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第597行：
第597行： + + + + + + + + + + + + 第602行：
第614行： − 所有这些过程都是以特征速率随时间发生的，这些速率对于工程来说非常重要。非平衡统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算最终结果，在外部不平衡被消除，整体回归到平衡状态之后。)+ 第610行：
第622行： − 原则上，非平衡态统计力学可以在数学上是精确的: 孤立系统的整体随时间演化，根据确定性方程，如刘维尔方程或其量子等价物冯 · 诺依曼方程。这些方程是将机械运动方程独立应用于整体中每个状态的结果。不幸的是，这些集合演化方程继承了潜在机械运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不破坏信息(系综的吉布斯熵被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。+ 第618行：
第630行： − 因此，非平衡力学是一个活跃的理论研究领域，因为这些额外的假设的有效范围继续探索。在下面的小节中描述了一些方法。+ 第745行：
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→Non-equilibrium statistical mechanics
* and irreversible processes in general.
* and irreversible processes in general.
* [[热传导|材料的内部运动来传导热量]]，由热度不平衡来驱动，
* [[导电|导体内电荷的运动产生电流]]，由电压不平衡来驱动，
* 自发的 [[化学反应]]，由自由能的下降驱动，
* [[摩擦]]，[[耗散]]，[[量子退相干]，
* 由外力泵送的系统([光泵]]等)
* 以及一般意义的可逆过程
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
All of these processes occur over time with characteristic rates, and these rates are of importance for engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)
所有这些过程都是以特征速率随时间发生的，这些速率对于工程来说非常重要。非平衡态统计力学研究领域关注的是在微观水平上理解这些非平衡过程。(统计热力学只能用来计算在外部不平衡被消除，整体回归到平衡状态之后的最终结果。)
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.
In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.
原则上，非平衡态统计力学在数学上可以是精确的: 孤立系统的系综根据确定性方程随时间演化，如刘维尔方程或其量子等价、冯·诺依曼方程。这些方程是将运动力学方程独立应用于系综中每个状态的结果。不幸的是，这些系综演化方程继承了潜在动力学运动的大部分复杂性，因此很难得到精确解。此外，系综演化方程是完全可逆的，不会破坏信息(系综的吉布斯熵被保留)。为了在模拟不可逆过程中取得进展，除了概率和可逆力学外，还必须考虑其他因素。
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.
Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.
因此，非平衡力学是一个活跃的理论研究领域，因为这些额外假设的有效范围仍将继续探索。在下面的小节中描述了一些方法。
一种先进的方法使用了随机方法和线性响应理论的结合。作为一个例子，计算电子系统电导中的量子相干效应(弱局域化，电导涨落)的一种方法是使用 Green-Kubo 关系，包括随机退相的各种电子之间的相互作用，使用凯尔迪什方法。
一种先进的方法使用了随机方法和线性响应理论的结合。作为一个例子，计算电子系统电导中的量子相干效应(弱局域化，电导涨落)的一种方法是使用 Green-Kubo 关系，包括随机退相的各种电子之间的相互作用，使用凯尔迪什方法。
==Applications outside thermodynamics==
==Applications outside thermodynamics==