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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).

{{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</math>表示相称性(见附录中的比例常数)。

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 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" />

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" />