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{{about|system energy states|particle energy levels and velocities|Maxwell–Boltzmann distribution}}

{{about|system energy states|particle energy levels and velocities|Maxwell–Boltzmann distribution}}

[[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''.]]

[[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]]

Boltzmann factor p_i / p_j (vertical axis) as a function of temperature T for several energy differences ε_i − ε_j.

Boltzmann factor p_i / p_j (vertical axis) as a function of temperature T for several energy differences ε_i − ε_j.

玻尔兹曼因子 p </sub > i </sub >/p </sub > j </sub > (垂直轴)作为温度 t 的函数，几个能量差异 ε </sub > i </sub >-ε </sub > j </sub > 。

玻尔兹曼因子 p_i&nbsp;/&nbsp;p_j (垂直轴) 作为温度 T 对于几个能量差异ε_i&nbsp;−&nbsp;ε_j的函数 。

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:

In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain 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) is a probability distribution or probability measure that gives the probability that a system will be in a certain 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)是一个概率分布或机率量测，它给出了一个系统处于某种状态的概率，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状态的概率，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  is the probability of the system being in state ,  is the energy of that state, and a constant  of the distribution is the product of Boltzmann's constant  and thermodynamic temperature . The symbol <math display="inline">\propto</math> denotes proportionality (see  for the proportionality constant).

where  is the probability of the system being in state ,  is the energy of that state, and a constant  of the distribution is the product of Boltzmann's constant  and thermodynamic temperature . The symbol <math display="inline">\propto</math> denotes proportionality (see  for the proportionality constant).

其中系统处于状态的概率，是该状态的能量，分布的一个常数是玻耳兹曼常数和热力学温度的乘积。符号 < math display = " inline" > propto </math > 表示相称性(比例常数见)。

{{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)。

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.

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.

其中 p < sub > i </sub > 是状态 i 的概率，ε < sub > i </sub > 是状态 i 的能量，k 是波兹曼常数，t 是系统的温度，m 是感兴趣的系统所能到达的所有状态的数目。

其中 p < sub > i 是状态 i 的概率，ε < sub > i 是状态 i 的能量，k 是波兹曼常数，t 是系统的温度，m 是感兴趣的系统所能到达的所有状态的数目。

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]]

where p_i is the probability of state i,  p_j the probability of state j, and ε_i and ε_j are the energies of states i and j, respectively.

where p_i is the probability of state i,  p_j the probability of state j, and ε_i and ε_j are the energies of states i and j, respectively.

其中 p < sub > i </sub > 是状态 i 的概率，p < sub > j </sub > 状态 j 的概率，ε </sub > i </sub > 和 ε </sub > j </sub > 分别是状态 i 和 j 的能量。

其中 p < sub > i 是状态 i 的概率，p < sub > j 状态 j 的概率，ε i 和 ε j 分别是状态 i 和 j 的能量。

The Boltzmann distribution is the distribution that maximizes the [[entropy]]

The Boltzmann distribution is the distribution that maximizes the [[entropy]]

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  In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not to be observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition.

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  In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not to be observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition.

其中 n < sub > i </sub > 是状态 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，就像我们看到的那样，等于处于 i 状态的粒子的比例。所以给出状态 i 的粒子比例，作为状态能量的函数的方程是，为了使这成为可能，必须有一些粒子处于第一个状态，才能发生跃迁。我们可以通过求第一态粒子的比例来满足这个条件。如果它可以忽略不计，那么在进行计算的温度下，极有可能不能观察到这种转变。一般来说，处于第一状态的分子比例越大，意味着向第二状态转变的次数越多。这就产生了一条更强的谱线。然而，还有其他因素影响谱线的强度，例如它是由允许的跃迁还是禁止的跃迁引起的。

其中 n < sub > i 是状态 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，就像我们看到的那样，等于处于 i 状态的粒子的比例。所以给出状态 i 的粒子比例，作为状态能量的函数的方程是，为了使这成为可能，必须有一些粒子处于第一个状态，才能发生跃迁。我们可以通过求第一态粒子的比例来满足这个条件。如果它可以忽略不计，那么在进行计算的温度下，极有可能不能观察到这种转变。一般来说，处于第一状态的分子比例越大，意味着向第二状态转变的次数越多。这就产生了一条更强的谱线。然而，还有其他因素影响谱线的强度，例如它是由允许的跃迁还是禁止的跃迁引起的。

The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database.<ref>[http://physics.nist.gov/PhysRefData/ASD/levels_form.html NIST Atomic Spectra Database Levels Form] at nist.gov</ref>

The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database.<ref>[http://physics.nist.gov/PhysRefData/ASD/levels_form.html NIST Atomic Spectra Database Levels Form] at nist.gov</ref>