Ito, Stratonovich, Hänggi and all the rest: The thermodynamics of interpretation

Abstract

To elucidate a question of the necessity and thermodynamical meaning of the interpretation of Langevin’s equations with multiplicative nose, we considered a simple model of particle diffusion in a slowly modulated periodic potential in the low-temperature limit. Our main point is to clarify, what kind of interpretation is appropriate to describe the situation under different types of modulation within the Langevin scheme. The existence of a free interpretation parameter α is a necessity connected with the fact, that the diffusion coefficient D is an amalgam of two kinetic parameters, the period length and the sojourn time in a potential well, which in their turn depend on the microscopic parameters of the model. The results show that the Ito and the Hänggi–Klimontovich interpretations correspond to a trap and barrier models respectively, and that the Fisk–Stratonovich one corresponds to a modulation of the potential’s period. All other situations are more complex and may correspond to position-dependent interpretation parameter.