Partition function (mathematics): Difference between revisions

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The role or meaning of the parameter <math>\beta</math> can be understood in a variety of different ways. In classical thermodynamics, it is an [[inverse temperature]]. More generally, one would say that it is the variable that is [[Conjugate variables (thermodynamics)|conjugate]] to some (arbitrary) function <math>H</math> of the random variables <math>X</math>. The word ''conjugate'' here is used in the sense of conjugate [[generalized coordinates]] in [[Lagrangian mechanics]], thus, properly <math>\beta</math> is a [[Lagrange multiplier]]. It is not uncommonly called the [[generalized force]]. All of these concepts have in common the idea that one value is meant to be kept fixed, as others, interconnected in some complicated way, are allowed to vary. In the current case, the value to be kept fixed is the [[expectation value]] of <math>H</math>, even as many different [[probability distribution]]s can give rise to exactly this same (fixed) value.

For the general case, one considers a set of functions <math>\{H_k(x_1,\cdots)\}</math> that each depend on the random variables <math>X_i</math>. These functions are chosen because one wants to hold their expectation values constant, for one reason or another. To constrain the expectation values in this way, one applies the method of [[Lagrange multiplier]]s. In the general case, [[maximum entropy method]]s illustrate the manner in which this is done.

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

Some specific examples are in order. In basic thermodynamics problems, when using the [[canonical ensemble]], the use of just one parameter <math>\beta</math> reflects the fact that there is only one expectation value that must be held constant: the [[free energy]] (due to [[conservation of energy]]). For chemistry problems involving chemical reactions, the [[grand canonical ensemble]] provides the appropriate foundation, and there are two Lagrange multipliers. One is to hold the energy constant, and another, the [[fugacity]], is to hold the particle count constant (as chemical reactions involve the recombination of atoms).

For the general case, one has

:<math>Z(\beta) = \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_kbeta) = \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]].

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with the angle brackets <math>\langle H_k \rangle</math> denoting the expected value of <math>H_k</math>; a precise definition of this expectation value is given below.

Although the value of <math>\beta</math> is commonly taken to be real, it need not be, in general; this is discussed in the section [[#Normalization|Normalization]] below. The values of <math>\beta</math> can be understood to be the coordinates of points in a space; this space is in fact a [[manifold]], as sketched below. The study of these spaces as manifolds constitutes the field of [[information geometry]].

== Symmetry ==