐喻思维之上的。

Local thermodynamic equilibrium

This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics, 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使用有许多共同点，连续热力学完全独立于统计力学和最大熵原理演化。

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.

Ponderable matter

Local thermodynamic equilibrium of matter (see also Keizer (1987) 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.

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偏离平衡状态的程度，除了用于确定平衡状态的本构变量 x _ 1，x _ 2，... ，x _ n </math > 之外，还引入了一组称为内部变量的变量 x _ 1，x _ 2，ldots </math > 。平衡态被认为是稳定的，内变量作为系统非平衡的度量，其主要性质是它们趋于消失，消失的局部规律可以写成每个内变量的松弛方程

[math]\displaystyle{ {{ NumBlk | : | \lt math \gt One can think here of two 'relaxation times' separated by order of magnitude.\lt ref name="Zubarev 1971/1974"\gt [[Dmitry Zubarev|Zubarev D. N.]],(1974). ''[https://books.google.com/books?id=SQy3AAAAIAAJ&hl=ru&source=gbs_ViewAPI Nonequilibrium Statistical Thermodynamics]'', translated from the Russian by P.J. Shepherd, New York, Consultants Bureau. {{ISBN|0-306-10895-X}}; {{ISBN|978-0-306-10895-2}}.\lt /ref\gt 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\lt ref name="Zubarev 1971/1974"/\gt 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. \frac{d\xi_i}{dt} = - \frac{1}{\tau_i} \, \left(\xi_i - \xi_i^{(0)} \right),\quad i =1,\,2,\ldots , 1} ，左(xi-xi-xi _ i ^ {(0)}右) ，quad i = 1，，2，ldots, }[/math]

()

[/math > | }

Milne's definition in terms of radiative equilibrium

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.

其中 < math > tau _ i = tau _ i (t，x _ 1，x _ 2，ldots，x _ n) </math > 是对应变量的松弛时间。考虑初始值 < math > xi _ i ^ 0 </math > 等于零是很方便的。上述方程适用于小偏离均衡的情况，Pokrovskii 考虑了一般情况下内变量的动力学。

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'. 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.

Entropy of the system in non-equilibrium is a function of the total set of variables

非平衡态系统的熵是总变量集的函数

Entropy in evolving systems

[math]\displaystyle{ {{ NumBlk | : | \lt math \gt It is pointed out by W.T. Grandy Jr,\lt ref\gt {{cite journal | doi = 10.1023/B:FOOP.0000012007.06843.ed | title = Time Evolution in Macroscopic Systems. I. Equations of Motion | year = 2004 | last1 = Grandy | first1 = W.T., Jr. | journal = Foundations of Physics | volume = 34 | issue = 1 | page = 1 |url=http://physics.uwyo.edu/~tgrandy/evolve.html |arxiv = cond-mat/0303290 |bibcode = 2004FoPh...34....1G }}\lt /ref\gt \lt ref\gt {{cite journal | url=http://physics.uwyo.edu/~tgrandy/entropy.html | doi=10.1023/B:FOOP.0000012008.36856.c1 | title=Time Evolution in Macroscopic Systems. II. The Entropy | year=2004 | last1=Grandy | first1=W.T., Jr. | journal=Foundations of Physics | volume=34 | issue=1 | page=21 |arxiv = cond-mat/0303291 |bibcode = 2004FoPh...34...21G | s2cid=18573684 }}\lt /ref\gt \lt ref\gt {{cite journal | url=http://physics.uwyo.edu/~tgrandy/applications.html | doi = 10.1023/B:FOOP.0000022187.45866.81 | title=Time Evolution in Macroscopic Systems. III: Selected Applications | year=2004 | last1=Grandy | first1=W. T., Jr | journal=Foundations of Physics | volume=34 | issue=5 | page=771 |bibcode = 2004FoPh...34..771G | s2cid = 119406182 }}\lt /ref\gt \lt ref\gt Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].\lt /ref\gt 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. S=S(T, x_1, x_2, , x_n; \xi_1, \xi_2, \ldots) S = s (t，x1，x2，，xn; xi _ 1，xi _ 2，ldots) }[/math]

()

[/math > | }

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

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, 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

对非平衡系统热力学的本质贡献是由普里高金提出的，当时他和他的合作者研究了化学反应物质系统。由于粒子和能量与环境的交换，这类系统的静止状态是存在的。在他的书的第三章的第8节中，普里高金详细说明了在给定的体积和恒定的温度下，被考虑系统的熵的变化有三个贡献。根据该公式可以计算出熵的增量 s </math >

[math]\displaystyle{ {{ NumBlk | : | \lt math \gt ===Entropy in non-equilibrium=== T\,dS = \Delta Q - \sum_{j} \, \Xi_{j} \,\Delta \xi_j + \sum_{\alpha =1}^k\, \mu_\alpha \, \Delta N_\alpha. T，dS = Delta q-sum { j } ，Xi { j } ，Delta Xi _ j + sum _ { alpha = 1} ^ k，mu _ alpha，Delta n _ alpha. }[/math]

()

[/math > | }

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

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,

方程式右边的第一项代表了进入系统的热能流; 最后一项ーー进入系统的能量流，伴随着粒子流进入系统，粒子流可以是正的也可以是负的。第一部分的中期描述了由于内部变量的松弛而引起的能量耗散(产生熵)。在化学反应物质的情况下，由普利戈金研究，内部变量似乎是测量不完全的化学反应，也就是测量多少考虑的体系与化学反应是不平衡的。这个理论可以推广,

1

({{{3}}})

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.

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].

表示系统熵的变化是密集量，温度，数学，压力，数学，化学势，以及大量能量的微分的函数。

Entropy of the system in non-equilibrium is a function of the total set of variables

Following Onsager (1931,I), 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) Theoretical analysis shows that chemical reactions do not obey extremal principles for the second differential of time rate of entropy production. The development of a general extremal principle seems infeasible in the current state of knowledge.

在 Onsager (1931，i)之后，她得出结论，大气动力学的一个模型有一个吸引子，它不是最大或最小耗散制度; 她说，这似乎排除了全球组织原则的存在，并评论说，这在某种程度上是令人失望的; 她还指出，很难找到一个热力学上一致的形式的产生熵。另一位顶尖专家对产生熵极值原理和能量耗散原理的可能性进行了广泛的讨论: Grandy 第12章(2008)理论分析表明，化学反应在产生熵的第二个时间速率微分中不遵守极值原理。在目前的知识状态下，发展一般的极值原理似乎是不可行的。

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

(1)

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

非平衡态热力学已成功地应用于描述蛋白质折叠/去折叠和通过膜转运等生物学过程。

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, 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

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.

它也被用来描述纳米颗粒的动力学，在涉及催化和电化学转化的系统中，纳米颗粒可以失去平衡。

[math]\displaystyle{ Also, ideas from non-equilibrium thermodynamics and the informatic theory of entropy have been adapted to describe general economic systems. 此外，来自非平衡态热力学的思想和熵的信息论已经被用来描述一般的经济系统。 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, 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.

Flows and forces

The fundamental relation of classical equilibrium thermodynamics

[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), 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), Onsager (1931, I) 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.

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) 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) 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 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. Theoretical analysis shows that chemical reactions do not obey extremal principles for the second differential of time rate of entropy production. 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.

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.

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

See also

• Time crystal

• Dissipative system

• Entropy production

• Extremal principles in non-equilibrium thermodynamics

• Self-organization

• Autocatalytic reactions and order creation

Category:Concepts in physics

分类: 物理概念

• Self-organizing criticality

Category:Branches of thermodynamics

分类: 热力学的分支

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