Time evolution towards q-Gaussian stationary states through unified Ito-Stratonovich stochastic equation

TL;DR

This paper studies how a unified stochastic equation with a parameter  can describe the evolution of systems towards q-Gaussian stationary states, analyzing the role of noise and external fields in shaping the distribution.

Contribution

It introduces a unified framework combining Itf4 and Stratonovich approaches to derive a linear Fokker-Planck equation with q-Gaussian stationary solutions, and explores their properties.

Findings

Stationary states are q-Gaussians with q depending on noise and external field parameters.

Time evolution of kurtosis measures helps track convergence to stationary states.

Calibration methods for q using kurtosis are proposed for experimental and numerical data.

Abstract

We consider a class of single-particle one-dimensional stochastic equations which include external field, additive and multiplicative noises. We use a parameter $\theta \in [0,1]$ which enables the unification of the traditional It\^o and Stratonovich approaches, now recovered respectively as the $\theta=0$ and $\theta=1/2$ particular cases to derive the associated Fokker-Planck equation (FPE). These FPE is a {\it linear} one, and its stationary state is given by a $q$-Gaussian distribution with $q = \frac{\tau + 2M (2 - \theta)}{\tau + 2M (1 - \theta)}<3$, where $\tau \ge 0$ characterizes the strength of the confining external field, and $M \ge 0$ is the (normalized) amplitude of the multiplicative noise. We also calculate the standard kurtosis $\kappa_1$ and the $q$-generalized kurtosis $\kappa_q$ (i.e., the standard kurtosis but using the escort distribution instead of the direct one). Through these two quantities we numerically follow the time evolution of the distributions. Finally, we exhibit how these quantities can be used as convenient calibrations for determining the index $q$ from numerical data obtained through experiments, observations or numerical computations.