非平衡动力学
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模板:Cleanup rewrite模板:Thermodynamics
Nonequilibrium 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 (nonequilibrium state variables) that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Nonequilibrium 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. Nonequilibrium thermodynamics is a work in progress, not an established edifice. This article will try to sketch some approaches to it and some concepts important for it.
Nonequilibrium 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 (nonequilibrium state variables) that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Nonequilibrium 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. Nonequilibrium thermodynamics is a work in progress, not an established edifice. This article will try to sketch some approaches to it and some concepts important for it.
非平衡态热力学热力学是热力学的一个分支，研究的物理系统不在热力学平衡中，但可以用变量(非平衡态变量)来描述，这些变量代表用来指定热力学平衡系统的变量的外推。非平衡态热力学与输运过程和化学反应速率有关。它依赖于被认为是或多或少接近热力学平衡的东西。非平衡态热力学是一项正在进行的工作，而不是一座已经建立的大厦。本文将试图勾勒出一些方法和一些重要的概念。
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 nonequilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of nonequilibrium 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 nonequilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope of nonequilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of free energy is lost .
自然界中发现的几乎所有系统都不在热力学平衡中，因为它们在不断变化，或者可以随着时间的推移而触发变化，并且不断不断地受制于物质和能量的流动，这些流动来自其他系统和化学反应。然而，一些系统和进程，在一个有用的意义上，足够接近热力学平衡，以允许目前已知的非平衡态热力学有用的精确描述。然而，由于非变分动力学的存在，许多自然系统和过程将永远远远超出非平衡热力学方法的范围，其中自由能的概念丢失。
The thermodynamic study of nonequilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. One fundamental difference between equilibrium thermodynamics and nonequilibrium 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 nonequilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. One fundamental difference between equilibrium thermodynamics and nonequilibrium 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.
非平衡体系的热力学研究比平衡态热力学研究需要更多的一般概念。平衡态热力学和非平衡态热力学之间的一个根本区别在于非均匀系统的行为，这就要求他们研究在均匀系统的平衡态热力学中没有考虑的反应速率的知识。下面将讨论这个问题。另一个基本的和非常重要的区别是，在一般情况下，难以或不可能用宏观条件来定义非热力学平衡的系统在瞬间的熵; 它可以做到，有用的近似，只有在精心选择的特殊情况下，即那些在整个局部热力学平衡。
Scope
Scope
范围
Difference between equilibrium and nonequilibrium thermodynamics
Difference between equilibrium and nonequilibrium thermodynamics
平衡和非平衡态热力学的区别
A profound difference separates equilibrium from nonequilibrium thermodynamics. Equilibrium thermodynamics ignores the timecourses of physical processes. In contrast, nonequilibrium thermodynamics attempts to describe their timecourses in continuous detail.
A profound difference separates equilibrium from nonequilibrium thermodynamics. Equilibrium thermodynamics ignores the timecourses of physical processes. In contrast, nonequilibrium thermodynamics attempts to describe their timecourses 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 timecourses 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 nonequilibrium 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 nonequilibrium thermodynamics.^{[4]} Equilibrium thermodynamics does, however, for theoretical development, use the idealized concept of the "quasistatic process". A quasistatic 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 timecourses 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 nonequilibrium 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 nonequilibrium 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。
Nonequilibrium thermodynamics, on the other hand, attempting to describe continuous timecourses, needs its state variables to have a very close connection with those of equilibrium thermodynamics.^{[7]} This profoundly restricts the scope of nonequilibrium thermodynamics, and places heavy demands on its conceptual framework.
Nonequilibrium thermodynamics, on the other hand, attempting to describe continuous timecourses, needs its state variables to have a very close connection with those of equilibrium thermodynamics. This profoundly restricts the scope of nonequilibrium 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。
Nonequilibrium state variables
Nonequilibrium state variables
Nonequilibrium state variables
The suitable relationship that defines nonequilibrium thermodynamic state variables is as follows. On occasions when the system happens to be in states that are sufficiently close to thermodynamic equilibrium, nonequilibrium 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, nonequilibrium thermodynamic systems are spatially and temporally nonuniform, but their nonuniformity still has a sufficient degree of smoothness to support the existence of suitable time and space derivatives of nonequilibrium state variables. Because of the spatial nonuniformity, nonequilibrium 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 nonequilibrium 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 nonuniformity. Further, the nonequilibrium 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 nonequilibrium thermodynamics is a work in progress.
The suitable relationship that defines nonequilibrium thermodynamic state variables is as follows. On occasions when the system happens to be in states that are sufficiently close to thermodynamic equilibrium, nonequilibrium 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, nonequilibrium thermodynamic systems are spatially and temporally nonuniform, but their nonuniformity still has a sufficient degree of smoothness to support the existence of suitable time and space derivatives of nonequilibrium state variables. Because of the spatial nonuniformity, nonequilibrium 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 nonequilibrium 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 nonuniformity. Further, the nonequilibrium 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 nonequilibrium thermodynamics is a work in progress.
定义非平衡热力学状态变量的合适关系如下。当系统处于足够接近热力学平衡的状态时，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，或者通过相应的时间和空间导数，包括物质和能量的通量，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，以支持存在适当的非平衡态变量的时间和空间导数。由于空间的非均匀性，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。在现实中，这些要求是非常苛刻的，并且可能很难或实际上，甚至在理论上，不可能满足它们。这就是为什么非平衡态热力学是一个进展中的工作的一部分。
Overview
Overview
概览
Nonequilibrium 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.
Nonequilibrium 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.
非平衡态热力学是一项正在进行的工作，而不是一座已经建立的大厦。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 nonequilibrium 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 nonlinear dynamical structure.^{[11]}
Some concepts of particular importance for nonequilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873, Onsager 1931, also), time rate of entropy production (Onsager 1931), dissipative structure, and nonlinear dynamical structure.
一些对非平衡态热力学特别重要的概念包括能量耗散的时间速率(Rayleigh 1873，Onsager 1931，也) ，产生熵耗散的时间速率(Onsager 1931) ，耗散结构和非线性动力结构。
One problem of interest is the thermodynamic study of nonequilibrium steady states, in which entropy production and some flows are nonzero, but there is no time variation of physical variables.
One problem of interest is the thermodynamic study of nonequilibrium steady states, in which entropy production and some flows are nonzero, but there is no time variation of physical variables.
一个有趣的问题是非平衡态的热力学研究，其中产生熵和一些流量是非零的，但没有物理变量的时间变化。
One initial approach to nonequilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'.^{[3]} There are other approaches to nonequilibrium thermodynamics, for example extended irreversible thermodynamics,^{[3]}^{[16]} and generalized thermodynamics,^{[17]} but they are hardly touched on in the present article.
One initial approach to nonequilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'. but they are hardly touched on in the present article.
非平衡态热力学的一个初始方法有时被称为经典不可逆热力学。但在本文中几乎没有涉及。
Quasiradiationless nonequilibrium thermodynamics of matter in laboratory conditions
Quasiradiationless nonequilibrium thermodynamics of matter in laboratory conditions
实验室条件下物质的准无辐射非平衡态热力学
According to Wildt^{[18]} (see also Essex^{[19]}^{[20]}^{[21]}), current versions of nonequilibrium 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.
According to Wildt (see also Essex), current versions of nonequilibrium 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; 它们之所以能够忽略辐射热，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。
Local equilibrium thermodynamics
Local equilibrium thermodynamics
局域平衡热力学
The terms 'classical irreversible thermodynamics'^{[3]} and 'local equilibrium thermodynamics' are sometimes used to refer to a version of nonequilibrium 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 wellmixed, or without an effective spatial structure, and without kinetic energy of bulk flow or of diffusive flux. Even within the thoughtframe 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 quasistatic local intensive variables.
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 quasistatic 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。
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
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，但是假定系统的全局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全局熵评估作出贡献，因为它应该对系统的全局熵评估作出恰当的贡献。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。
In other writings, local flow variables are considered; these might be considered as classical by analogy with the timeinvariant longterm timeaverages 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 twodimensional surfaces, with no spatial volume, and no spatial variation.
In other writings, local flow variables are considered; these might be considered as classical by analogy with the timeinvariant longterm timeaverages 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 twodimensional surfaces, with no spatial volume, and no spatial variation.
在其他著作中，考虑了局部流动变量; 这些可以被视为经典的，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19世纪和拉斯昂萨格尔在20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。
Local equilibrium thermodynamics with materials with "memory"
Local equilibrium thermodynamics with materials with "memory"
具有“记忆”材料的局域平衡热力学
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 timedependent local equilibrium thermodynamics with memoryless materials, but fluxes are not independent variables of state.^{[29]}
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 timedependent 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，时间比具有无记忆材料的时变局域平衡热力学更深入地研究了这个问题，但是通量并不是状态的独立变量。
Extended irreversible thermodynamics
Extended irreversible thermodynamics
扩展不可逆热力学
Extended irreversible thermodynamics is a branch of nonequilibrium 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.
Extended irreversible thermodynamics is a branch of nonequilibrium 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。通过包括质量、动量和能量的通量以及最终的高阶通量，状态变量的空间被扩大了。
The formalism is wellsuited for describing highfrequency processes and smalllength scales materials.
The formalism is wellsuited for describing highfrequency processes and smalllength scales materials.
形式主义非常适合于描述高频过程和小尺度材料。
Basic concepts
Basic concepts
基本概念
There are many examples of stationary nonequilibrium 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 nonequilibrium conditions at the walls. Laser action is also a nonequilibrium 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 nonstationary nonequilibrium processes. Driven complex fluids, turbulent systems and glasses are other examples of nonequilibrium systems.
There are many examples of stationary nonequilibrium 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 nonequilibrium conditions at the walls. Laser action is also a nonequilibrium 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 nonstationary nonequilibrium processes. Driven complex fluids, turbulent systems and glasses are other examples of nonequilibrium 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以相反方向运动，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。
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.
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，其他所有能量都严格保持恒定，那么系统的温度是可以测量和有意义的。系统的性质可以用热动力位能亥姆霍兹自由能(a uTS)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 + PVTS) ，其中系统的性质由温度和压力决定。
Nonequilibrium 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 nonequilibrium 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.
Nonequilibrium 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 nonequilibrium 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。
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:
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)是一个函数的集合的大量数学 ei / 数学。每个扩展量都有一个共轭密集变量 math 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限制定义) ，因此:
 [math]\displaystyle{ I_i = \frac{\partial{S}}{\partial{E_i}}. }[/math]
[math]\displaystyle{ I_i = \frac{\partial{S}}{\partial{E_i}}. }[/math]
数学 ifrac { partial { ei } . / math
We then define the extended Massieu function as follows:
We then define the extended Massieu function as follows:
然后我们将扩展的 Massieu 函数定义如下:
 [math]\displaystyle{ \ k_{\rm B} M = S  \sum_i( I_i E_i), }[/math]
[math]\displaystyle{ \ k_{\rm B} M = S  \sum_i( I_i E_i), }[/math]
M ssum i (i i e i) ，/ math
where [math]\displaystyle{ \ k_{\rm B} }[/math] is Boltzmann's constant, whence
where [math]\displaystyle{ \ k_{\rm B} }[/math] is Boltzmann's constant, whence
数学是波尔兹曼常数，从哪里来的
 [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{ \ k_{\rm B} \, dM = \sum_i (E_i \, dI_i). }[/math]
The independent variables are the intensities.
The independent variables are the intensities.
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.
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。
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.
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，对于静态态，在扩展的 Massieu 函数的最小条件下，勒壤得转换改变了熵的最大条件(在平衡时有效)。
Stationary states, fluctuations, and stability
Stationary states, fluctuations, and stability
稳态、涨落和稳定性
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 subprocesses and to exchange of matter or energy with the system's surroundings that create the constraints that define the process.
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 subprocesses 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，允许定态包括在系统状态中出现不可预测和实验上不可重现的涨落。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。
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.
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。
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.
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引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。
Local thermodynamic equilibrium
Local thermodynamic equilibrium
当地热力学平衡
The scope of presentday nonequilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in nonequilibrium thermodynamics of matter is that they deal with what is known as local thermodynamic equilibrium.
The scope of presentday nonequilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in nonequilibrium 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。
Ponderable matter
Ponderable matter
值得思考的问题
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 'microphases' 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.
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 'microphases' 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.
物质的局部热力学平衡(参见 Keizer (1987))意味着，从概念上来说，为了研究和分析，系统可以在空间上和时间上分为小(无限小)尺寸的‘细胞’或‘微相’ ，在这种情况下，物质的经典热力学平衡条件得到了很好的近似。例如，在非常稀薄的气体中，分子碰撞很少发生; 在恒星的边界层中，辐射将能量传递到空间; 在非常低的温度下，相互作用的费米子中，耗散过程变得无效。当这些细胞被定义时，人们承认物质和能量可以在相邻的细胞之间自由通过，慢到足以让细胞在它们各自关于强化变量的局部热力学平衡中离开。
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 nonequilibrium 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.
One can think here of two 'relaxation times' separated by order of magnitude. 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 nonequilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning 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.
你可以在这里想象一下两个放松的时间被数量级分开了。较长的弛豫时间是系统宏观动力学结构改变所需时间的数量级。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非平衡热力学概念就失去了意义，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，声速远远大于风速; 这就有利于大气热传递研究的本地物质热力学平衡的想法，在60公里以下的高度，声音传播，但不超过100公里，在那里，由于分子间的碰撞，声音不传播。
Milne's definition in terms of radiative equilibrium
Milne's definition in terms of radiative equilibrium
米尔恩对辐射平衡的定义
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.
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.
考虑到恒星，Edward a. Milne 给出了局部热力学平衡的定义，即每个局部细胞中物质的热辐射。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放射辐射，就好像它处于细胞物质温度下的辐射平衡中一样。然后严格遵循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。
Entropy in evolving systems
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 nonequilibrium 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.
It is pointed out by W.T. Grandy Jr,, that entropy, though it may be defined for a nonequilibrium 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。虽然熵可以被定义为一个非平衡系统，但严格考虑时，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值就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 maximumentropy principles.
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 maximumentropy 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。
Entropy in nonequilibrium
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 nonequilibrium 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情况，除了用于确定平衡状态的本构变量 math x 1，x 2，... ，xn / math 之外，还引入了一组称为内部变量的变量 math xi 1， xi 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{ \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]}
是对应变量的松弛时间。考虑初始值 math xi i ^ 0 / math 等于零是比较方便的。上述方程对于平衡点的小偏差是有效的，Pokrovskii 考虑了一般情况下内变量的动力学。
Entropy of the system in nonequilibrium 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

[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 nonequilibrium 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
普里高金和他的合作者在研究化学反应物质系统时，提出了非平衡系统热力学的本质贡献。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体积下考虑系统的熵变化的三个贡献和恒温数学 t / math。根据这个公式可以计算出熵的增量 s / 数

[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.
方程式右边的第一项表示一股热能流进入系统，最后一项表示一股能量流进入系统，这股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，math mu alpha math 表示物质的化学势。中间项描述了由于内部变量 math xi j / math 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，即测量多少考虑的体系与化学反应是不平衡的。该理论可以推广，将任何偏离平衡态的情况作为一个内变量，因此我们认为方程(1)中的内变量集合 math / math 不仅包含了定义体系中所有化学反应完全程度的量，而且还包含了体系的结构、温度梯度、物质浓度差等。
Flows and forces
流动与力量
The fundamental relation of classical equilibrium thermodynamics ^{[44]}
The fundamental relation of classical equilibrium thermodynamics
经典平衡态热力学的基本关系
 [math]\displaystyle{ dS=\frac{1}{T}dU+\frac{p}{T}dV\sum_{i=1}^s\frac{\mu_i}{T}dN_i }[/math]
[math]\displaystyle{ dS=\frac{1}{T}dU+\frac{p}{T}dV\sum_{i=1}^s\frac{\mu_i}{T}dN_i }[/math]
数学 d frac {1}{ t } dU + frac { p }{ t } dV sum { i } ^ s frac { mu i } 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].
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].
将系统的熵数学 dS / 数学的变化表示为密集量、温度数学 t / 数学、压力数学 p / 数学和数学 i ^ (th) / 数学化学势数学 μ / 数学和广义量的微分数学 u / 数学、体积数学 v / 数学和数学 i ^ (th) / 数学粒子数数学 i / 数学。
Following Onsager (1931,I),^{[10]} let us extend our considerations to thermodynamically nonequilibrium 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].
Following Onsager (1931,I), let us extend our considerations to thermodynamically nonequilibrium 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].
在昂萨格(1931，i)之后，让我们将我们的考虑扩展到热力学非平衡系统。作为基础，我们需要局部定义的宏观量数学 u / math，math v / math 和 math n / math，以及密集宏观量数学 t / math，math p / math 和 math mu i / math。
For classical nonequilibrium 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.
For classical nonequilibrium 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.
对于经典的非平衡研究，我们将考虑一些新的局部定义的强烈的宏观变量。我们可以在适当的条件下，通过局部定义基本局部定义的宏观量的梯度和通量密度，导出这些新的变量。
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.
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.
这种局部定义的强烈宏观变量梯度被称为“热力学力”。它们“驱动”磁通密度，也许经常被误称为“磁通量” ，这是力的双重作用。这些量在关于昂萨格互反关系的文章中定义。
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 nearsteady thermodynamically nonequilibrium regime, which has dynamics linear in the forces and flux densities.
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 nearsteady thermodynamically nonequilibrium regime, which has dynamics linear in the forces and flux densities.
建立这种力和通量密度之间的关系在统计力学是一个问题。通量密度(数学 j i / math)可以耦合。本文在昂萨格互反关系的基础上，考虑了稳定的近稳态热力学非平衡态，它在力和流量密度方面具有线性动力学性质。
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 nonequilibrium state, it organizes itself so as to minimize total entropy production defined locally. This is considered further below.
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 nonequilibrium state, it organizes itself so as to minimize total entropy production defined locally. This is considered further below.
在稳态条件下，这样的力和相关的通量密度是定义的时间不变的，就像系统的局部定义的熵和产生熵一样。值得注意的是，根据 Ilya Prigogine 和其他人的研究，当一个开放系统处于允许它达到稳定的热力学非平衡状态的条件下时，它会自我组织以使局部定义的总产生熵最小化。下文将进一步讨论这个问题。
One wants to take the analysis to the further stage of describing the behaviour of surface and volume integrals of nonstationary 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.
One wants to take the analysis to the further stage of describing the behaviour of surface and volume integrals of nonstationary 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.
人们希望进一步分析描述非平稳局部量的表面积分和体积积分的行为; 这些积分是宏观通量和产生率。一般来说，这些积分的动力学不能用线性方程来充分描述，尽管在特殊情况下它们可以这样描述。
Onsager reciprocal relations
Onsager reciprocal relations
昂萨格互反关系
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:
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:
在瑞利(1873)第三节之后，昂萨格(1931，i)指出，在两种流动(数学 j i / math)都很小且热力学力(数学 f i / math)变化缓慢的情况下，熵的产生速率(数学 σ) / 数学与流动呈线性关系:
 [math]\displaystyle{ \sigma = \sum_i J_i\frac{\partial F_i}{\partial x_i} }[/math]
[math]\displaystyle{ \sigma = \sum_i J_i\frac{\partial F_i}{\partial x_i} }[/math]
数学部分和数学部分和数学
and the flows are related to the gradient of the forces, parametrized by a matrix of coefficients conventionally denoted [math]\displaystyle{ L }[/math]:
and the flows are related to the gradient of the forces, parametrized by a matrix of coefficients conventionally denoted [math]\displaystyle{ L }[/math]:
流动与力的梯度有关，通过一个系数矩阵参数化，传统上用数学 l / math 表示:
 [math]\displaystyle{ J_i = \sum_{j} L_{ij} \frac{\partial F_j}{\partial x_j} }[/math]
[math]\displaystyle{ J_i = \sum_{j} L_{ij} \frac{\partial F_j}{\partial x_j} }[/math]
数学部分数学部分数学部分数学
from which it follows that:
from which it follows that:
由此可见:
 [math]\displaystyle{ \sigma = \sum_{i,j} L_{ij} \frac{\partial F_i}{\partial x_i}\frac{\partial F_j}{\partial x_j} }[/math]
[math]\displaystyle{ \sigma = \sum_{i,j} L_{ij} \frac{\partial F_i}{\partial x_i}\frac{\partial F_j}{\partial x_j} }[/math]
数学部分数学部分数学部分数学部分数学部分数学
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 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.
热力学第二定律要求矩阵数学 l / 数学是正定的。考虑到动力学的微观可逆性，统计力学表明矩阵数学 l / math 是对称的。这个事实被称为昂萨格互反关系。
The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.^{[45]}
The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.
上述方程的熵产生率的推广是由 Pokrovskii 给出的。
Speculated extremal principles for nonequilibrium processes
Speculated extremal principles for nonequilibrium processes
推测非平衡过程的极值原理
Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999)^{[46]} 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 CasasVásquez (2008), and Šilhavý (1997).
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 CasasVásquez (2008), and Šilhavý (1997).
直到最近，这个领域中有用的极端原理的前景似乎还很模糊。Nicolis (1999)得出结论，大气动力学的一个模型有一个吸引子，它不是一个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。另一位顶级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 的第12章(2008年)非常谨慎，并且发现在许多情况下难以定义内部产生熵的速率，并且发现有时为了预测一个过程的进程，一个叫做能量耗散速率的极值可能比产生熵的速率更有用; 这个量出现在昂萨格尔1931年创立这个主题时。其他作家也认为，一般的全球极端原则的前景是模糊的。这些作家包括格兰斯多夫和普里戈金(1971年) ，莱邦，乔和卡萨斯v 斯奎兹(2008年) ，和伊尔哈夫(1997年)。
A recent proposal may perhaps bypass those clouded prospects.^{[47]}^{[48]}
A recent proposal may perhaps bypass those clouded prospects.
最近的一项提议或许可以绕过那些阴云密布的前景。
Applications
Applications
申请
Nonequilibrium thermodynamics has been successfully applied to describe biological processes such as protein folding/unfolding and transport through membranes^{[49]}^{[50]}.
Nonequilibrium 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诸如蛋白质折叠 / 去折叠和通过膜转运等生物学过程。
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.^{[51]}
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可能失去平衡。
Also, ideas from nonequilibrium thermodynamics and the informatic theory of entropy have been adapted to describe general economic systems.^{[52]}
Also, ideas from nonequilibrium 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。
^{[53]}
See also
See also
参见
References
References
参考资料
 ↑ Bodenschatz, Eberhard; Cannell, David S.; de Bruyn, John R.; Ecke, Robert; Hu, YuChou; Lerman, Kristina; Ahlers, Guenter (December 1992). "Experiments on three systems with nonvariational aspects". Physica D: Nonlinear Phenomena. 61 (1–4): 77–93. doi:10.1016/01672789(92)90150L.
 ↑ ^{2.0} ^{2.1} Grandy, W.T., Jr (2008).
 ↑ ^{3.0} ^{3.1} ^{3.2} ^{3.3} Lebon, G., Jou, D., CasasVázquez, J. (2008). Understanding Nonequilibrium Thermodynamics: Foundations, Applications, Frontiers, SpringerVerlag, Berlin, e .
 ↑ ^{4.0} ^{4.1} Lieb, E.H., Yngvason, J. (1999), p. 5.
 ↑ Gyarmati, I. (1967/1970), pp. 8–12.
 ↑ Callen, H.B. (1960/1985), § 4–2.
 ↑ Glansdorff, P., Prigogine, I. (1971), Ch. II,§ 2.
 ↑ ^{8.0} ^{8.1} ^{8.2} ^{8.3} Gyarmati, I. (1967/1970).
 ↑ ^{9.0} ^{9.1} Strutt, J. W. (1871). "Some General Theorems relating to Vibrations". Proceedings of the London Mathematical Society. s14: 357–368. doi:10.1112/plms/s14.1.357.
 ↑ ^{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.
 ↑ ^{11.0} ^{11.1} ^{11.2} ^{11.3} Lavenda, B.H. (1978). Thermodynamics of Irreversible Processes, Macmillan, London, .
 ↑ Gyarmati, I. (1967/1970), pages 414.
 ↑ Ziegler, H., (1983). An Introduction to Thermomechanics, NorthHolland, Amsterdam, .
 ↑ Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics, WileyInterscience, New York, , Section 3.2, pages 6472.
 ↑ ^{15.0} ^{15.1} ^{15.2} Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability, and Fluctuations, WileyInterscience, London, 1971, .
 ↑ ^{16.0} ^{16.1} Jou, D., CasasVázquez, J., Lebon, G. (1993). Extended Irreversible Thermodynamics, Springer, Berlin, , .
 ↑ Eu, B.C. (2002).
 ↑ 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.
 ↑ Essex, C. (1984a). "Radiation and the irreversible thermodynamics of climate". Journal of the Atmospheric Sciences. 41 (12): 1985–1991. Bibcode:1984JAtS...41.1985E. doi:10.1175/15200469(1984)041<1985:RATITO>2.0.CO;2..
 ↑ 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.
 ↑ 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:10.1016/00320633(84)900606.
 ↑ Prigogine, I., Defay, R. (1950/1954). Chemical Thermodynamics, Longmans, Green & Co, London, page 1.
 ↑ ^{23.0} ^{23.1} De Groot, S.R., Mazur, P. (1962). Nonequilibrium Thermodynamics, NorthHolland, Amsterdam.
 ↑ ^{24.0} ^{24.1} Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons, New York, .
 ↑ ^{25.0} ^{25.1} Mihalas, D., WeibelMihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press, New York .
 ↑ ^{26.0} ^{26.1} Schloegl, F. (1989). Probability and Heat: Fundamentals of Thermostatistics, Freidr. Vieweg & Sohn, Braunschweig, .
 ↑ ^{27.0} ^{27.1} Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, SpringerVerlag, New York, .
 ↑ Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester UK, , pages 333338.
 ↑ 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.
 ↑ Kondepudi, D., Prigogine, I, (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures, Wiley, Chichester, 1998, .
 ↑ ^{31.0} ^{31.1} Zubarev D. N.,(1974). Nonequilibrium Statistical Thermodynamics, translated from the Russian by P.J. Shepherd, New York, Consultants Bureau. ; .
 ↑ 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.
 ↑ Grandy, W.T., Jr. (2004). "Time Evolution in Macroscopic Systems. I. Equations of Motion". Foundations of Physics. 34 (1): 1. arXiv:condmat/0303290. Bibcode:2004FoPh...34....1G. doi:10.1023/B:FOOP.0000012007.06843.ed.
 ↑ Grandy, W.T., Jr. (2004). "Time Evolution in Macroscopic Systems. II. The Entropy". Foundations of Physics. 34 (1): 21. arXiv:condmat/0303291. Bibcode:2004FoPh...34...21G. doi:10.1023/B:FOOP.0000012008.36856.c1.
 ↑ 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.
 ↑ Grandy 2004 see also [1].
 ↑ Truesdell, Clifford (1984). Rational Thermodynamics (2 ed.). Springer.
 ↑ Maugin, Gérard A. (2002). Continuum Thermomechanics. Kluwer.
 ↑ Gurtin, Morton E. (2010). The Mechanics and Thermodynamics of Continua. Cambridge University Press.
 ↑ Amendola, Giovambattista (2012). Thermodynamics of Materials with Memory: Theory and Applications. Springer.
 ↑ ^{41.0} ^{41.1} Pokrovskii V.N. (2013) A derivation of the main relations of nonequilibrium thermodynamics. Hindawi Publishing Corporation: ISRN Thermodynamics, vol. 2013, article ID 906136, 9 p. https://dx.doi.org/10.1155/2013/906136.
 ↑ Prigogine, I. (1955/1961/1967). Introduction to Thermodynamics of Irreversible Processes. 3rd edition, Wiley Interscience, New York.
 ↑ Pokrovskii V.N. (2005) Extended thermodynamics in a discretesystem approach, Eur. J. Phys. vol. 26, 769781.
 ↑ W. Greiner, L. Neise, and H. Stöcker (1997), Thermodynamics and Statistical Mechanics (Classical Theoretical Physics) ,SpringerVerlag, New York, P85, 91, 101,108,116, .
 ↑ Pokrovskii V.N. (2013) A derivation of the main relations of nonequilibrium thermodynamics. Hindawi Publishing Corporation: ISRN Thermodynamics, vol. 2013, article ID 906136, 9 p. https://dx.doi.org/10.1155/2013/906136.
 ↑ Nicolis, C. (1999). "Entropy production and dynamical complexity in a loworder atmospheric model". Quarterly Journal of the Royal Meteorological Society. 125 (557): 1859–1878. Bibcode:1999QJRMS.125.1859N. doi:10.1002/qj.49712555718.
 ↑ Attard, P. (2012). "Optimising Principle for NonEquilibrium Phase Transitions and Pattern Formation with Results for Heat Convection". arXiv:1208.5105 [condmat.statmech].
 ↑ Attard, P. (2012). NonEquilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications, Oxford University Press, Oxford UK, .
 ↑ Kimizuka, Hideo; Kaibara, Kozue (September 1975). "Nonequilibrium thermodynamics of ion transport through membranes". Journal of Colloid and Interface Science. 52 (3): 516–525. doi:10.1016/00219797(75)902763.
 ↑ Baranowski, B. (April 1991). "Nonequilibrium thermodynamics as applied to membrane transport". Journal of Membrane Science. 57 (2–3): 119–159. doi:10.1016/S03767388(00)806754.
 ↑ 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.
 ↑ Pokrovskii, Vladimir (2011). Econodynamics. The Theory of Social Production.. https://www.springer.com/physics/complexity/book/9789400720954: Springer, DordrechtHeidelbergLondonNew York..
 ↑ Chen, Jing (2015). The Unity of Science and Economics: A New Foundation of Economic Theory. https://www.springer.com/us/book/9781493934645: Springer.
Sources
Sources
资料来源
 Callen, H.B. (1960/1985). Thermodynamics and an Introduction to Thermostatistics, (1st edition 1960) 2nd edition 1985, Wiley, New York, .
 Eu, B.C. (2002). Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics, Kluwer Academic Publishers, Dordrecht, .
 Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability, and Fluctuations, WileyInterscience, London, 1971, .
 Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems. Oxford University Press. .
 Gyarmati, I. (1967/1970). Nonequilibrium Thermodynamics. Field Theory and Variational Principles, translated from the Hungarian (1967) by E. Gyarmati and W.F. Heinz, Springer, Berlin.
 Lieb, E.H., Yngvason, J. (1999). 'The physics and mathematics of the second law of thermodynamics', Physics Reports, 310: 1–96. See also this.
Further reading
Further reading
进一步阅读
 Ziegler, Hans (1977): An introduction to Thermomechanics. North Holland, Amsterdam. . Second edition (1983) .
 Kleidon, A., Lorenz, R.D., editors (2005). Nonequilibrium Thermodynamics and the Production of Entropy, Springer, Berlin. .
 Prigogine, I. (1955/1961/1967). Introduction to Thermodynamics of Irreversible Processes. 3rd edition, Wiley Interscience, New York.
 Zubarev D. N. (1974): Nonequilibrium Statistical Thermodynamics. New York, Consultants Bureau. ; .
 Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, SpringerVerlag, New York, .
 Zubarev D. N., Morozov V., Ropke G. (1996): Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory. John Wiley & Sons. .
 Zubarev D. N., Morozov V., Ropke G. (1997): Statistical Mechanics of Nonequilibrium Processes: Relaxation and Hydrodynamic Processes. John Wiley & Sons. .
 Tuck, Adrian F. (2008). Atmospheric turbulence : a molecular dynamics perspective. Oxford University Press. .
 Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems. Oxford University Press. .
 Kondepudi, D., Prigogine, I. (1998). Modern Thermodynamics: From Heat Engines to Dissipative Structures. John Wiley & Sons, Chichester. .
 de Groot S.R., Mazur P. (1984). NonEquilibrium Thermodynamics (Dover).
 Ramiro Augusto Salazar La Rotta. (2011). The NonEquilibrium Thermodynamics, Perpetual
External links
External links
外部链接
 Nonequilibrium Statistical Thermodynamics applied to Fluid Dynamics and Laser Physics  1992 book by Xavier de Hemptinne.
 Nonequilibrium Thermodynamics of Small Systems  PhysicsToday.org
 Into the Cool  2005 book by Dorion Sagan and Eric D. Schneider, on nonequilibrium thermodynamics and evolutionary theory.
Category:Nonequilibrium thermodynamics
类别: 非平衡态热力学
Category:Concepts in physics
分类: 物理概念
Category:Branches of thermodynamics
分类: 热力学的分支
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