更改

[[File:Boltzmann distribution graph.svg|upright=1.75|right|thumb|Boltzmann factor ''p_i'' / ''p_j'' (vertical axis) as a function of temperature ''T'' for several energy differences ''ε_i'' − ''ε_j''.|链接=Special:FilePath/Boltzmann_distribution_graph.svg]]

[[File:Boltzmann distribution graph.svg|upright=1.75|right|thumb|Boltzmann factor ''p_i'' / ''p_j'' (vertical axis) as a function of temperature ''T'' for several energy differences ''ε_i'' − ''ε_j''.|链接=Special:FilePath/Boltzmann_distribution_graph.svg]]

In [[statistical mechanics]] and [[mathematics]], a '''Boltzmann distribution''' (also called '''Gibbs distribution'''<ref name="landau">{{cite book | author=Landau, Lev Davidovich |author2=Lifshitz, Evgeny Mikhailovich |name-list-style=amp | title=Statistical Physics |volume=5 |series=Course of Theoretical Physics |edition=3 |origyear=1976 |year=1980 |place=Oxford |publisher=Pergamon Press|isbn=0-7506-3372-7|author-link=Lev Landau |author2-link=Evgeny Lifshitz }} Translated by J.B. Sykes and M.J. Kearsley. See section 28</ref>) is a [[probability distribution]] or [[probability measure]] that gives the probability that a system will be in a certain [[microstate (statistical mechanics)|state]] as a function of that state's energy and the temperature of the system. The distribution is expressed in the form:

In [[statistical mechanics]] and [[mathematics]], a '''Boltzmann distribution''' (also called '''Gibbs distribution'''<ref name="landau">{{cite book | author=Landau, Lev Davidovich |author2=Lifshitz, Evgeny Mikhailovich |name-list-style=amp | title=Statistical Physics |volume=5 |series=Course of Theoretical Physics |edition=3 |origyear=1976 |year=1980 |place=Oxford |publisher=Pergamon Press|isbn=0-7506-3372-7|author-link=Lev Landau |author2-link=Evgeny Lifshitz }} Translated by J.B. Sykes and M.J. Kearsley. See section 28</ref>) is a [[probability distribution]] or [[probability measure]] that gives the probability that a system will be in a certain [[microstate (statistical mechanics)|state]] as a function of that state's energy and the temperature of the system. The distribution is expressed in the form:

在统计力学和数学中，波兹曼分布分布(也称为吉布斯分布<ref name="landau" />)是一个概率分布或机率量测，它给出了一个系统处于某种量子态的概率，这个概率是该状态的能量和系统温度的函数。分布情况以下列形式表示:

在统计力学和数学中，波兹曼分布分布(也称为吉布斯分布<ref name="landau" />)是一个概率分布或机率量测，它给出了一个系统处于某种量子态的概率，这个概率是该状态的能量和系统温度的函数。分布情况以下列形式表示:

where {{mvar|p_i}} is the probability of the system being in state {{mvar|i}}, {{mvar|ε_i}} is the energy of that state, and a constant {{mvar|kT}} of the distribution is the product of [[Boltzmann's constant]] {{mvar|k}} and [[thermodynamic temperature]] {{mvar|T}}. The symbol <math display="inline">\propto</math> denotes [[proportionality (mathematics)|proportionality]] (see {{section link||The distribution}} for the proportionality constant).

where {{mvar|p_i}} is the probability of the system being in state {{mvar|i}}, {{mvar|ε_i}} is the energy of that state, and a constant {{mvar|kT}} of the distribution is the product of [[Boltzmann's constant]] {{mvar|k}} and [[thermodynamic temperature]] {{mvar|T}}. The symbol <math display="inline">\propto</math> denotes [[proportionality (mathematics)|proportionality]] (see {{section link||The distribution}} for the proportionality constant).

{{mvar|p_i}} 是其中系统处于状态{{mvar|i}}的概率, {{mvar|ε_i}} 是该状态的能量，还有关于这个分布的一个常数 {{mvar|kT}} ，它是玻耳兹曼常数{{mvar|k}}和热力学温度 {{mvar|T}} 的乘积。符号 <math display="inline">\propto</math>表示相称性(见附录中的比例常数)。

{{mvar|p_i}} 是其中系统处于状态{{mvar|i}}的概率, {{mvar|ε_i}} 是该状态的能量，还有关于这个分布的一个常数 {{mvar|kT}} ，它是玻耳兹曼常数{{mvar|k}}和热力学温度 {{mvar|T}} 的乘积。符号 <math display="inline">\propto</math>表示相称性(见附录中的比例常数)。

The term system here has a very wide meaning; it can range from a single atom to a macroscopic system such as a [[Natural gas storage|natural gas storage tank]]. Because of this the Boltzmann distribution can be used to solve a very wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied .

The term system here has a very wide meaning; it can range from a single atom to a macroscopic system such as a [[Natural gas storage|natural gas storage tank]]. Because of this the Boltzmann distribution can be used to solve a very wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied .

The term system here has a very wide meaning; it can range from a single atom to a macroscopic system such as a natural gas storage tank. Because of this the Boltzmann distribution can be used to solve a very wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied .

这里的术语系统含义广泛; 它可以从单个原子到宏观系统，如天然气储罐。正因为如此，波兹曼分布可以用来解决各种各样的问题。玻尔兹曼分布表明，能量较低的状态被占据的概率总是较高的。

The ''ratio'' of probabilities of two states is known as the '''Boltzmann factor''' and characteristically only depends on the states' energy difference:

The ''ratio'' of probabilities of two states is known as the '''Boltzmann factor''' and characteristically only depends on the states' energy difference:

两种状态的概率之比被称为玻尔兹曼因子，取决于两种状态的能量差:

两种状态的概率之比被称为玻尔兹曼因子，取决于两种状态的能量差:

:<math>\frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}}</math>

:<math>\frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}}</math>

[数学][数学]

[数学][数学]

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>

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>

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>

The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of [[entropy]] (The [[Entropy_(statistical_thermodynamics)#Gibbs_entropy_formula | Gibbs entropy formula]] <math>S = -k_{\mathrm{B}}\sum_i p_i \log p_i</math>) and the thermodynamic definition of entropy (<math>d S = \frac{\delta Q_\text{rev}}{T}</math>, and the [[fundamental thermodynamic relation]]).<ref name=":0">{{cite journal |last1= Gao |first1= Xiang |last2= Gallicchio |first2= Emilio |first3= Adrian |last3= Roitberg  |date= 2019 |title= The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy |journal= The Journal of Chemical Physics|volume= 151|issue= 3|pages= 034113|doi= 10.1063/1.5111333|pmid= 31325924 |arxiv= 1903.02121 |s2cid= 118981017 }}</ref>

The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of [[entropy]] (The [[Entropy_(statistical_thermodynamics)#Gibbs_entropy_formula | Gibbs entropy formula]] <math>S = -k_{\mathrm{B}}\sum_i p_i \log p_i</math>) and the thermodynamic definition of entropy (<math>d S = \frac{\delta Q_\text{rev}}{T}</math>, and the [[fundamental thermodynamic relation]]).<ref name=":0">{{cite journal |last1= Gao |first1= Xiang |last2= Gallicchio |first2= Emilio |first3= Adrian |last3= Roitberg  |date= 2019 |title= The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy |journal= The Journal of Chemical Physics|volume= 151|issue= 3|pages= 034113|doi= 10.1063/1.5111333|pmid= 31325924 |arxiv= 1903.02121 |s2cid= 118981017 }}</ref>

The Boltzmann distribution should not be confused with the Maxwell–Boltzmann distribution. The former gives the probability that a system will be in a certain state as a function of that state's energy; in contrast, the latter is used to describe particle speeds in idealized gases.

The Boltzmann distribution should not be confused with the [[Maxwell–Boltzmann distribution]]. The former gives the probability that a system will be in a certain state as a function of that state's energy;<ref name="Atkins, P. W. 2010">Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York</ref> in contrast, the latter is used to describe particle speeds in idealized gases.

值得一提的是，玻尔兹曼分布不应与麦克斯韦-玻尔兹曼分布混淆。前者给出了系统处于某种状态的概率，作为该状态能量的函数; 后者则是用于描述理想气体中的粒子速度。

值得一提的是，玻尔兹曼分布不应与麦克斯韦-玻尔兹曼分布混淆。前者给出了系统处于某种状态的概率，作为该状态能量的函数; 后者则是用于描述理想气体中的粒子速度。

==The distribution==

==分布==

The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state's energy and  temperature of the system to which the distribution is applied. It is given as

The Boltzmann distribution is a [[probability distribution]] that gives the probability of a certain state as a function of that state's energy and  temperature of the [[system]] to which the distribution is applied.<ref name="McQuarrie, A. 2000">McQuarrie, A. (2000) Statistical Mechanics, University Science Books, California</ref> It is given as

波兹曼分布是一个概率分布，它给出了出于某种量子态处于某种能量和温度的时候的概率。它被这样定义：

波兹曼分布是一个概率分布，它给出了出于某种量子态处于某种能量和温度的时候的概率。它被这样定义：

<math>

<math>

</math>

</math>

where ''p_i'' is the probability of state ''i'', ''ε_i'' the energy of state ''i'', ''k'' the Boltzmann constant, ''T'' the temperature of the system and ''M'' is the number of all states accessible to the system of interest.<ref name="McQuarrie, A. 2000" /><ref name="Atkins, P. W. 2010" /> Implied parentheses around the denominator ''kT'' are omitted for brevity. The normalization denominator ''Q'' (denoted by some authors by ''Z'') is the [[canonical partition function]]

where ''p_i'' is the probability of state ''i'', ''ε_i'' the energy of state ''i'', ''k'' the Boltzmann constant, ''T'' the temperature of the system and ''M'' is the number of all states accessible to the system of interest.<ref name="McQuarrie, A. 2000" /><ref name="Atkins, P. W. 2010" /> Implied parentheses around the denominator ''kT'' are omitted for brevity. The normalization denominator ''Q'' (denoted by some authors by ''Z'') is the [[canonical partition function]]

其中 ''p_i'' 是状态 i 的概率，''ε_i'' 是状态 i 的能量，''k'' 是波兹曼常数，''T'' 是系统的温度，''M'' 是系统所能到达的所有量子态的数目。<ref name="Atkins, P. W. 2010" /> <ref name="McQuarrie, A. 2000" />这里为了简洁美观，省略了''kT''周围的括弧。归一化的分母''Q''（被有些作者写为''Z''）是对于系统中所有量子态进行总和，此部分又被称为[[Index.php?title=Canonical partition function|正则配分函数]]''。''

其中 ''p_i'' 是状态 i 的概率，''ε_i'' 是状态 i 的能量，''k'' 是波兹曼常数，''T'' 是系统的温度，''M'' 是系统所能到达的所有量子态的数目。<ref name="Atkins, P. W. 2010" /> <ref name="McQuarrie, A. 2000" />这里为了简洁美观，省略了''kT''周围的括弧。归一化的分母''Q''（被有些作者写为''Z''）是对于系统中所有量子态进行总和，此部分又被称为[[Index.php?title=Canonical partition function|正则配分函数]]''。''

It results from the constraint that the probabilities of all accessible states must add up to 1.

It results from the constraint that the probabilities of all accessible states must add up to 1.

在知道量子态能量的情况下，这个配分函数可以被计算。对于原子来说，配分函数的值可以在NIST Atomic Spectra Database中找到。<ref name=":1" />

在知道量子态能量的情况下，这个配分函数可以被计算。对于原子来说，配分函数的值可以在NIST Atomic Spectra Database中找到。<ref name=":1" />

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" />。

The Boltzmann distribution is related to the [[softmax function]] commonly used in machine learning.

The Boltzmann distribution is related to the [[softmax function]] commonly used in machine learning.

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。

 

{{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)'''

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 statistics]] of classical gases (systems of non-interacting particles)

[[Maxwell–Boltzmann statistics]] of classical gases (systems of non-interacting particles)

: 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。

* 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。

== In mathematics ==

== 数学 ==

In more general mathematical settings, the Boltzmann distribution is also known as the [[Gibbs measure]]. In statistics and machine learning, it is called a [[log-linear model]]. In [[deep learning]], the Boltzmann distribution is used in the sampling distribution of [[stochastic neural network]]s such as the [[Boltzmann machine]], [[Restricted Boltzmann machine]], [[Energy based model|Energy-Based models]] and [[Deep Boltzmann Machine|deep Boltzmann machine]]

In more general mathematical settings, the Boltzmann distribution is also known as the [[Gibbs measure]]. In statistics and machine learning, it is called a [[log-linear model]]. In [[deep learning]], the Boltzmann distribution is used in the sampling distribution of [[stochastic neural network]]s such as the [[Boltzmann machine]], [[Restricted Boltzmann machine]], [[Energy based model|Energy-Based models]] and [[Deep Boltzmann Machine|deep Boltzmann machine]]

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。

== In economics ==

== 经济 ==

The Boltzmann distribution can be introduced to allocate permits in emissions trading.<ref name="Park, J.-W. 2012">Park, J.-W., Kim, C. U. and Isard, W. (2012) Permit allocation in emissions trading using the Boltzmann distribution. Physica A 391: 4883–4890</ref><ref name=":5">[http://www.technologyreview.com/view/425051/the-thorny-problem-of-fair-allocation/ The Thorny Problem Of Fair Allocation]. ''Technology Review'' blog. August 17, 2011. Cites and summarizes Park, Kim and Isard (2012).</ref> The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications.

The Boltzmann distribution can be introduced to allocate permits in emissions trading.<ref name="Park, J.-W. 2012">Park, J.-W., Kim, C. U. and Isard, W. (2012) Permit allocation in emissions trading using the Boltzmann distribution. Physica A 391: 4883–4890</ref><ref name=":5">[http://www.technologyreview.com/view/425051/the-thorny-problem-of-fair-allocation/ The Thorny Problem Of Fair Allocation]. ''Technology Review'' blog. August 17, 2011. Cites and summarizes Park, Kim and Isard (2012).</ref> The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications.

在经济学中，玻尔兹曼分布被用来分配排放交易中的许可。<ref name="Park, J.-W. 2012" /><ref name=":5" />  使用玻尔兹曼分布的新分配方法可以创造出在多个国家之间最可能的、自然的和无偏见的排放许可分配。这种新方法简单而通用，在经济和环保中有很大的应用潜力。

在经济学中，玻尔兹曼分布被用来分配排放交易中的许可。<ref name="Park, J.-W. 2012" /><ref name=":5" />  使用玻尔兹曼分布的新分配方法可以创造出在多个国家之间最可能的、自然的和无偏见的排放许可分配。这种新方法简单而通用，在经济和环保中有很大的应用潜力。

The Boltzmann distribution has the same form as the [[Multinomial logistic regression|multinomial logit]] model. As a [[discrete choice]] model, this is very well known in economics since [[Daniel McFadden]] made the connection to random utility maximization.

The Boltzmann distribution has the same form as the [[Multinomial logistic regression|multinomial logit]] model. As a [[discrete choice]] model, this is very well known in economics since [[Daniel McFadden]] made the connection to random utility maximization.

玻尔兹曼分布与多项逻辑回归模型具有相同的形式。在丹尼尔 · 麦克法登将它与随机效用最大化联系起来之后，成为了大家所熟知的一种离散选择模型。

玻尔兹曼分布与多项逻辑回归模型具有相同的形式。在丹尼尔 · 麦克法登将它与随机效用最大化联系起来之后，成为了大家所熟知的一种离散选择模型。