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

Graphical abstract: Slowly modulated periodic potentials corresponding to trap, accordion and barrier models, from top to bottom. The diffusion in such potentials on large scales needs for using Ito, Stratonovich and Haenggi interpretation rules, respectively, when described within the Langevin scheme. - 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 Haenggi-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.