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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>
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 p_i \log p_i</math>))与熵的热力学定义(<math>d S = \frac{\delta Q_\text{rev}}{T}</math>)等价的充要条件。<ref name=":0" />
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 p_i \log p_i</math>)与熵的热力学定义(<math>d S = \frac{\delta Q_\text{rev}}{T}</math>)等价的充要条件。<ref name=":0" />
subject to the constraint that <math>\sum {p_i {\varepsilon}_i} </math> equals a particular mean energy value (which can be proven using [[Lagrange multipliers]]).
当 ΣΡ<sub>i</sub>ε<sub>i</sub>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。
当<math>\sum {p_i {\varepsilon}_i} </math>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。
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 name=":2">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 name=":3">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 name=":2">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 name=":3">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]]
这个等式对于光谱学来说很重要。在光谱中，我们期待能观察到一条原子或分子跃迁的谱线。<ref name="Atkins, P. W. 2010" /><ref name=":2" /> 在为了使这成为可能，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。<ref name=":3" />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.