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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。

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<font color="#FF8000">吉布斯熵 Gibbs entropy</font>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。

One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

处理非平衡态统计力学的一个方法是将随机行为引入系统。随机行为可以破坏系综中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，黑洞系统本身不会导致信息丢失) ，但这种随机性是为了反映出，随着时间的推移，感兴趣的信息会在系统内部转化为微妙的相关性，或者系统与环境之间的相关性。这些关联表现为对感兴趣的变量的混沌或伪随机的影响。用适当的随机性取代这些相关性，计算可以变得容易得多。

处理非平衡态统计力学的一个方法是将随机行为引入系统。随机行为可以破坏系综中包含的信息。虽然这在技术上是不准确的(除了涉及黑洞的假设情况外，黑洞系统本身不会导致信息丢失) ，但这种随机性是为了反映出，随着时间的推移，感兴趣的信息会在系统内部转化为微妙的相关性，或者系统与环境之间的相关性。这些关联表现为对感兴趣的变量的<font color="#FF8000">混沌 Chaos</font>或伪随机的影响。用适当的随机性取代这些相关性，计算可以变得容易得多。

|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).

|1 = Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).

玻尔兹曼输运方程: 在动力学理论研究中，早期的随机力学形式甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。

<font color="#FF8000">玻尔兹曼输运方程 Boltzmann transport equation</font>: 在动力学理论研究中，早期的随机力学形式甚至在“统计力学”一词被创造之前就已经出现了。詹姆斯·克拉克·麦克斯韦已经证明分子碰撞会导致气体内部明显的混沌运动。路德维希·玻尔兹曼随后证明，如果把这种分子混沌理所当然地看作是一种完全的随机化，那么气体中粒子的运动将遵循一个简单的玻尔兹曼输运方程，这个方程将使气体迅速恢复到平衡状态(见H-定理)。

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=== Near-equilibrium methods ===

=== Near-equilibrium methods 近平衡态方法 ===

 

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.<ref name="balescu"/>{{rp|664}}

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.<ref name="balescu"/>{{rp|664}}

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation-dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.

非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用线性响应理论进行分析。涨落-耗散定理是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。

非平衡态统计力学模型处理的另一类重要的系统，是对平衡态仅有非常轻微扰动的系统。在很小的扰动下，响应可以用<font color="#FF8000">线性响应理论 linear response theory</font>进行分析。<font color="#FF8000">涨落-耗散定理 fluctuation-dissipation theorem</font>是其中一个重要的结果，近平衡态系统的响应与系统总体平衡时的涨落准确相关。从本质上讲，一个系统如果稍微偏离平衡，无论是由于外力还是由于涨落，都会以同样的方式向平衡方向弛豫。因为这个系统无法区分偏离和回归，也“不知道”它是如何偏离平衡的。

* [[Mori–Zwanzig 公式]]

* [[Mori–Zwanzig 公式]]

=== Hybrid methods ===

=== Hybrid methods 组合方法===

 

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.<ref>{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}</ref><ref>{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}</ref>

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.<ref>{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}</ref><ref>{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}</ref>