大正準集団

${\displaystyle p(\omega )={\frac {1}{\Xi (\beta ,{\boldsymbol {\mu }})}}e^{-\beta E(\omega )+\beta \sum _{i}\mu _{i}N_{i}(\omega )}}$

${\displaystyle \Xi (\beta ,{\boldsymbol {\mu }})=\sum _{\omega }e^{-\beta E(\omega )+\beta \sum _{i}\mu _{i}N_{i}(\omega )}}$

${\displaystyle O(\beta ,{\boldsymbol {\mu }})=\langle O(\omega )\rangle =\sum _{\omega }O(\omega )p(\omega )={\frac {1}{\Xi (\beta ,{\boldsymbol {\mu }})}}\sum _{\omega }O(\omega )e^{-\beta E(\omega )+\beta \sum _{i}\mu _{i}N_{i}(\omega )}}$

${\displaystyle N_{i}(\beta ,{\boldsymbol {\mu }})={\frac {1}{\Xi (\beta ,{\boldsymbol {\mu }})}}\sum _{\omega }N_{i}(\omega )e^{-\beta E(\omega )+\beta \sum _{i}\mu _{i}N_{i}(\omega )}={\frac {1}{\beta }}{\frac {\partial }{\partial \mu _{i}}}\ln \Xi (\beta ,{\boldsymbol {\mu }})}$

${\displaystyle E(\beta ,{\boldsymbol {\mu }})={\frac {1}{\Xi (\beta ,{\boldsymbol {\mu }})}}\sum _{\omega }E(\omega )e^{-\beta E(\omega )+\beta \sum _{i}\mu _{i}N_{i}(\omega )}=-{\frac {\partial }{\partial \beta }}\ln \Xi (\beta ,{\boldsymbol {\mu }})+\sum _{i}{\frac {\mu _{i}}{\beta }}{\frac {\partial }{\partial \mu _{i}}}\ln \Xi (\beta ,{\boldsymbol {\mu }})}$

${\displaystyle J(\beta ,{\boldsymbol {\mu }})=-{\frac {1}{\beta }}\ln \Xi (\beta ,{\boldsymbol {\mu }})}$

${\displaystyle N(n_{1},\ldots ,n_{j},\ldots )=\sum _{j=1}^{\infty }n_{j},~E(n_{1},\ldots ,n_{j},\ldots )=\sum _{j=1}^{\infty }\epsilon _{j}n_{j}}$

${\displaystyle \Xi (\beta ,\mu )=\sum _{n_{1}}\cdots \sum _{n_{j}}\cdots \left(\prod _{j=1}^{\infty }\exp[-\beta (\epsilon _{j}-\mu )n_{j}]\right)=\prod _{j=1}^{\infty }\left(\sum _{n_{j}}\exp[-\beta (\epsilon _{j}-\mu )n_{j}]\right)=\prod _{j=1}^{\infty }\Xi ^{(j)}(\beta ,\mu )}$

${\displaystyle \Xi ^{(j)}(\beta ,\mu )=\sum _{n_{j}=0}^{\infty }e^{-\beta (\epsilon _{j}-\mu )n_{j}}={\frac {1}{1-e^{-\beta (\epsilon _{j}-\mu )}}}}$

${\displaystyle \langle n_{j}\rangle ={\frac {1}{\beta }}{\frac {\partial }{\partial \mu }}\ln \Xi ^{(j)}(\beta ,\mu )=-{\frac {1}{\beta }}{\frac {\partial }{\partial \mu }}\ln(1-e^{-\beta (\epsilon _{j}-\mu )})={\frac {1}{e^{\beta (\epsilon _{j}-\mu )}-1}}}$

${\displaystyle \Xi ^{(j)}(\beta ,\mu )=\sum _{n_{j}=0}^{1}e^{-\beta (\epsilon _{j}-\mu )n_{j}}=1+e^{-\beta (\epsilon _{j}-\mu )}}$

${\displaystyle \langle n_{j}\rangle ={\frac {1}{\beta }}{\frac {\partial }{\partial \mu }}\ln \Xi ^{(j)}(\beta ,\mu )={\frac {1}{\beta }}{\frac {\partial }{\partial \mu }}\ln(1+e^{-\beta (\epsilon _{j}-\mu )})={\frac {1}{e^{\beta (\epsilon _{j}-\mu )}+1}}}$

${\displaystyle \mathbf {Tr} [{\hat {A}}]=\sum _{i}\langle e_{i}|{\hat {A}}|e_{i}\rangle }$

${\displaystyle \Xi (\beta ,\mu )=\mathbf {Tr} [e^{-\beta ({\hat {H}}-\mu {\hat {N}})}]}$