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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 ...
The canonical ensemble is the only ensemble with constant N, V, and T that reproduces the fundamental thermodynamic relation. [9] Statistical equilibrium (steady state): A canonical ensemble does not evolve over time, despite the fact that the underlying system is in constant motion. This is because the ensemble is only a function of a ...
As such, the partition function can be understood to provide a measure (a probability measure) on the probability space; formally, it is called the Gibbs measure. It generalizes the narrower concepts of the grand canonical ensemble and canonical ensemble in statistical mechanics. There exists at least one configuration (,, …
Canonical ensemble (or NVT ensemble)—a statistical ensemble where the energy is not known exactly but the number of particles is fixed. In place of the energy, the temperature is specified. The canonical ensemble is appropriate for describing a closed system which is in, or has been in, weak thermal contact with a heat bath. In order to be in ...
The grand canonical ensemble is the only ensemble with constant , V, and T that reproduces the fundamental thermodynamic relation. [5] Statistical equilibrium (steady state): A grand canonical ensemble does not evolve over time, despite the fact that the underlying system is in constant motion. Indeed, the ensemble is only a function of the ...
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 partition function for the -ensemble can be derived from statistical mechanics by beginning with a system of identical atoms described by a Hamiltonian of the form / + and contained within a box of volume =.
Constructing a density matrix using a canonical ensemble gives a result of the form = / (), where is the inverse temperature () and is the system's Hamiltonian. The normalization condition that the trace of ρ {\displaystyle \rho } be equal to 1 defines the partition function to be Z ( β ) = t r exp ( − β H ) {\displaystyle Z(\beta ...