Partition function (mathematics): Difference between revisions

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Practically speaking, it can be understood by examining the derivation of the partition function using [[maximum entropy method]]s. Here, the parameter appears as a [[Lagrange multiplier]]; the multiplier is used to guarantee that the [[expected value]] of some quantity is preserved by the distribution of probabilities. Thus, in chemistry problems, the use of just one parameter <math>\beta</math> reflects the fact that there is only one expectation value that must be held constant: this is the energy. For the [[grand canonical ensemble]], there are two Lagrange multipliers: one to hold the energy constant, and another (the [[fugacity]]) to hold the particle count constant. In the general case, there are a set of parameters taking the place of <math>\beta</math>, one for each constraint enforced by the multiplier. Thus, for the general case, one has

:<math>Z(\beta_kbeta) = \sum_{x_i} \exp \left(-\sum_k\beta_k H_k(x_i) \right)</math>

with <math>\beta=(\beta_1, \beta_2,\cdots)</math> a point in a space. This space can be understood as a [[manifold]], explained below.

For a collection of observables <math>H_k</math>, one would write

:<math>Z(\beta_k) = \mbox{tr} \exp \left(-\sum_k\beta_k H_k\right) </math>

As before, it is presumed that the argument of tr is [[trace class]].

The corresponding [[Gibbs measure]] then provides a probability distribution such that the expectation value of each <math>H_k</math> is a fixed value. More precisely, one has