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The Boltzmann distribution is named after [[Ludwig Boltzmann]] who first formulated it in 1868 during his studies of the [[statistical mechanics]] of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"<ref>http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf</ref>

The Boltzmann distribution is named after [[Ludwig Boltzmann]] who first formulated it in 1868 during his studies of the [[statistical mechanics]] of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"<ref>http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf</ref>. The distribution was later investigated extensively, in its modern generic form, by [[Josiah Willard Gibbs]] in 1902.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]]

波兹曼分布是根据路德维希·玻尔兹曼1868年在研究统计力学中气体热平衡的时候首次提出的一个公式命名的。波尔兹曼的统计工作在他1877年的论文《论热力学第二基本定理与热平衡条件的概率计算之间的关系》中得到了体现。<ref>“On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). </ref>

|location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref>

|location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref>

波兹曼分布是根据路德维希·玻尔兹曼1868年在研究统计力学中气体热平衡的时候首次提出的一个公式命名的。波尔兹曼的统计工作在他1877年的论文《论热力学第二基本定理与热平衡条件的概率计算之间的关系》中得到了体现。<ref>“On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). </ref> 在玻尔兹曼分布被发明之后，约西亚·华纳德·吉布森充分地研究了它并在1902年提出了它的一般形式。<ref name="gibbs" />

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<nowiki>subject to the constraint that {\sum{p_i {\varepsilon}_i}} equals a particular mean energy value (which can be proven using </nowiki>[[Lagrange multipliers]]).

subject to the constraint that ΣΡ_iε_i equals a particular mean energy value (which can be proven using [[Lagrange multipliers]]).

<nowiki>当 {\sum{p_i {\varepsilon}_i}} 等于平均能量值时，玻尔兹曼分布是这种情况下能让熵最大化的分布。我们可以通过拉格朗日乘数法来证明。</nowiki>

当 ΣΡ_iε_i等于平均能量值时，玻尔兹曼分布是这种情况下能让熵最大化的分布。我们可以通过拉格朗日乘数法来证明。

其中 p_i 是量子态 i 的概率， p_j 是量子态 j 的概率，ε_i 和 ε_j 分别是状态 i 和 j 的能量。

其中 p_i 是量子态 i 的概率， p_j 是量子态 j 的概率，ε_i 和 ε_j 分别是状态 i 和 j 的能量。

玻尔兹曼分布通常被用来描述粒子的分布，比如原子或分子，在能量状态上的分布情况。如果我们有一个由许多粒子组成的系统，某个粒子处于量子态 i 的概率就等同于当我们从该系统中选择一个随机的粒子并观察它处于什么状态，发现它处于状态 i 的概率。这个概率等于量子态 i 的粒子数除以系统中粒子的总数，即那些占据量子态i的粒子的比例。

玻尔兹曼分布通常被用来描述粒子的分布，比如原子或分子，在能量状态上的分布情况。如果我们有一个由许多粒子组成的系统，某个粒子处于量子态 i 的概率就等同于当我们从该系统中选择一个随机的粒子并观察它处于什么状态，发现它处于状态 i 的概率。这个概率等于量子态 i 的粒子数除以系统中粒子的总数，即那些占据量子态i的粒子的比例。

 

where N_i is the number of particles in state i and N is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state i as a function of the energy of that state is

where N_i is the number of particles in state i and N is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state i as a function of the energy of that state is

N_i 是位于量子态 i 的粒子数，N是系统中粒子的总数。我们可以用波兹曼分布来求出这个概率，它等于处于量子态i的粒子的比例。所以这个方程给出位于量子态 i 的粒子比例关于这个状态能量的函数<ref name="Atkins, P. W. 2010" />。

N_i 是位于量子态 i 的粒子数，N是系统中粒子的总数。我们可以用波兹曼分布来求出这个概率，它等于处于量子态i的粒子的比例。所以这个方程给出位于量子态 i 的粒子比例关于这个状态能量的函数<ref name="Atkins, P. W. 2010" />。

玻尔兹曼分布与机器学习中常用的[[Index.php?title=Softmax function|归一化指数函数]]（Softmax 函数）有关。

玻尔兹曼分布与机器学习中常用的[[Index.php?title=Softmax function|归一化指数函数]]（Softmax 函数）有关。

=='''统计力学''' ==

=='''统计力学'''==

{{main|Canonical ensemble|Maxwell–Boltzmann statistics}}The Boltzmann distribution appears in statistical mechanics when considering isolated (or nearly-isolated) systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble, but also some special cases (derivable from the canonical ensemble) also show the Boltzmann distribution in different aspects:

{{main|Canonical ensemble|Maxwell–Boltzmann statistics}}The Boltzmann distribution appears in statistical mechanics when considering isolated (or nearly-isolated) systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble, but also some special cases (derivable from the canonical ensemble) also show the Boltzmann distribution in different aspects:

当独立的(或者几乎独立的)固定体系处于统计力学热平衡状态时，玻尔兹曼分布就会出现。最普遍的情况是概率分布的正则系综，但也有一些特殊的情况(从正则系综衍生)也显示了不同形式的玻尔兹曼分布:

当独立的(或者几乎独立的)固定体系处于统计力学热平衡状态时，玻尔兹曼分布就会出现。最普遍的情况是概率分布的正则系综，但也有一些特殊的情况(从正则系综衍生)也显示了不同形式的玻尔兹曼分布:

; [[Canonical ensemble]] (general case)

;[[Canonical ensemble]] (general case)

; '''正则系综(一般情况)'''

;'''正则系综(一般情况)'''

; '''The [[canonical ensemble]] gives the [[probabilities]] of the various possible states of a closed system of fixed volume, in thermal equilibrium with a [[heat bath]]. The canonical ensemble is a probability distribution with the Boltzmann form.'''

;'''The [[canonical ensemble]] gives the [[probabilities]] of the various possible states of a closed system of fixed volume, in thermal equilibrium with a [[heat bath]]. The canonical ensemble is a probability distribution with the Boltzmann form.'''

正则系综模型给出了在一个封闭固定体积，带有热浴的热平衡系统中的各种可能状态的概率。我们称正则系综为一个玻尔兹曼概率分布。

正则系综模型给出了在一个封闭固定体积，带有热浴的热平衡系统中的各种可能状态的概率。我们称正则系综为一个玻尔兹曼概率分布。

;  

;

; Statistical frequencies of subsystems' states (in a non-interacting collection)

;Statistical frequencies of subsystems' states (in a non-interacting collection)

'''子系统状态的统计频率(在一个无交互的集合中)'''

'''子系统状态的统计频率(在一个无交互的集合中)'''

: When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the [[statistical frequency]] of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the [[expectation value|expected]] statistical frequency distribution of subsystem states has the Boltzmann form.

:When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the [[statistical frequency]] of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the [[expectation value|expected]] statistical frequency distribution of subsystem states has the Boltzmann form.

当系统是一个较小子系统的许多非交互副本的集合时，有时我们需要在某个子系统集合中找到某个子系统的统计频率。当应用于这样一个集合时，这个子系统的正则系综具有可分离性: 只要不相互作用的子系统的组成是固定的，那么每个子系统的状态都是独立于其他子系统的，它们也拥有自己的正则系综。因此，子系统状态的期望统计频率分布形式是玻尔兹曼分布。

当系统是一个较小子系统的许多非交互副本的集合时，有时我们需要在某个子系统集合中找到某个子系统的统计频率。当应用于这样一个集合时，这个子系统的正则系综具有可分离性: 只要不相互作用的子系统的组成是固定的，那么每个子系统的状态都是独立于其他子系统的，它们也拥有自己的正则系综。因此，子系统状态的期望统计频率分布形式是玻尔兹曼分布。

'''Maxwell-Boltzmann 统计中的经典气体系统(非相互作用粒子系统)'''

'''Maxwell-Boltzmann 统计中的经典气体系统(非相互作用粒子系统)'''

: In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. [[Maxwell–Boltzmann statistics]] give the expected number of particles found in a given single-particle state, in a [[classical mechanics|classical]] gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.

:In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. [[Maxwell–Boltzmann statistics]] give the expected number of particles found in a given single-particle state, in a [[classical mechanics|classical]] gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.

在粒子系统中，许多粒子共享同一空间，并且相互之间有规律地改变位置; 它们所占据的单粒子状态空间是一个共享空间。[[Index.php?title=Maxwell–Boltzmann statistics|麦克斯韦-玻尔兹曼统计]]给出了在一个给定的单粒子态，在一个处于平衡状态的非相互作用粒子的经典气体中所发现的粒子的预期数量。这个预期的数量分布具有玻尔兹曼分布形式。

在粒子系统中，许多粒子共享同一空间，并且相互之间有规律地改变位置; 它们所占据的单粒子状态空间是一个共享空间。[[Index.php?title=Maxwell–Boltzmann statistics|麦克斯韦-玻尔兹曼统计]]给出了在一个给定的单粒子态，在一个处于平衡状态的非相互作用粒子的经典气体中所发现的粒子的预期数量。这个预期的数量分布具有玻尔兹曼分布形式。

虽然这些案例有很多相似之处，但是当关键假设发生变化时，我们需要对它们以不同的方式进行归纳总结，因此区分它们是有帮助的:

虽然这些案例有很多相似之处，但是当关键假设发生变化时，我们需要对它们以不同的方式进行归纳总结，因此区分它们是有帮助的:

* When a system is in thermodynamic equilibrium with respect to both energy exchange ''and particle exchange'', the requirement of fixed composition is relaxed and a [[grand canonical ensemble]] is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a [[microcanonical ensemble]] applies instead.

*When a system is in thermodynamic equilibrium with respect to both energy exchange ''and particle exchange'', the requirement of fixed composition is relaxed and a [[grand canonical ensemble]] is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a [[microcanonical ensemble]] applies instead.

* 当一个系统在能量交换和粒子交换方面都处于热力学平衡时，固定组成的要求就放宽了。这时我们就得到了一个[[Index.php?title=Microcanonical ensemble|巨正则系综]]而不是正则系综。另一方面，如果成分和能量都是固定的，则可以使用[[微正则系综Index.php?title=Microcanonical ensemble|微正则系综]]。

*当一个系统在能量交换和粒子交换方面都处于热力学平衡时，固定组成的要求就放宽了。这时我们就得到了一个[[Index.php?title=Microcanonical ensemble|巨正则系综]]而不是正则系综。另一方面，如果成分和能量都是固定的，则可以使用[[微正则系综Index.php?title=Microcanonical ensemble|微正则系综]]。

* If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an [[analytical solution]].<ref name=":4">A classic example of this is [[magnetic ordering]]. Systems of non-interacting [[spin (physics)|spins]] show [[paramagnetic]] behaviour that can be understood with a single-particle canonical ensemble (resulting in the [[Brillouin function]]). Systems of ''interacting'' spins can show much more complex behaviour such as [[ferromagnetism]] or [[antiferromagnetism]].</ref> The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.

*If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an [[analytical solution]].<ref name=":4">A classic example of this is [[magnetic ordering]]. Systems of non-interacting [[spin (physics)|spins]] show [[paramagnetic]] behaviour that can be understood with a single-particle canonical ensemble (resulting in the [[Brillouin function]]). Systems of ''interacting'' spins can show much more complex behaviour such as [[ferromagnetism]] or [[antiferromagnetism]].</ref> The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.

* 如果集合中的子系统确实相互交互，则子系统状态的预期频率不再遵循玻尔兹曼 分布，甚至可能没有解析解。<ref name=":4" />然而，如果整个系统是独立的并且处于热平衡状态，则正则系综仍然可以应用于作为一个整体考虑。

*如果集合中的子系统确实相互交互，则子系统状态的预期频率不再遵循玻尔兹曼 分布，甚至可能没有解析解。<ref name=":4" />然而，如果整个系统是独立的并且处于热平衡状态，则正则系综仍然可以应用于作为一个整体考虑。

* With ''[[quantum mechanics|quantum]]'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]], depending on whether the particles are [[fermion]]s or [[boson]]s respectively.

*With ''[[quantum mechanics|quantum]]'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]], depending on whether the particles are [[fermion]]s or [[boson]]s respectively.

* 对于处于平衡状态的非相互作用粒子的量子气体，在给定的单粒子状态中发现的粒子数量并不遵循麦克斯韦-玻尔兹曼统计。并且正则系综中的量子气体没有简单的封闭表达式。在巨正则系综中，量子气体的统计分布状态由费米-狄拉克统计或玻色-爱因斯坦统计来描述，具体则取决于该粒子是费米子还是玻色子。

*对于处于平衡状态的非相互作用粒子的量子气体，在给定的单粒子状态中发现的粒子数量并不遵循麦克斯韦-玻尔兹曼统计。并且正则系综中的量子气体没有简单的封闭表达式。在巨正则系综中，量子气体的统计分布状态由费米-狄拉克统计或玻色-爱因斯坦统计来描述，具体则取决于该粒子是费米子还是玻色子。

== 数学 ==

==数学==

在更一般的数学环境中，玻尔兹曼分布也被称为吉布斯量度。在统计学和机器学习中，它被称为对数线性回归。在深度学习中，玻尔兹曼分布被用于随机神经网络的抽样分布，如玻尔兹曼机、受限玻尔兹曼机、基于能量的模型和深度波茨曼机。

在更一般的数学环境中，玻尔兹曼分布也被称为吉布斯量度。在统计学和机器学习中，它被称为对数线性回归。在深度学习中，玻尔兹曼分布被用于随机神经网络的抽样分布，如玻尔兹曼机、受限玻尔兹曼机、基于能量的模型和深度波茨曼机。

== 经济 ==

==经济==

[[Category:待整理页面]]

[[Category:待整理页面]]