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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<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:

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:

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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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

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 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。

 

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]]''。''

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

分布表明，能量较低的状态被占据的概率总是高于能量较高的状态。它还可以给出两个状态被占领概率之间的定量关系。给出了状态 i 和状态 j 的概率比为

玻尔兹曼分布表明，能量较低的状态被占据的概率总是高于能量较高的状态被占据的概率。玻尔兹曼分布还给出两个量子态被占据的概率之间的定量关系。状态 i 和状态 j 的概率比为

:<math>

:<math>

{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}

{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}

{ 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>

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.

 

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 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。

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。

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

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

The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over energy states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state i is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state i. This probability is equal to the number of particles in state i divided by the total number of particles in the system, that is the fraction of particles that occupy state i.

The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over energy states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state i is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state i. This probability is equal to the number of particles in state i divided by the total number of particles in the system, that is the fraction of particles that occupy state i.

波兹曼分布通常被用来描述粒子，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，粒子处于状态 i 的概率实际上等于，如果我们从该系统中选择一个随机粒子并检查它处于什么状态，我们会发现它处于状态 i 的概率。这个概率等于状态 i 的粒子数除以系统中粒子的总数，即占据状态 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。

:<math>H(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i</math>

:<math>H(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i</math>

</math> equals a particular mean energy value (which can be proven using [[Lagrange multipliers]]).

</math> equals a particular mean energy value (which can be proven using [[Lagrange multipliers]]).

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 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。

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。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。

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>