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[[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]]

{{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 <nowiki></math ></nowiki> 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 <nowiki></math ></nowiki> 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 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 .

这里的术语系统有非常广泛的含义; 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。分布表明，能量较低的状态被占据的概率总是较高的。

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，其特点仅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:

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"

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"

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这种分布后来在1902年由约西亚·威拉德·吉布斯进行了广泛的调查，以其现代通用形式。

这种分布后来在1902年由约西亚·威拉德·吉布斯进行了广泛的调查，以其现代通用形式。

|location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref>{{rp|Ch.IV}}

|location=New York|title-link=Elementary Principles in Statistical Mechanics }}</ref>{{rp|Ch.IV}}、

 

 

The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of entropy (The  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).

The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of entropy (The  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).

广义波兹曼分布是熵的统计力学定义(吉布斯熵公式 < math > s =-k { mathrm { b } sum i log pi i </math >)与熵的热力学定义(< math > ds = frac { delta q { rev }{ t } </math >)等价的充要条件。

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>{{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>

广义波兹曼分布是熵的统计力学定义(吉布斯熵公式 < math > s =-k { mathrm { b } sum i log pi i <nowiki></math ></nowiki>)与熵的热力学定义(< math > ds = frac { delta q { rev }{ t } <nowiki></math ></nowiki>)等价的充要条件。<ref name=":0" />

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; in contrast, the latter is used to describe particle speeds in idealized gases.

波兹曼分布不应与麦克斯韦-波兹曼分布混淆。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; 相反，后者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 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.

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.

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. 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

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