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* Page found: Das ideale Bosegas (eq math.2556.3)

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TeX (original user input): 

\begin{align}
  & P\left( {{N}_{1}},{{N}_{2}},... \right)={{Y}^{-1}}\exp \left( -\beta \left( {{N}_{j}}{{E}_{j}}-\mu {{N}_{j}} \right) \right)=\prod\limits_{j=1}^{l}{{}}\left( 1-{{t}_{j}} \right){{t}_{j}}^{{{N}_{j}}}=\prod\limits_{j=1}^{l}{{}}p\left( {{N}_{j}} \right) \\ 
 & (separiert) \\ 
 &  \\ 
 & p\left( {{N}_{j}} \right)=\left( 1-{{t}_{j}} \right){{t}_{j}}^{{{N}_{j}}}=\left( 1-\exp \left( \beta \left( \mu -{{E}_{j}} \right) \right) \right)\exp \left( -\beta \left( {{N}_{j}}{{E}_{j}}-\mu {{N}_{j}} \right) \right) \\ 
 & 1-\exp \left( \beta \left( \mu -{{E}_{j}} \right) \right):={{e}^{{{\Psi }_{j}}}} \\ 
 & p\left( {{N}_{j}} \right)={{e}^{{{\Psi }_{j}}}}\exp \left( -\beta \left( {{N}_{j}}{{E}_{j}}-\mu {{N}_{j}} \right) \right) \\ 
\end{align}

TeX (checked): 

{\begin{aligned}&P\left({{N}_{1}},{{N}_{2}},...\right)={{Y}^{-1}}\exp \left(-\beta \left({{N}_{j}}{{E}_{j}}-\mu {{N}_{j}}\right)\right)=\prod \limits _{j=1}^{l}{}\left(1-{{t}_{j}}\right){{t}_{j}}^{{N}_{j}}=\prod \limits _{j=1}^{l}{}p\left({{N}_{j}}\right)\\&(separiert)\\&\\&p\left({{N}_{j}}\right)=\left(1-{{t}_{j}}\right){{t}_{j}}^{{N}_{j}}=\left(1-\exp \left(\beta \left(\mu -{{E}_{j}}\right)\right)\right)\exp \left(-\beta \left({{N}_{j}}{{E}_{j}}-\mu {{N}_{j}}\right)\right)\\&1-\exp \left(\beta \left(\mu -{{E}_{j}}\right)\right):={{e}^{{\Psi }_{j}}}\\&p\left({{N}_{j}}\right)={{e}^{{\Psi }_{j}}}\exp \left(-\beta \left({{N}_{j}}{{E}_{j}}-\mu {{N}_{j}}\right)\right)\\\end{aligned}}

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MathML (62.469 KB / 6.159 KB) :

P ⁢ ( N 1 , N 2 , … ) = Y - 1 ⁢ exp ⁡ ( - β ⁢ ( N j ⁢ E j - μ ⁢ N j ) ) = ∏ j = 1 l ( 1 - t j ) ⁢ t j N j = ∏ j = 1 l p ⁢ ( N j ) ( s ⁢ e ⁢ p ⁢ a ⁢ r ⁢ i ⁢ e ⁢ r ⁢ t ) p ⁢ ( N j ) = ( 1 - t j ) ⁢ t j N j = ( 1 - exp ⁡ ( β ⁢ ( μ - E j ) ) ) ⁢ exp ⁡ ( - β ⁢ ( N j ⁢ E j - μ ⁢ N j ) ) 1 - exp ⁡ ( β ⁢ ( μ - E j ) ) := e Ψ j p ⁢ ( N j ) = e Ψ j ⁢ exp ⁡ ( - β ⁢ ( N j ⁢ E j - μ ⁢ N j ) ) missing-subexpression 𝑃 subscript 𝑁 1 subscript 𝑁 2 … superscript 𝑌 1 𝛽 subscript 𝑁 𝑗 subscript 𝐸 𝑗 𝜇 subscript 𝑁 𝑗 superscript subscript product 𝑗 1 𝑙 1 subscript 𝑡 𝑗 superscript subscript 𝑡 𝑗 subscript 𝑁 𝑗 superscript subscript product 𝑗 1 𝑙 𝑝 subscript 𝑁 𝑗 missing-subexpression 𝑠 𝑒 𝑝 𝑎 𝑟 𝑖 𝑒 𝑟 𝑡 missing-subexpression missing-subexpression missing-subexpression 𝑝 subscript 𝑁 𝑗 1 subscript 𝑡 𝑗 superscript subscript 𝑡 𝑗 subscript 𝑁 𝑗 1 𝛽 𝜇 subscript 𝐸 𝑗 𝛽 subscript 𝑁 𝑗 subscript 𝐸 𝑗 𝜇 subscript 𝑁 𝑗 missing-subexpression assign 1 𝛽 𝜇 subscript 𝐸 𝑗 superscript 𝑒 subscript Ψ 𝑗 missing-subexpression 𝑝 subscript 𝑁 𝑗 superscript 𝑒 subscript Ψ 𝑗 𝛽 subscript 𝑁 𝑗 subscript 𝐸 𝑗 𝜇 subscript 𝑁 𝑗 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle P\left({{N}_{1}},{{% N}_{2}},...\right)={{Y}^{-1}}\exp\left(-\beta\left({{N}_{j}}{{E}_{j}}-\mu{{N}_% {j}}\right)\right)=\prod\limits_{j=1}^{l}{{}}\left(1-{{t}_{j}}\right){{t}_{j}}% ^{{{N}_{j}}}=\prod\limits_{j=1}^{l}{{}}p\left({{N}_{j}}\right)\\ &\displaystyle(separiert)\\ &\\ &\displaystyle p\left({{N}_{j}}\right)=\left(1-{{t}_{j}}\right){{t}_{j}}^{{{N}% _{j}}}=\left(1-\exp\left(\beta\left(\mu-{{E}_{j}}\right)\right)\right)\exp% \left(-\beta\left({{N}_{j}}{{E}_{j}}-\mu{{N}_{j}}\right)\right)\\ &\displaystyle 1-\exp\left(\beta\left(\mu-{{E}_{j}}\right)\right):={{e}^{{{% \Psi}_{j}}}}\\ &\displaystyle p\left({{N}_{j}}\right)={{e}^{{{\Psi}_{j}}}}\exp\left(-\beta% \left({{N}_{j}}{{E}_{j}}-\mu{{N}_{j}}\right)\right)\\ \end{aligned}}}

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P(N1,N2,...)=Y−1exp⁡(−β(NjEj−μNj))=∏j=1l(1−tj)tjNj=∏j=1lp(Nj)(separiert)p(Nj)=(1−tj)tjNj=(1−exp⁡(β(μ−Ej)))exp⁡(−β(NjEj−μNj))1−exp⁡(β(μ−Ej)):=eΨjp(Nj)=eΨjexp⁡(−β(NjEj−μNj))

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