Stochastic Process and a Generalized Fluctuation-Dissipation Theorem in Two-Dimensional Non-Linear Brownian Oscillator

Abstract

A two-dimensional (2D) non-linear Brownian oscillator in the potential; V ( x , y )= A 1 ( x + y )+ A 2 ( x 2 + y 2 )/2+ A 4 ( x 4 + y 4 )/4+ C x 2 y 2 /2, is numerically studied from the statistical mechanical viewpoint. We found that: (i) under large dissipation, the probability density function of the oscillator takes a form W s ( x , y )∝exp (- V ( x , y )/ T * ) although the coupling term C x 2 y 2 exists; (ii) the occurrence of jump events in a strongly dissipative case ( A 2 0) has a quasi-Poisson character even when the Kramer's theory of activation process is unavailable; (iii) as far as the Smoluchowski's approximation is valid, a generalized fluctuation-dissipation theorem (FDT) for the 2D non-linear Brownian oscillator can be constructed as a natural extension of Okabe's theory for Kubo's problem. System identification with use of the generalized FDT is also discussed.