User:Egm6321.f10/configuration integral

by

Prof. Loc Vu-Quoc

Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28 and also Partition function (statistical mechanics).

this page has been under reconstruction below ....

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The classical configuration integral, sometimes referred to as the configurational partition function , for a system with $$\displaystyle N$$ particles is defined as follows:
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$$  \displaystyle Z_N :=  \int\limits_V \exp \left[ - \beta U (x_1, \cdots , x_N) \right] d^3 x_1 \cdots d^3 x_N $$ where $$\displaystyle V$$ is the volume enclosing the $$\displaystyle N$$ particles, $$\displaystyle \beta$$ a constant defined as
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$$  \displaystyle \beta :=  \frac {1}  {k_B T} $$ with $$\displaystyle k_B$$ being the Boltzmann constant,
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$$\displaystyle T$$ the

thermodynamic temperature $$\displaystyle U$$ the potential energy of interparticle forces, $$\displaystyle \{ x_1, \cdots , x_N \}$$ the positions in the 3-D space $$\displaystyle \mathbb R ^3$$ of the $$\displaystyle N$$ particles, with $$\displaystyle x_i = (x_i^1, x_i^2 , x_i^3)$$ and $$\displaystyle x_i^j$$ the $$\displaystyle jth$$ coordinate of the $$\displaystyle ith$$ particle, and $$\displaystyle d^3 x_i = d x_i^1 d x_i^2 d x_i^3$$ an infinitesimal volume. An example for the potential energy $$\displaystyle U$$ is the Lennard-Jones potential.