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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
Two important examples are the partitions restricted to only odd integer parts or only even integer parts, with the corresponding partition functions often denoted () and (). A theorem from Euler shows that the number of strict partitions is equal to the number of partitions with only odd parts: for all n , q ( n ) = p o ( n ) {\displaystyle q ...
The partition function is commonly used as a probability-generating function for expectation values of various functions of the random variables. So, for example, taking β {\displaystyle \beta } as an adjustable parameter, then the derivative of log ( Z ( β ) ) {\displaystyle \log(Z(\beta ))} with respect to β {\displaystyle \beta }
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 ]
The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part. The number of partitions of n is given by the partition ...
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 general, there are two theoretical approaches to determining the statistical mechanical properties of materials. The first is the direct calculation of the overall partition function of the system, which directly yields the system free energy. The second approach, known as the Widom insertion method, instead derives from calculations ...
The normalization condition that the trace of be equal to 1 defines the partition function to be () = (). If the number of particles involved in the system is itself not certain, then a grand canonical ensemble can be applied, where the states summed over to make the density matrix are drawn from a Fock space .