非平衡系统

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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。


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

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

几乎所有在自然界中发现的系统都不是在热力学平衡中，因为它们正在变化或者可以随着时间而被触发变化，并且不断不断地受制于物质和能量的流动，这些物质和能量来自其他系统和化学反应。然而，某些系统和进程在某种意义上是有用的，它们接近于热力学平衡，足以允许目前已知的非平衡态热力学对其进行有用的精确描述。然而，由于非变分动力学的存在，许多自然系统和过程总是远远超出非平衡热力学方法的范围，自由能的概念丢失了。平衡态热力学，然而，为了理论的发展，使用理想化的概念“准静态过程”。准静态过程是一个概念上的(永恒的和物理上不可能的)沿着热力学平衡状态的连续路径平滑的数学通道。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。


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


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

非平衡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。

Scope

Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873, thermodynamic fields, and non-linear dynamical structure.

一些对非平衡态热力学特别重要的概念包括时间能量耗散率(瑞利1873，热力学场，和非线性动力学结构。

Difference between equilibrium and non-equilibrium thermodynamics

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.

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。

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.


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, 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 (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.

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。还有其他的方法来处理非平衡态热力学热力学，例如扩展不可逆热力学，和广义热力学，和局部平衡热力学有时被用来指代一个版本的非平衡态热力学，要求某些简化的假设，如下所示。这些假设的效果是使系统中的每一个非常小的体积元有效地均匀化，或混合良好，或没有有效的空间结构，没有体积流动的动能或扩散通量的动能。即使在经典不可逆热力学的思想框架内，在选择自变量时也需要谨慎(另见 Keizer (1987)) ，这意味着从概念上讲，为了研究和分析，该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” ，其中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。

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


One can think here of two 'relaxation times' separated by order of magnitude. 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。和其他方法必须被提出，例如扩展不可逆热力学。例如，在大气中，声速远远大于风速; 这就有利于在大气热传递研究中使用局部物质热力学平衡的想法，在低于60公里的高度进行声音传播研究，但不能超过100公里，在那里，由于分子间的碰撞很少，声音不能传播。

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


Non-equilibrium state variables

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'. 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.

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。将任何偏离平衡态的情况作为一个内变量来考虑，所以我们认为方程式(1)中的内变量集不仅包含了定义系统中所有化学反应完全程度的量，而且还包含了系统的结构、温度梯度、物质浓度差等。

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


Overview

The fundamental relation of classical equilibrium thermodynamics 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].

经典平衡态热力学的基本关系使我们把注意力扩展到热力学非平衡态系统。作为基础，我们需要大量宏观量的局部定义版本。

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.


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.

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。在适当的条件下，我们可以通过局部定义基本局部定义的宏观量的梯度和通量密度来推导这些新的变量。

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


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”通量密度，也许常被误称为“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。

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.


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.

建立这种力量和通量密度之间的关系在统计力学是一个问题。通量密度(< math > j _ i </math >)可能是耦合的。本文在昂萨格互反关系的基础上，考虑了稳定的近稳态热动力学非平衡态，它在力和流量密度方面具有线性动力学性质。

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


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.

在稳态条件下，这样的力和相关的磁通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。

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

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

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.

人们想要把分析带到描述非平稳局部量的表面积分和体积积分的行为的更深一步; 这些积分是宏观的通量和产生率。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。


Local equilibrium thermodynamics

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


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)第三部分之后，昂萨格(1931，i)指出，在流量(< math > j _ i </math >)较小且热力学力(< math > f _ i </math >)变化缓慢的情况下，熵的产生率与流量呈线性关系:

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


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

[数学][数学]

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


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，常规表示为:

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.[29]

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

部分数学部分数学


Extended irreversible thermodynamics

from which it follows that:

由此可见:

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.

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

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

部分 f _ i } frac { partial f _ j }{ partial x _ j }{ partial x _ j } </math >


Basic concepts

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是正定的。考虑到统计力学动力学的微观可逆性暗示了矩阵是对称的。这个事实被称为昂萨格互反关系。

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.


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。

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.


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.

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:


Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999) 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).

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。尼古利斯(1999)非常谨慎，在许多情况下难以定义内产生熵，他发现有时为了预测一个过程的进程，一个叫做能量耗散率的极值可能比产生熵的极值更有用; 这个量出现在昂萨格尔1931年创立的这个主题中。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-瓦斯奎斯(2008年) ，以及伊尔哈维(1997年)。

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

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.

有很好的实验证据表明，热对流不服从时间产生熵的极值原理。理论分析表明，化学反应不服从产生熵二阶微分的极值原理。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。


We then define the extended Massieu function as follows:


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。

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

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。


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。

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]


The independent variables are the intensities.


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.


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

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.


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 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.


Local thermodynamic equilibrium

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


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


Milne's definition in terms of radiative equilibrium

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.


Entropy in evolving systems

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


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


Entropy in non-equilibrium

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

[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]


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

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

 

 

 

 

(1)

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

[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.


Flows and forces

The fundamental relation of classical equilibrium thermodynamics [44]


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


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


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


Category:Concepts in physics

分类: 物理概念

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.

Category:Branches of thermodynamics

分类: 热力学的分支


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

  1. ↑ Bodenschatz, Eberhard; Cannell, David S.; de Bruyn, John R.; Ecke, Robert; Hu, Yu-Chou; Lerman, Kristina; Ahlers, Guenter (December 1992). "Experiments on three systems with non-variational aspects". Physica D: Nonlinear Phenomena. 61 (1–4): 77–93. doi:10.1016/0167-2789(92)90150-L.
  2. ↑ Grandy, W.T., Jr (2008).
  3. ↑ 3.0 3.1 3.2 3.3 Lebon, G., Jou, D., Casas-Vázquez, J. (2008). Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers, Springer-Verlag, Berlin, e-
    .
  4. ↑ 4.0 4.1 Lieb, E.H., Yngvason, J. (1999), p. 5.
  5. ↑ Gyarmati, I. (1967/1970), pp. 8–12.
  6. ↑ Callen, H.B. (1960/1985), § 4–2.
  7. ↑ Glansdorff, P., Prigogine, I. (1971), Ch. II,§ 2.
  8. ↑ 8.0 8.1 8.2 8.3 Gyarmati, I. (1967/1970).
  9. ↑ Strutt, J. W. (1871). "Some General Theorems relating to Vibrations". Proceedings of the London Mathematical Society. s1-4: 357–368. doi:10.1112/plms/s1-4.1.357.
  10. ↑ 10.0 10.1 10.2 Onsager, L. (1931). "Reciprocal relations in irreversible processes, I". Physical Review. 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405.
  11. ↑ 11.0 11.1 11.2 11.3 Lavenda, B.H. (1978). Thermodynamics of Irreversible Processes, Macmillan, London,
    .
  12. ↑ Gyarmati, I. (1967/1970), pages 4-14.
  13. ↑ Ziegler, H., (1983). An Introduction to Thermomechanics, North-Holland, Amsterdam,
    .
  14. ↑ Balescu, R. (1975). Equilibrium and Non-equilibrium Statistical Mechanics, Wiley-Interscience, New York,
    , Section 3.2, pages 64-72.
  15. ↑ 15.0 15.1 15.2 Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley-Interscience, London, 1971,
    .
  16. ↑ 16.0 16.1 Jou, D., Casas-Vázquez, J., Lebon, G. (1993). Extended Irreversible Thermodynamics, Springer, Berlin,
    ,
    .
  17. ↑ Eu, B.C. (2002).
  18. ↑ Wildt, R. (1972). "Thermodynamics of the gray atmosphere. IV. Entropy transfer and production". Astrophysical Journal. 174: 69–77. Bibcode:1972ApJ...174...69W. doi:10.1086/151469.
  19. ↑ Essex, C. (1984a). "Radiation and the irreversible thermodynamics of climate". Journal of the Atmospheric Sciences. 41 (12): 1985–1991. Bibcode:1984JAtS...41.1985E. doi:10.1175/1520-0469(1984)041<1985:RATITO>2.0.CO;2..
  20. ↑ Essex, C. (1984b). "Minimum entropy production in the steady state and radiative transfer". Astrophysical Journal. 285: 279–293. Bibcode:1984ApJ...285..279E. doi:10.1086/162504.
  21. ↑ Essex, C. (1984c). "Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes". Planetary and Space Science. 32 (8): 1035–1043. Bibcode:1984P&SS...32.1035E. doi:10.1016/0032-0633(84)90060-6.
  22. ↑ Prigogine, I., Defay, R. (1950/1954). Chemical Thermodynamics, Longmans, Green & Co, London, page 1.
  23. ↑ 23.0 23.1 De Groot, S.R., Mazur, P. (1962). Non-equilibrium Thermodynamics, North-Holland, Amsterdam.
  24. ↑ 24.0 24.1 Balescu, R. (1975). Equilibrium and Non-equilibrium Statistical Mechanics, John Wiley & Sons, New York,
    .
  25. ↑ 25.0 25.1 Mihalas, D., Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press, New York
    .
  26. ↑ 26.0 26.1 Schloegl, F. (1989). Probability and Heat: Fundamentals of Thermostatistics, Freidr. Vieweg & Sohn, Braunschweig,
    .
  27. ↑ 27.0 27.1 Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York,
    .
  28. ↑ Kondepudi, D. (2008). Introduction to Modern Thermodynamics, Wiley, Chichester UK,
    , pages 333-338.
  29. ↑ Coleman, B.D.; Noll, W. (1963). "The thermodynamics of elastic materials with heat conduction and viscosity". Arch. Ration. Mach. Analysis. 13 (1): 167–178. Bibcode:1963ArRMA..13..167C. doi:10.1007/bf01262690. S2CID 189793830.
  30. ↑ Kondepudi, D., Prigogine, I, (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures, Wiley, Chichester, 1998,
    .
  31. ↑ 31.0 31.1 Zubarev D. N.,(1974). Nonequilibrium Statistical Thermodynamics, translated from the Russian by P.J. Shepherd, New York, Consultants Bureau.
    ;
    .
  32. ↑ Milne, E.A. (1928). "The effect of collisions on monochromatic radiative equilibrium". Monthly Notices of the Royal Astronomical Society. 88 (6): 493–502. Bibcode:1928MNRAS..88..493M. doi:10.1093/mnras/88.6.493.
  33. ↑ Grandy, W.T., Jr. (2004). "Time Evolution in Macroscopic Systems. I. Equations of Motion". Foundations of Physics. 34 (1): 1. arXiv:cond-mat/0303290. Bibcode:2004FoPh...34....1G. doi:10.1023/B:FOOP.0000012007.06843.ed.
  34. ↑ Grandy, W.T., Jr. (2004). "Time Evolution in Macroscopic Systems. II. The Entropy". Foundations of Physics. 34 (1): 21. arXiv:cond-mat/0303291. Bibcode:2004FoPh...34...21G. doi:10.1023/B:FOOP.0000012008.36856.c1. S2CID 18573684.
  35. ↑ Grandy, W. T., Jr (2004). "Time Evolution in Macroscopic Systems. III: Selected Applications". Foundations of Physics. 34 (5): 771. Bibcode:2004FoPh...34..771G. doi:10.1023/B:FOOP.0000022187.45866.81. S2CID 119406182.
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  37. ↑ Truesdell, Clifford (1984). Rational Thermodynamics (2 ed.). Springer. 
  38. ↑ Maugin, Gérard A. (2002). Continuum Thermomechanics. Kluwer. 
  39. ↑ Gurtin, Morton E. (2010). The Mechanics and Thermodynamics of Continua. Cambridge University Press. 
  40. ↑ Amendola, Giovambattista (2012). Thermodynamics of Materials with Memory: Theory and Applications. Springer. 
  41. ↑ 41.0 41.1 Pokrovskii V.N. (2013) A derivation of the main relations of non-equilibrium thermodynamics. Hindawi Publishing Corporation: ISRN Thermodynamics, vol. 2013, article ID 906136, 9 p. https://dx.doi.org/10.1155/2013/906136.
  42. ↑ Prigogine, I. (1955/1961/1967). Introduction to Thermodynamics of Irreversible Processes. 3rd edition, Wiley Interscience, New York.
  43. ↑ Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach, Eur. J. Phys. vol. 26, 769-781.
  44. ↑ W. Greiner, L. Neise, and H. Stöcker (1997), Thermodynamics and Statistical Mechanics (Classical Theoretical Physics) ,Springer-Verlag, New York, P85, 91, 101,108,116,
    .
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