# Partition Function (statistical Mechanics) - Canonical Partition Function - Definition

Definition

As a beginning assumption, assume that a thermodynamically large system is in constant thermal contact with the environment, with a temperature T, and both the volume of the system and the number of constituent particles fixed. This kind of system is called a canonical ensemble. Let us label with s ( s = 1, 2, 3, ...) the exact states (microstates) that the system can occupy, and denote the total energy of the system when it is in microstate s as Es . Generally, these microstates can be regarded as analogous to discrete quantum states of the system.

The canonical partition function is

,

where the "inverse temperature", β, is conventionally defined as

with kB denoting Boltzmann's constant. The term exp(–β·Es) is known as the Boltzmann factor. In systems with multiple quantum states s sharing the same Es, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j ) as follows:

,

where gj is the degeneracy factor, or number of quantum states s which have the same energy level defined by Ej = Es .

The above treatment applies to quantum statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above. In classical statistical mechanics, it is not really correct to express the partition function as a sum of discrete terms, as we have done. In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In this case we must describe the partition function using an integral rather than a sum. For instance, the partition function of a gas of N identical classical particles is

$Z=frac{1}{N! h^{3N}} int , exp[-beta H(p_1 cdots p_N, x_1 cdots x_N)] ; mathrm{d}^3p_1 cdots mathrm{d}^3p_N , mathrm{d}^3x_1 cdots mathrm{d}^3x_N$

where

indicate particle momenta
indicate particle positions
is a shorthand notation serving as a reminder that the and are vectors in three dimensional space, and
is the classical Hamiltonian.

The reason for the factor is discussed below. For simplicity, we will use the discrete form of the partition function in this article. Our results will apply equally well to the continuous form. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. To make it into a dimensionless quantity, we must divide it by where is some quantity with units of action (usually taken to be Planck's constant).

In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis):

,

where is the quantum Hamiltonian operator. The exponential of an operator can be defined using the exponential power series. The classical form of is recovered when the trace is expressed in terms of coherent states and when quantum-mechanical uncertainties in the position and momentum of a particle are regarded as negligible. Formally, one inserts under the trace for each degree of freedom a resolution of the identity

$boldsymbol{1} = int |x,prangle,langle x,p|~frac{mathrm{d}x,mathrm{d}p}{h}$

where is a normalised Gaussian wavepacket centered at position and momentum . Thus,

$Z = int operatorname{tr} left( e^{-betahat{H}} |x,prangle,langle x,p| right) frac{mathrm{d}x,mathrm{d}p}{h} = intlangle x,p|e^{-betahat{H}}|x,prangle ~frac{mathrm{d}x,mathrm{d}p}{h}$

A coherent state is an approximate eigenstate of both operators and, hence also of the Hamiltonian, with errors of the size of the uncertainties. If and can be regarded as zero, the action of reduces to multiplication by the classical Hamiltonian, and reduces to the classical configuration integral.

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