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Statistical-Mechanical Partition Function

A statistical-mechanical partition function is the normalizing integral of an exponential energy weight against a specified reference measure. For energy H, inverse temperature beta, and reference measure nu, it is

Z(beta)=inte^(-betaH(sigma))dnu(sigma).

When this integral is finite and positive, dividing the integrand by Z gives a Gibbs measure. For binary spins, counting measure and uniform probability give partition functions differing by a factor of 2^N. At zero energy they give 2^N and 1, respectively. The logarithm divided by N is thermodynamic pressure. The reference measure is therefore essential when comparing formulas.

See also

Canonical Free Energy, Gibbs Measure, Pair Potential, Thermodynamic Pressure

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References

Friedli, S. and Velenik, Y. Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge, England: Cambridge University Press, 2017. https://doi.org/10.1017/9781316882603.

Cite this as:

Weisstein, Eric W. "Statistical-Mechanical Partition Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Statistical-MechanicalPartitionFunction.html

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