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( Partition function (statistical mechanics)) 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 functions, 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.

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

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