Brownian Motion with Restoring Drift: Micro-canonical Ensemble and the Thermodynamic Limit

Abstract

We take up the old problem of micro-canonical conditioning in the context of diffusion. Starting with a potential \(\), the Schrodinger operator \(\) with ground state \(\) is carried by a conjugation into the diffusion generator \(\) with invariant density \(\). The latter motion \(\) is made micro-canonical by first conditioning the path to be periodic, \(\), and then further conditioning on the empirical mean-square or ``particle number'' \(\). The thermodynamics are then studied by taking \(\) while D remains fixed. The problem in this form owes its inception to McKean-Vaninsky \cite{MV2} who obtained the following result. For \(\) with \(\), they showed the same type of diffusion appears in the thermodynamic limit, but with drift arising from the shifted potential \(\) being such that the limiting mean-square equals D. Their method of proof predicts the same outcome for \(\), so long as D is smaller than the canonical mean-square \(\), while if \(\), the matter was unresolved. The purpose of this note is to show a type of phase transition takes place in this case: the conditioning is overcome in the limit and one sees the original (stationary) diffusion on the line. The proof employs an entropy inequality due to Csiszar [1].