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This is almost the partition function for the -ensemble, but it has units of volume, an unavoidable consequence of taking the above sum over volumes into an integral. Restoring the constant C {\displaystyle C} yields the proper result for Δ ( N , P , T ) {\displaystyle \Delta (N,P,T)} .
Consider the sum = = where >0 for all N.Since all the terms are positive, the value of S must be greater than the value of the largest term, , and less than the product of the number of terms and the value of the largest term.
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
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.
The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., = (+ +). This special case of SSP is known as the partition problem . SSP can also be regarded as an optimization problem : find a subset whose sum is at most T , and subject to that, as close as possible to T .
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then partition function can be understood to be a sum or integral over Gaussians. The correlation function C ( x j , x k ) {\displaystyle C(x_{j},x_{k})} can be understood to be the Green's function for the differential operator (and generally giving rise to Fredholm theory ).
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