Loewner time conversion for q-generalized stochastic dynamics

Abstract

Generalized statistical mechanics based on q-Gaussian has been demonstrated to be an effective theoretical framework for the analysis of non-equilibrium systems. Since q-generalized (non-extensive) statistical mechanics reduces the nonlinearity in the system into deformed entropy and probability distributions, we introduce an alternative method based on the time conversion method using the Loewner equation by investigating the statistical physical properties of one-dimensional stochastic dynamics described by the Langevin equation with multiplicative noise. We demonstrate that a randomized time transformation using Loewner time enables the conversion of the multiplicative Langevin dynamics into an equilibrium system obeying a conventional microcanonical ensemble. For the equilibrium Langevin system after the Loewner time conversion, the fluctuation-dissipation relation and path integral fluctuation theorem were discussed to derive the response function under a nonlinear perturbation and an extended Jarzynski equality. The present results suggest the efficacy of the introducing randomized time for analyzing non-equilibrium systems, and indicate a novel connection between q-generalized (non-extensive) and Boltzmann–Gibbs statistical mechanics.