Partition function (mathematics): Difference between revisions

Revision as of 19:20, 10 July 2012

Revision as of 19:21, 10 July 2012

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:<math>Z(\beta) = \sum_{x_i} \exp \left(-\beta H(x_1,x_2,\dots) \right)</math>

The function ''H'' is understood to be a real-valued function on the space of states <math>\{X_1,X_2,\cdots\}</math>, while <math>\beta</math> is a real-valued free parameter (conventionally, the [[inverse temperature]]). The sum over the <math>x_i</math> is understood to be a sum over all possible values that each of the random variablevariables <math>X_i</math> may take. Thus, the sum is to be replaced by an [[integral]] when the <math>X_i</math> are continuous, rather than discrete. Thus, one writes

:<math>Z(\beta) = \int \exp \left(-\beta H(x_1,x_2,\dots) \right) dx_1 dx_2 \cdots</math>