Probabilistic Formulation of Classical Mechanics

Abstract

Starting axiomatically with a system of finite degrees of freedom whose logic ℒ _c is an atomic Boolean σ-algebra, we prove the existence of phase space Ω _c , as a separable metric space, and a natural (weak) topology on the set of states <Emphasis FontCategory=“NonProportional”>S</Emphasis> (all the probability measures on ℒ _c ) such that Ω _c , the subspace of pure states <Emphasis FontCategory=“NonProportional”>P</Emphasis>, the set of atoms of ℒ _c and the space <Emphasis FontCategory=“NonProportional”>P</Emphasis>(Ω _c ) of all the atomic measures on Ω _c , are all homeomorphic. The only physically accessible states are the points of Ω _c . This probabilistic formulation is shown to be reducible to a purely deterministic theory.