Partition function (statistical mechanics) Guide, Meaning , Facts, Information and Description

In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas. Most of the thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives.

There are actually several different types of partition function, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy.) The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances.

Table of contents

1 Canonical partition function 1.1 Definition 1.2 Meaning and significance 1.3 Calculating the thermodynamic total energy 1.4 Relation to thermodynamic variables 1.5 Partition functions of subsystems 2 Grand canonical partition function 2.6 Definition

Canonical partition function

Definition

Suppose we have a thermodynamically large system that is in constant thermal contact with the environment, which has temperature T, with 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 the exact states (microstates) that the system can occupy by j (j = 1, 2, 3, ...), and denote the total energy of the system when it is in microstate j as E_j. Generally, these microstates can be regarded as discrete quantum states of the system.

The canonical partition function is

where the "inverse temperature" β is conventionally defined as

with k_B denoting Boltzmann's constant.

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, some form of coarse graining procedure must be carried out, which essentially amounts to treating two mechanical states as the same microstate if the differences in their position and momentum variables are "not too large". The partition function then takes the form of an integral. For instance, the partition function of a gas of N classical particles is

where h is some infinitesimal quantity with units of action (usually taken to be Planck's constant, in order to be consistent with quantum mechanics), and H is the classical Hamiltonian. The reason for the N! factor is discussed below. For simplicity, we will use the discrete form of the partition function in this article, but our results will apply equally well to the continuous form.

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 H is the quantum Hamiltonian operator. Note that the exponential of an operator is defined using the usual power series.

Meaning and significance

It may not be obvious why the partition function, as we have defined it above, is an important quantity. Firstly, let us consider what goes into it. The partition function is a function of, firstly, the temperature T; and, secondly, the microstate energies E₁, E₂, E₃, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.

The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability P_j that the system occupies microstate j is

This is the well-known Boltzmann factor. (For a detailed derivation of this result, see Derivation of the partition function.) The partition function thus plays the role of a normalizing constant (note that it does not depend on j), ensuring that the probabilities add up to one:

This is the reason for calling Z the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. The letter Z stands for the German word Zustandsumme, "sum over states".

Calculating the thermodynamic total energy

In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the expected value for the energy, which is the sum of the microstate energies weighted by their probabilities:

or, equivalently,

Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner

then the expected value of A is

This provides us with a trick for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set λ to zero in the final expression. This is analogous to the source field method used in the path integral formulation of quantum field theory.

Relation to thermodynamic variables

In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.

As we have already seen, the thermodynamic energy is

The variance in the energy (or "energy fluctuation") is

The heat capacity is

The entropy is

The above expression allows us to identify the Helmholtz free energy F (defined as F = E - TS, where E is the total energy and S is the entropy) as

Partition functions of subsystems

Suppose a system is subdivided into N sub-systems with negligible interaction energy. If the partition functions of the sub-systems are ζ₁, ζ₂, ... ζ_N, then the partition function of the entire system is the product of the individual partition functions:

If the sub-systems have the same physical properties, then their partition functions are equal, ζ₁ = ζ₂ = ... = ζ, in which case

However, there is a well-known exception to this rule. If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a N! (N factorial):

This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the Gibbs paradox.

Grand canonical partition function

Definition

We can similarly define a grand canonical partition function for a grand canonical ensemble, a system that can exchange both heat and particles with the environment, which has a constant temperature T and chemical potential μ. The volume of the system is fixed. For the moment, we will assume there is only one species of particle present, and denote the number of particles in the system by N. Once again, we denote the energy of microstate j as E_j. We also denote the number of particles present in microstate j as N_j.

The grand canonical partition function is

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