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 Schrödinger 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 Csiszár [1].