Gibbs Canonical Ensemble
Consider an ensemble of systems described by a Hamiltonian H with average energy E. If H has pure-point spectrum and the eigenvalues of H go to + ∞ sufficiently fast, e-r H will be a non-negative trace-class operator for every positive r.
The Gibbs canonical ensemble is described by the state
Where β is such that the ensemble average of energy satisfies
and
This is called the partition function; it is the quantum mechanical version of the canonical partition function of classical statistical mechanics. The probability that a system chosen at random from the ensemble will be in a state corresponding to energy eigenvalue is
Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.
Read more about this topic: Quantum Statistical Mechanics
Famous quotes containing the word canonical:
“If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.”
—Thomas Hobbes (1579–1688)