Partition function (mathematics): Difference between revisions

Revision as of 11:18, 26 August 2018

Revision as of 22:24, 14 November 2018

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The partition function is commonly used as a [[generating function]] for [[expectation value]]s of various functions of the random variables. So, for example, taking <math>\beta</math> as an adjustable parameter, then the derivative of <math>\log(Z(\beta))</math> with respect to <math>\beta</math>

:<math>\boldmathbf{E}[H] = \langle H \rangle = -\frac {\partial \log(Z(\beta))} {\partial \beta}</math>

gives the average (expectation value) of ''H''. In physics, this would be called the average [[energy]] of the system.

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one then has

:<math>\boldmathbf{E}[x_k] = \langle x_k \rangle = \left.

\frac{\partial}{\partial J_k}

\log Z(\beta,J)\right|_{J=0}