Partition function (statistical mechanics): Difference between revisions

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This is the reason for calling ''Z'' the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. Other partition functions for different ensembles divide up the probabilities based on other macrostate variables. As an example: the partition function for the [[isothermal-isobaric ensemble]], the [[Boltzmann distribution#Generalized Boltzmann distribution|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 letter ''Z'' stands for the [[German language|German]] word ''Zustandssumme'', "sum over states". The usefulness of the partition function stems from the fact that it can be used to relatethe macroscopic [[thermodynamic state|thermodynamic quantities]] of a system can be related to theits microscopic details of a system through the derivatives of its partition function. Finding the partition function is also equivalent to performing a [[Laplace transform]] of the density of states function from the energy ___domain to the β ___domain, and the [[inverse Laplace transform]] of the partition function reclaims the state density function of energies.

==Grand canonical partition function==