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

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

==Entropy in evolving systems==

== Entropy in evolving systems 进化系统中的熵 ==

 

 

It is pointed out by W.T. Grandy Jr,<ref>{{cite journal | doi = 10.1023/B:FOOP.0000012007.06843.ed | title = Time Evolution in Macroscopic Systems. I. Equations of Motion | year = 2004 | last1 = Grandy | first1 = W.T., Jr. | journal = Foundations of Physics | volume = 34 | issue = 1 | page = 1 |url=http://physics.uwyo.edu/~tgrandy/evolve.html |arxiv = cond-mat/0303290 |bibcode = 2004FoPh...34....1G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/entropy.html | doi=10.1023/B:FOOP.0000012008.36856.c1 | title=Time Evolution in Macroscopic Systems. II. The Entropy | year=2004 | last1=Grandy | first1=W.T., Jr. | journal=Foundations of Physics | volume=34 | issue=1 | page=21 |arxiv = cond-mat/0303291 |bibcode = 2004FoPh...34...21G | s2cid=18573684 }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/applications.html | doi = 10.1023/B:FOOP.0000022187.45866.81 | title=Time Evolution in Macroscopic Systems. III: Selected Applications | year=2004 | last1=Grandy | first1=W. T., Jr | journal=Foundations of Physics | volume=34 | issue=5 | page=771 |bibcode = 2004FoPh...34..771G | s2cid = 119406182 }}</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref> 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.

It is pointed out by W.T. Grandy Jr,<ref>{{cite journal | doi = 10.1023/B:FOOP.0000012007.06843.ed | title = Time Evolution in Macroscopic Systems. I. Equations of Motion | year = 2004 | last1 = Grandy | first1 = W.T., Jr. | journal = Foundations of Physics | volume = 34 | issue = 1 | page = 1 |url=http://physics.uwyo.edu/~tgrandy/evolve.html |arxiv = cond-mat/0303290 |bibcode = 2004FoPh...34....1G }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/entropy.html | doi=10.1023/B:FOOP.0000012008.36856.c1 | title=Time Evolution in Macroscopic Systems. II. The Entropy | year=2004 | last1=Grandy | first1=W.T., Jr. | journal=Foundations of Physics | volume=34 | issue=1 | page=21 |arxiv = cond-mat/0303291 |bibcode = 2004FoPh...34...21G | s2cid=18573684 }}</ref><ref>{{cite journal | url=http://physics.uwyo.edu/~tgrandy/applications.html | doi = 10.1023/B:FOOP.0000022187.45866.81 | title=Time Evolution in Macroscopic Systems. III: Selected Applications | year=2004 | last1=Grandy | first1=W. T., Jr | journal=Foundations of Physics | volume=34 | issue=5 | page=771 |bibcode = 2004FoPh...34..771G | s2cid = 119406182 }}</ref><ref>Grandy 2004 see also [http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html].</ref> 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,<ref>{{cite book| title = Rational Thermodynamics | year = 1984 | last1 = Truesdell | first1 = Clifford | publisher = Springer | edition = 2 }}</ref><ref>{{cite book| title = Continuum Thermomechanics | year = 2002 | last1 = Maugin | first1 = Gérard A. | publisher = Kluwer }}</ref><ref>{{cite book| title = The Mechanics and Thermodynamics of Continua | year = 2010 | last1 = Gurtin | first1 = Morton E. | publisher = Cambridge University Press }}</ref><ref>{{cite book| title = Thermodynamics of Materials with Memory: Theory and Applications | year = 2012| last1 = Amendola | first1 = Giovambattista | publisher = Springer }}</ref> which evolved completely independently of statistical mechanics and maximum-entropy principles.

This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics,<ref>{{cite book| title = Rational Thermodynamics | year = 1984 | last1 = Truesdell | first1 = Clifford | publisher = Springer | edition = 2 }}</ref><ref>{{cite book| title = Continuum Thermomechanics | year = 2002 | last1 = Maugin | first1 = Gérard A. | publisher = Kluwer }}</ref><ref>{{cite book| title = The Mechanics and Thermodynamics of Continua | year = 2010 | last1 = Gurtin | first1 = Morton E. | publisher = Cambridge University Press }}</ref><ref>{{cite book| title = Thermodynamics of Materials with Memory: Theory and Applications | year = 2012| last1 = Amendola | first1 = Giovambattista | publisher = Springer }}</ref> which evolved completely independently of statistical mechanics and maximum-entropy principles.

 

 

 

 

 

 

 

 

 

To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables <math>x_1, x_2, ..., x_n</math> that are used to fix the equilibrium state, as was described above, a set of variables <math>\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

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

The first term on the right hand side of the equation presents a stream of thermal energy into the system; the last term—a stream of energy into the system coming with the stream of particles of substances <math> \Delta N_\alpha </math> that can be positive or negative, <math> \mu_\alpha</math> is chemical potential of substance <math> \alpha</math>. The middle term in (1) depicts energy dissipation (entropy production) due to the relaxation of internal variables <math> \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,

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

{{NumBlk|:|<math>

{{NumBlk|:|<math>

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

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

where <math> \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> \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.<ref name="dx.doi.org">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.</ref>

where <math> \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> \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.<ref name="dx.doi.org">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.</ref>

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

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

 

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

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

{{NumBlk|:|<math>

{{NumBlk|:|<math>

S=S(T, x_1, x_2, , x_n; \xi_1, \xi_2,  \ldots)  

S=S(T, x_1, x_2, , x_n; \xi_1, \xi_2,  \ldots)  

</math>|{{EquationRef|1}}}}

</math>|{{EquationRef|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,<ref>[[Ilya Prigogine|Prigogine, I.]] (1955/1961/1967). ''Introduction to Thermodynamics of Irreversible Processes''. 3rd edition, Wiley Interscience, New York.</ref> Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature <math> T</math> . The increment of [[entropy]] <math> S</math> can be calculated according to the formula  

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,<ref>[[Ilya Prigogine|Prigogine, I.]] (1955/1961/1967). ''Introduction to Thermodynamics of Irreversible Processes''. 3rd edition, Wiley Interscience, New York.</ref> Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature <math> T</math> . The increment of [[entropy]] <math> S</math> can be calculated according to the formula  

{{NumBlk|:|<math>

{{NumBlk|:|<math>

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

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

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> \Delta N_\alpha </math> that can be positive or negative, <math> \mu_\alpha</math> is [[chemical potential]] of substance <math> \alpha</math>. The middle term in (1) depicts [[energy dissipation]] ([[entropy production]]) due to the relaxation of internal variables <math> \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,<ref>Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach,  Eur. J. Phys.  vol. 26,  769-781.</ref><ref name="dx.doi.org"/> to consider any deviation from the equilibrium state as an internal variable, so that we consider the set of internal variables <math> \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.

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> \Delta N_\alpha </math> that can be positive or negative, <math> \mu_\alpha</math> is [[chemical potential]] of substance <math> \alpha</math>. The middle term in (1) depicts [[energy dissipation]] ([[entropy production]]) due to the relaxation of internal variables <math> \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,<ref>Pokrovskii V.N. (2005) Extended thermodynamics in a discrete-system approach,  Eur. J. Phys.  vol. 26,  769-781.</ref><ref name="dx.doi.org"/> to consider any deviation from the equilibrium state as an internal variable, so that we consider the set of internal variables <math> \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==

==Flows and forces==