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第303行： + − 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 (thermodynamics)|volume]] <math>V</math> and <math>i^{th}</math> [[particle number]] <math>N_i</math>.+ + + + + 第317行：
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第370行： + − The [[second law of thermodynamics]] requires that the matrix <math>L</math> be [[Positive-definite matrix|positive definite]]. [[Statistical mechanics]] considerations involving microscopic reversibility of dynamics imply that the matrix <math>L</math> is [[symmetric matrix|symmetric]]. This fact is called the ''Onsager reciprocal relations''.+ 第365行：
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→Flows and forces
等式右边的第一项表示进入系统的热能流。最后一项是进入系统的能量流与可能为正或负的物质粒子流<math> \Delta N_\alpha </math>一起出现，<math> \mu_\alpha</math>为物质<math> \alpha</math>的'''<font color="#ff8000"> 化学势Chemical potential</font>'''。（1）中的中间项描述了由于内部变量<math> \xi_j</math>的松弛而导致的'''<font color="#ff8000"> 能量耗散Energy dissipation</font>'''（熵产生）。就化学反应物质而言（由Prigogine调查），内部变量似乎是化学反应不完整的量度，即所考虑的具有化学反应的系统失衡程度。该理论可以推广为，与平衡状态的任何偏差视为内部变量，因此我们认为方程式（1）中的内部变量<math> \xi_j</math>不仅由定义系统中发生的所有化学反应的完成度量组成，而且还有系统的结构，温度梯度，物质浓度的差异等等组成。
等式右边的第一项表示进入系统的热能流。最后一项是进入系统的能量流与可能为正或负的物质粒子流<math> \Delta N_\alpha </math>一起出现，<math> \mu_\alpha</math>为物质<math> \alpha</math>的'''<font color="#ff8000"> 化学势Chemical potential</font>'''。（1）中的中间项描述了由于内部变量<math> \xi_j</math>的松弛而导致的'''<font color="#ff8000"> 能量耗散Energy dissipation</font>'''（熵产生）。就化学反应物质而言（由Prigogine调查），内部变量似乎是化学反应不完整的量度，即所考虑的具有化学反应的系统失衡程度。该理论可以推广为，与平衡状态的任何偏差视为内部变量，因此我们认为方程式（1）中的内部变量<math> \xi_j</math>不仅由定义系统中发生的所有化学反应的完成度量组成，而且还有系统的结构，温度梯度，物质浓度的差异等等组成。
==Flows and forces==
== Flows and forces 流量与力 ==
The fundamental relation of classical equilibrium thermodynamics <ref name="W. Greiner et. al. 1997">W. Greiner, L. Neise, and H. Stöcker (1997), ''Thermodynamics and Statistical Mechanics (Classical Theoretical Physics)'' ,Springer-Verlag, New York, '''P85, 91, 101,108,116''', {{ISBN|0-387-94299-8}}.</ref>
The fundamental relation of classical equilibrium thermodynamics <ref name="W. Greiner et. al. 1997">W. Greiner, L. Neise, and H. Stöcker (1997), ''Thermodynamics and Statistical Mechanics (Classical Theoretical Physics)'' ,Springer-Verlag, New York, '''P85, 91, 101,108,116''', {{ISBN|0-387-94299-8}}.</ref>
经典平衡热力学的基本关系:
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 (thermodynamics)|volume]] <math>V</math> and <math>i^{th}</math> [[particle number]] <math>N_i</math>.
其中：表示系统的熵<math>dS</math>随强度温度<math>T</math>，压力<math>p</math>和第<math>i^{th}</math>个化学势<math>\mu_i</math>以及大量能量<math>U</math>，体积<math>V</math>和第<math>i^{th}</math>个粒子数<math>N_i</math>的微分而变化。
Following Onsager (1931,I),<ref name="Onsager 1931 I"/> let us extend our considerations to thermodynamically non-equilibrium systems. As a basis, we need locally defined versions of the extensive macroscopic quantities <math>U</math>, <math>V</math> and <math>N_i</math> and of the intensive macroscopic quantities <math>T</math>, <math>p</math> and <math>\mu_i</math>.
Following Onsager (1931,I),<ref name="Onsager 1931 I"/> let us extend our considerations to thermodynamically non-equilibrium systems. As a basis, we need locally defined versions of the extensive macroscopic quantities <math>U</math>, <math>V</math> and <math>N_i</math> and of the intensive macroscopic quantities <math>T</math>, <math>p</math> and <math>\mu_i</math>.
继Onsager（1931，I）之后，我们需要考虑将范围扩展到热力学非平衡系统。我们需要将局部定义的广延宏观量math>U</math>, <math>V</math> 和 <math>N_i</math>，以及强度宏观量<math>T</math>, <math>p</math> 和 <math>\mu_i</math>的形式作为基础。
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.
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广延宏观变量。我们可以在合适的条件下，通过局部定义梯度和基本局部定义的宏观量通量密度来导出这些新变量。
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]].
这样的局部宏观强度变量的梯度被称为“热力学力”。它们的“驱动”通量密度，可能被误导为“通量”，这对力是双重的。这些数量在关于Onsager互惠关系的文章中有所定义。
Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities (<math>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.
Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities (<math>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.
在统计力学中建立这样一个连接力与通量密度之间的关系，是一个问题。因为通量密度（<math>J_i</math>）可以耦合。关于Onsager互惠关系的文章考虑了稳定的近稳态热力学非平衡态，该态在力和通量密度上具有线性关系。
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.
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.
在静止状态下，这种力和相关的通量密度根据定义是时间不变的，系统局部定义的熵和熵的产生率也是如此。值得注意的是，根据Ilya Prigogine伊利亚·普里戈吉因等人的说法，当开放系统处于允许其达到稳定的静态热力学非平衡状态条件下，它会自我组织，以使局部定义的总熵最小化。这在下面进一步考虑。
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.
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.
有人希望将分析带入到描述非平稳局部量的表面和体积积分行为的下一步阶段。其中这些积分是指宏观通量和生产率。通常这些动力学方程并不能用线性方程式充分描述，但是在特殊情况下仍然是可以的。
===Onsager reciprocal relations===
=== Onsager reciprocal relations 昂萨格倒易关系 ===
{{Main|Onsager reciprocal relations}}
{{Main|Onsager reciprocal relations}}
Following Section III of Rayleigh (1873),<ref name="Rayleigh 1873"/> Onsager (1931, I)<ref name="Onsager 1931 I"/> showed that in the regime where both the flows (<math>J_i</math>) are small and the thermodynamic forces (<math>F_i</math>) vary slowly, the rate of creation of entropy <math>(\sigma)</math> is [[linear relation|linearly related]] to the flows:
Following Section III of Rayleigh (1873),<ref name="Rayleigh 1873"/> Onsager (1931, I)<ref name="Onsager 1931 I"/> showed that in the regime where both the flows (<math>J_i</math>) are small and the thermodynamic forces (<math>F_i</math>) vary slowly, the rate of creation of entropy <math>(\sigma)</math> is [[linear relation|linearly related]] to the flows:
根据Rayleigh（1873）的文章《Some General Theorems relating to Vibrations》第三节，Onsager（1931，I）的文章《Reciprocal Relations in Irreversible Processes. I.》，在流量（<math>J_i</math>）都较小且热力学力（<math>F_i</math>）缓慢变化的状态下，熵的产生速率（<math>(\sigma)</math>）为与流量线性相关：
:<math>\sigma = \sum_i J_i\frac{\partial F_i}{\partial x_i} </math>
:<math>\sigma = \sum_i J_i\frac{\partial F_i}{\partial x_i} </math>
and the flows are related to the gradient of the forces, parametrized by a [[matrix (mathematics)|matrix]] of coefficients conventionally denoted <math>L</math>:
and the flows are related to the gradient of the forces, parametrized by a [[matrix (mathematics)|matrix]] of coefficients conventionally denoted <math>L</math>:
流量与力的梯度有关，由通常表示为<math>L</math>的系数矩阵参数化
从中得出：
The [[second law of thermodynamics]] requires that the matrix <math>L</math> be [[Positive-definite matrix|positive definite]]. [[Statistical mechanics]] considerations involving microscopic reversibility of dynamics imply that the matrix <math>L</math> is [[symmetric matrix|symmetric]]. This fact is called the ''Onsager reciprocal relations''.
热力学第二定律要求矩阵<math>L</math>为正定。涉及动力学的微观可逆性的统计力学考虑意味着矩阵L是对称的。关于动力学的微观可逆性，涉及到统计力学考虑因素认为矩阵<math>L</math>是对称的。这个事实称为昂萨格倒易关系。
The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.<ref name="dx.doi.org"/>
The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.<ref name="dx.doi.org"/>
Pokrovskii给出了上述熵产生速率方程的广义概括。
==Speculated extremal principles for non-equilibrium processes==
==Speculated extremal principles for non-equilibrium processes==