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Partition Function

The partition function of a system with states s, energies E_s, and parameter beta is

Z(beta)=sum_(s)e^(-betaE_s).

It normalizes the weights into probabilities p_s=e^(-betaE_s)/Z(beta) and generates thermodynamic quantities through derivatives of lnZ.

In a continuous or field-theoretic system, the sum is replaced formally by a functional integral. In Euclidean quantum field theory, for example,

Z=inte^(-S_E[phi])Dphi,

where S_E is the Euclidean action. This statistical-mechanical partition function is distinct from the number-theoretic partition functions that count integer partitions.

See also

Euclidean Action, Euclidean Quantum Field Theory, Functional Integral, Partition, Partition Function P, Random-Cluster Model

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References

Baxter, R. J. Exactly Solved Models in Statistical Mechanics. New York: Academic Press, 1982.Glimm, J. and Jaffe, A. Quantum Physics: A Functional Integral Point of View, 2nd ed. New York: Springer-Verlag, 1987.

Cite this as:

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

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