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It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.

It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.

Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.

Statistical mechanics describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.

统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如热容)与微观行为联系起来，而在经典热力学中，唯一可行的选择就是测量和列出各种材料的热力学量。

统计力学描述了宏观观测量(如温度和压强)与围绕平均值波动的微观参数的关系。它将热力学量(比如<font color="#FFD700">热容</font>)与微观行为联系起来，而在经典热力学中，唯一可行的选择就是测量和列出各种材料的热力学量。

Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

Statistical mechanics can also be used to study systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics (sometimes called statistical dynamics) deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

统计力学也可以用来研究非平衡的系统。非平衡统计力学(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。涨落-耗散定理是人们从非平衡态统计力学中获得的基本知识，这是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。

统计力学也可以用来研究非平衡的系统。<font color="#FFD700">非平衡统计力学</font>(有时称为统计动力学)是统计力学的重要分支，它涉及的问题是对由非平衡导致的不可逆过程的速度进行微观模拟。例如化学反应或粒子流和热流。<font color="#FFD700">涨落-耗散定理</font>是人们从非平衡态统计力学中获得的基本知识，这是在应用非平衡态统计力学来研究多粒子系统中稳态电流流动这样的最简单的非平衡态情况下所发现的。

{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}

{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}

In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:

In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:

In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:

In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:

在物理学中，通常有两种力学被研究: 经典力学和量子力学。对于这两种类型的力学，标准的数学方法是考虑两个概念:

在物理学中，通常有两种力学被研究: <font color="#FFD700">经典力学</font>和<font color="#FFD700">量子力学</font>。对于这两种类型的力学，标准的数学方法是考虑两个概念:

# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).

# The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).

  The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).

  The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a pure quantum state vector (quantum mechanics).

力学系统在给定时间内的完整状态，用数学表示为相空间中的点(经典力学)或纯量子态矢量(量子力学)。

力学系统在给定时间内的完整状态，用数学表示为<font color="#FFD700">相空间</font>中的点(经典力学)或纯量子态矢量(量子力学)。

# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)

# An equation of motion which carries the state forward in time: [[Hamilton's equations]] (classical mechanics) or the [[time-dependent Schrödinger equation]] (quantum mechanics)

  An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)

  An equation of motion which carries the state forward in time: Hamilton's equations (classical mechanics) or the time-dependent Schrödinger equation (quantum mechanics)

一个运动方程描述状态在时间上的演化: 哈密尔顿方程(经典力学)或含时薛定谔方程(量子力学)

一个运动方程描述状态在时间上的演化: <font color="#FFD700">哈密尔顿方程</font>(经典力学)或<font color="#FFD700">含时薛定谔方程</font>(量子力学)

Using these two concepts, the state at any other time, past or future, can in principle be calculated.

Using these two concepts, the state at any other time, past or future, can in principle be calculated.

However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.

However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.

然而，这种概率是被解释的，系综中的每个状态随时间的演化都可以由运动方程给出。因此，系综本身(状态的概率分布概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由刘维尔方程(经典力学)或冯·诺依曼方程(量子力学)给出。这些方程是简单地通过分别应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统随时间演化过程中概率是守恒的。

然而，这种概率是被解释的，系综中的每个状态随时间的演化都可以由运动方程给出。因此，系综本身(状态的概率分布概率分布)也在随时间演化，因为系综中的虚拟系统不断地离开一个状态进入另一个状态。系综演化由<font color="#FFD700">刘维尔方程</font>(经典力学)或<font color="#FFD700">冯·诺依曼方程</font>(量子力学)给出。这些方程是简单地通过分别应用力学运动方程到系综中的每个虚拟系统而导出的，虚拟系统随时间演化过程中概率是守恒的。

One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.

One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.

== Statistical thermodynamics ==

== Statistical thermodynamics ==