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{{Use American English|date = March 2019}}
{{Use American English|date = March 2019}}
It results from the constraint that the probabilities of all accessible states must add up to 1.
所有量子态的概率之和为1.
The Boltzmann distribution is the distribution that maximizes the [[entropy]]，
<math>H(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i</math>
<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]]).
<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>
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 name=":1">[http://physics.nist.gov/PhysRefData/ASD/levels_form.html NIST Atomic Spectra Database Levels Form] at nist.gov</ref>
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" />
</math>
</math>
{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}
<nowiki>{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}</nowiki>
<nowiki>{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}</nowiki>
<nowiki>{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}</nowiki>
</math>
<nowiki></math></nowiki>
数学
数学
where p<sub>i</sub> is the probability of state i, p<sub>j</sub> the probability of state j, and ε<sub>i</sub> and ε<sub>j</sub> are the energies of states i and j, respectively.
where p<sub>i</sub> is the probability of state i, p<sub>j</sub> the probability of state j, and ε<sub>i</sub> and ε<sub>j</sub> are the energies of states i and j, respectively.
其中 p<sub>i</sub> 是量子态 i 的概率， p<sub>j</sub> 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<sub>i</sub> 是量子态 i 的概率， p<sub>j</sub> 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。
<nowiki><math display="inline"></nowiki>
<nowiki>p_i={\frac{N_i}{N}}</nowiki>
<math>
p_i={\frac{N_i}{N}}
P _ i = { frac { n _ i }{ n }
P _ i = { frac { n _ i }{ n }
{\sum{p_i {\varepsilon}_i}}
<nowiki>{\sum{p_i {\varepsilon}_i}}</nowiki>
</math> equals a particular mean energy value (which can be proven using [[Lagrange multipliers]]).
where N<sub>i</sub> 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<sub>i</sub> 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.
N<sub>i</sub> 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。为了使这成为可能，必须有一些粒子处于第一个量子态，等待着发生跃迁。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。
N<sub>i</sub> 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。为了使这成为可能，必须有一些粒子处于第一个量子态，等待着发生跃迁。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，那么在当前计算的温度下，则很难观察到这种跃迁。一般来说，处于第一状态的分子比例越大，意味着向第二状态跃迁的次数越多。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，例如它是否是被允许的跃迁还是被禁止的跃迁。
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归一化指数函数（Softmax 函数）有关。
The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states ''i'' and ''j'' is given as
The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states ''i'' and ''j'' is given as
玻尔兹曼分布指出，能量低的量子态总是有比能量高的量子态更高的概率被粒子占据。它同时也能让我们定量地比较两个量子态分布概率的关系。
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:
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:
Canonical ensemble (general case)
'''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。
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)'''
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 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 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。
Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles)
'''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 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。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。这个预期的数分布具有玻耳兹曼形式。
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。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。
Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:
Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:
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 ensembleis obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a microcanonical ensemble applies instead.
* 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. 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 in thermal equilibrium.
* With ''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 fermions or bosons, respectively.
This equation is of great importance to [[spectroscopy]]. In spectroscopy we observe a [[spectral line]] of atoms or molecules that we are interested in going from one state to another.<ref name="Atkins, P. W. 2010"/><ref>Atkins, P. W.; de Paula J. (2009) Physical Chemistry, 9th edition, Oxford University Press, Oxford, UK</ref> 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.<ref>Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006) Principles of Instrumental Analysis, Brooks/Cole, Boston, MA</ref> 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]].
This equation is of great importance to [[spectroscopy]]. In spectroscopy we observe a [[spectral line]] of atoms or molecules that we are interested in going from one state to another.<ref name="Atkins, P. W. 2010"/><ref>Atkins, P. W.; de Paula J. (2009) Physical Chemistry, 9th edition, Oxford University Press, Oxford, UK</ref> 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.<ref>Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006) Principles of Instrumental Analysis, Brooks/Cole, Boston, MA</ref> 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]].