Characteristic state function
Encyclopedia
The characteristic state function in statistical mechanics
refers to a particular relationship between the partition function
of an ensemble
.
In particular, if the partition function P satisfies
or 
in which Q is a thermodynamic quantity, then Q is known as the "characteristic state function" of the ensemble corresponding to "P". Beta refers to the thermodynamic beta.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
refers to a particular relationship between the partition function
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
of an ensemble
Statistical ensemble (mathematical physics)
In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1878, an ensemble is an idealization consisting of a large number of mental copies of a system, considered all at once, each of which represents a possible state that the real...
.
In particular, if the partition function P satisfies
or in which Q is a thermodynamic quantity, then Q is known as the "characteristic state function" of the ensemble corresponding to "P". Beta refers to the thermodynamic beta.
Examples
- The microcanonical ensembleMicrocanonical ensembleIn statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
satisfies
hence, its characteristic state function is 
This quantity roughly speaking, denotes the energy of the entropy at a particular temperature. - The canonical ensembleCanonical ensembleThe canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
satisfies
hence, its characteristic state function is the Helmholtz free energyHelmholtz free energyIn thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...
. - The grand canonical ensembleGrand canonical ensembleIn statistical mechanics, a grand canonical ensemble is a theoretical collection of model systems put together to mirror the calculated probability distribution of microscopic states of a given physical system which is being maintained in a given macroscopic state...
satisfies
, so its characteristic state function is the total Pressure-volume work. - The isothermal-isobaric ensembleIsothermal-isobaric ensembleThe isothermal–isobaric ensemble is a statistical mechanical ensemble that maintains constant temperature T \, and constant pressure P \, applied. It is also called the NpT-ensemble, where the number of particles N \, is also kept as a constant...
satisfies
so its characteristic function is the Gibbs free energyGibbs free energyIn thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
.