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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; see partition function (mathematics) for
These three equations, along with the free energy in terms of the partition function, = , allow an efficient way of calculating thermodynamic variables of interest given the partition function and are often used in density of state calculations. One can also do Legendre transformations for different systems. For example, for a system with a ...
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution.
The generalized version of the partition function provides the complete framework for working with ensemble averages in thermodynamics, information theory, statistical mechanics and quantum mechanics. The microcanonical ensemble represents an isolated system in which energy (E), volume (V) and the number of particles (N) are all constant.
The total canonical partition function of a system of identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions : [1] =! with: = /, where is the degeneracy of the jth quantum level of an individual particle, is the Boltzmann constant, and is the absolute temperature of system.
In statistical mechanics, the translational partition function, is that part of the partition function resulting from the movement (translation) of the center of mass. For a single atom or molecule in a low pressure gas, neglecting the interactions of molecules , the canonical ensemble q T {\displaystyle q_{T}} can be approximated by: [ 1 ]
What has been presented above is essentially a derivation of the canonical partition function. As one can see by comparing the definitions, the Boltzmann sum over states is equal to the canonical partition function. Exactly the same approach can be used to derive Fermi–Dirac and Bose–Einstein statistics.
In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the Kirkwood's coupling parameter method. [1]