Microcanonical Distributions, Gibbs States, and the Equivalence of Ensembles

Abstract

Consider an ideal gas of n identical particles which have achieved equilibrium subject only to the constraint that their average energy is some specified constant. For the purposes of this discussion, we will interpret this sentence as saying that we have

• i) a Polish space E (the phase space of the individual particles),

• ii) a σ-finite measure λ (the Liouville measure for the dynamical system governing the motion of the individual particles) on (E,B_E) (B_E is the Borel field over E),

• iii) a function U:E → [0, ∞) (the energy),

and that we are looking at is a regular version A_n of the conditional probability under λⁿ given that $\frac{1}{n}\sum\nolimits_{m = 1}^n U \left( {{x_m}} \right) = 1$.