Partition function (mathematics): Difference between revisions

Revision as of 19:53, 10 July 2012

Revision as of 19:59, 10 July 2012

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:<math>P(x_1,x_2,\dots) = \frac{1}{Z(\beta)} \exp \left(-\beta H(x_1,x_2,\dots) \right)</math>

is the [[probability density function|probability]] of the configuration <math>(x_1,x_2,\dots)</math> occurring in the system, which is now properly normalized so that <math>0\le P(x_1,x_2,\dots)\le 1</math>, and such that the sum over all configurations totals to one. As such, the partition function can be understood to provide a [[measure (mathematics)|measure]] (a [[probability measure]]) on the [[phaseprobability space|space of states]]; formally, it is sometimes called the [[Gibbs measure]]. MoreIt narrowly,generalizes itthe isnarrower knownconcepts of as the [[grand canonical ensemble]] and [[canonical ensemble]] in statistical mechanics.

There exists at least one configuration <math>(x_1,x_2,\dots)</math> for which the probability is maximized; this configuration is conventionally called the [[ground state]]. If the configuration is unique, the ground state is said to be '''non-degenerate''', and the system is said to be [[ergodic]]; otherwise the ground state is '''degenerate'''. The ground state may or may not commute with the generators of the symmetry; if commutes, it is said to be an [[invariant measure]]. When it does not commute, the symmetry is said to be [[spontaneously broken]].