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As an example: the partition function for the isothermal-isobaric ensemble, the generalized Boltzmann distribution, divides up probabilities based on particle number, pressure, and temperature. The energy is replaced by the characteristic potential of that ensemble, the Gibbs Free Energy .
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 other words, each single-particle level is a separate, tiny grand canonical ensemble. By the Pauli exclusion principle, there are only two possible microstates for the single-particle level: no particle (energy E = 0), or one particle (energy E = ε). The resulting partition function for that single-particle level therefore has just two terms:
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 ]
For partition functions with Grassmann valued fermion fields, the sources are also Grassmann valued. [7] For example, a theory with a single Dirac fermion requires the introduction of two Grassmann currents and ¯ so that the partition function is
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.
That is, the number of particles within the overall system that occupy a given single particle state form a sub-ensemble that is also grand canonical ensemble; hence, it may be analysed through the construction of a grand partition function. Every single-particle state is of a fixed energy, .