Signature of nonlinear damping in geometric structure of a nonequilibrium process

We investigate the effect of nonlinear interaction on the geometric structure of a nonequilibrium process. Specifically, by considering a driven-dissipative system where a stochastic variable x is damped either linearly (x) or nonlinearly (x3) while driven by a white noise, we compute the time-dependent probability density functions (PDFs) during the relaxation towards equilibrium from an initial nonequilibrium state. From these PDFs, we quantify the information change by the information length L, which is the total number of statistically distinguishable states which the system passes through from the initial state to the final state. By exploiting different initial PDFs and the strength D of the white-noise forcing, we show that for a linear system, L increases essentially linearly with an initial mean value y0 of x as Ly0, demonstrating the preservation of a linear geometry. In comparison, in the case of a cubic damping, L has a power-law scaling as Ly0m, with the exponent m depending on D and the width of the initial PDF. The rate at which information changes also exhibits a robust power-law scaling with time for the cubic damping.