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Display information for equation id:math.2508.17 on revision:2508

* Page found: Reale Gase (eq math.2508.17)

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Reale Gase Eq: math.2511.17

Hash: 177541c59d7c8aee21cfff609a3430fb

TeX (original user input):

\begin{align}

& {{Z}_{N}}=\frac{1}{N!}\Phi {{\left( \beta  \right)}^{N}}\left[ {{V}^{N}}+{{V}^{N-2}}\sum\limits_{i<j}^{{}}{{}}\int_{{}}^{{}}{{{d}^{3}}{{q}_{i}}}\int_{{}}^{{}}{{{d}^{3}}{{q}_{j}}}{{f}_{ij}}({{r}_{ij}})+..... \right] \\

& \int_{{}}^{{}}{{{d}^{3}}{{q}_{i}}}\int_{{}}^{{}}{{{d}^{3}}{{q}_{j}}}{{f}_{ij}}({{r}_{ij}})=\int_{V}^{{}}{{}}{{d}^{3}}q\int_{{}}^{{}}{{}}{{d}^{3}}rf(r) \\

& \Rightarrow {{Z}_{N}}=\frac{1}{N!}\Phi {{\left( \beta  \right)}^{N}}\left[ {{V}^{N}}+{{V}^{N-1}}\frac{N(N-1)}{2}\int_{{}}^{{}}{{}}{{d}^{3}}rf(r)+O\left( {{V}^{N-2}} \right) \right] \\

\end{align}

TeX (checked):

{\begin{aligned}&{{Z}_{N}}={\frac {1}{N!}}\Phi {{\left(\beta \right)}^{N}}\left[{{V}^{N}}+{{V}^{N-2}}\sum \limits _{i<j}^{}{}\int _{}^{}{{{d}^{3}}{{q}_{i}}}\int _{}^{}{{{d}^{3}}{{q}_{j}}}{{f}_{ij}}({{r}_{ij}})+.....\right]\\&\int _{}^{}{{{d}^{3}}{{q}_{i}}}\int _{}^{}{{{d}^{3}}{{q}_{j}}}{{f}_{ij}}({{r}_{ij}})=\int _{V}^{}{}{{d}^{3}}q\int _{}^{}{}{{d}^{3}}rf(r)\\&\Rightarrow {{Z}_{N}}={\frac {1}{N!}}\Phi {{\left(\beta \right)}^{N}}\left[{{V}^{N}}+{{V}^{N-1}}{\frac {N(N-1)}{2}}\int _{}^{}{}{{d}^{3}}rf(r)+O\left({{V}^{N-2}}\right)\right]\\\end{aligned}}

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\begin{aligned} Z_{N} = \frac{1}{N!} Φ {(β)}^{N} [V^{N} + V^{N - 2} \sum_{i < j}^{} \int_{}^{} d^{3} q_{i} \int_{}^{} d^{3} q_{j} f_{i j} (r_{i j}) + . . . . .] \\ \int_{}^{} d^{3} q_{i} \int_{}^{} d^{3} q_{j} f_{i j} (r_{i j}) = \int_{V}^{} d^{3} q \int_{}^{} d^{3} r f (r) \\ \Rightarrow Z_{N} = \frac{1}{N!} Φ {(β)}^{N} [V^{N} + V^{N - 1} \frac{N (N - 1)}{2} \int_{}^{} d^{3} r f (r) + O (V^{N - 2})] \end{aligned}

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Translations to Computer Algebra Systems

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In Mathematica:

Identifiers

$Z_{N}$

$N$

$Φ$

$β$

$N$

$V$

$N$

$V$

$N$

$i$

$j$

$q_{i}$

$q_{j}$

$f_{i j}$

$r_{i j}$

$q_{i}$

$q_{j}$

$f_{i j}$

$r_{i j}$

$V$

$q$

$r$

$f$

$r$

$Z_{N}$

$N$

$Φ$

$β$

$N$

$V$

$N$

$V$

$N$

$N$

$N$

$r$

$f$

$r$

$O$

$V$

$N$

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