Time-average for physical quantities in infinite classical systems

Abstract

Existence of a time-average for space-translation invariant physical quantities (pq) is proven. Pq are defined [1] as some compatible and measurable systems of functions on the phase space X . We consider a topological space Y built over the phase space so that pq may be regarded as real Borel functions on Y . The dynamics on X which satisfies the so called “particle” condition may be transferred on Y . Conditions equivalent to measurability of the transferred dynamics are studied. A σ-finite Borel measure v is defined on the quotient space Y ⧸τ of Y modulo the space-translation group τ. It is invariant with respect to transferred dynamics on Y ⧸τ, if the Gibbs measure is invariant with respect to the initial dynamics on X and if the probability of “the existence of infinitely long particle path, at finite time, is zero”. Using Birkhoff Ergodic Theorem on the quotient space Y ⧸τ with measure v, we obtain the existence of time-average for τ-invariant pq. If the transferred dynamics is ergodic on n-particle level Y n ⧸τ, where n ⩾ 2, then the time-average for n-pq equals zero.