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第479行：
第479行： − 建立这种力和流密度之间的关系是一个统计力学的问题。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。+ 第487行：
第487行： − 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。+ 第495行：
第495行： − 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。+ + 第507行：
第508行： − + 第523行：
第524行： − 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:+ 第531行：
第532行： − 部分数学部分数学 第547行：
第547行： − 部分 f _ i } frac { partial f _ j }{ partial x _ j }{ partial x _ j } </math > 第555行：
第554行： − 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。+ 第563行：
第562行： − 上述方程的熵产生率的推广是由 Pokrovskii 给出的。+
→Flows and forces
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.
建立这种力和流密度之间的关系是一个统计力学的问题。流密度(< math > j _ i </math >)可能是耦合的。昂萨格倒易关系的文章考虑了稳定的近定态热力学非平衡态，其中力和流密度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 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.
在稳态条件下，这样的力和对应的流密度根据定义是不随时间变化的，就像系统的局部定义的熵和熵产生率一样。值得注意的是，根据普利高津和其他人的研究，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熵产生最小化。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 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.
人们想要进一步分析非静态局部量的表面和体积积分的行为，这些积分是宏观的流和产生率。一般来说，这些积分的动力学用线性方程不能充分描述，尽管在特殊情况下它们可以这样描述。
===Onsager reciprocal relations===
===Onsager reciprocal relations===
昂萨格倒易关系
{{Main|Onsager reciprocal relations}}
{{Main|Onsager reciprocal relations}}
Following Section III of Rayleigh (1873), 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 linearly related to the flows:
Following Section III of Rayleigh (1873), 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 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:
在瑞利(1873)第三部分之后，昂萨格(1931，i)指出，在流(< 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<math>(\sigma)</math>与流呈线性关系:
and the flows are related to the gradient of the forces, parametrized by a matrix of coefficients conventionally denoted <math>L</math>:
and the flows are related to the gradient of the forces, parametrized by a matrix of coefficients conventionally denoted <math>L</math>:
并且流与力的梯度有关，通过一个系数矩阵参数化，通常表示为:
<math>J_i = \sum_{j} L_{ij} \frac{\partial F_j}{\partial x_j} </math>
<math>J_i = \sum_{j} L_{ij} \frac{\partial F_j}{\partial x_j} </math>
<math>\sigma = \sum_{i,j} L_{ij} \frac{\partial F_i}{\partial x_i}\frac{\partial F_j}{\partial x_j} </math>
<math>\sigma = \sum_{i,j} L_{ij} \frac{\partial F_i}{\partial x_i}\frac{\partial F_j}{\partial x_j} </math>
The second law of thermodynamics requires that the matrix <math>L</math> be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix <math>L</math> is symmetric. This fact is called the Onsager reciprocal relations.
The second law of thermodynamics requires that the matrix <math>L</math> be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix <math>L</math> is symmetric. This fact is called the Onsager reciprocal relations.
热力学第二定律要求矩阵<math>L</math>是正定的。统计力学动力学的微观可逆性的考虑暗示了矩阵是对称的。这个事实被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.
The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.
上述熵产生率方程的推广由 Pokrovskii 给出。
==Speculated extremal principles for non-equilibrium processes==
==Speculated extremal principles for non-equilibrium processes==