Multiplicative Ornstein Uhlenbeck Noise in Nonequilibrium Phenomena

Abstract

Stochastic differential equations of the Langevin type for a finite set of variables are a common tool to study a variety of physical, chemical and biological systems. The more recent interest in this type of equation is mainly due to its success in describing nonequilibrium situations of open systems. When dealing with these equations it is often assumed that the fluctuating term does not depend on the state of the system (“additive noise”) and, invoking a difference in time scale, that the white noise idealization is appropriate. Nevertheless remarkable novel features of these equations appear when removing these two constraints. We are here precisely concerned with this last situation. That is, we consider stochastic differential equations of the form

$$ {{\rm{\dot q}}_{\rm{\mu }}}({\rm{t}})\; = \;{{\rm{v}}_{\rm{\mu }}}[{\rm{q(t)] + }}{{\rm{g}}_{{\rm{\mu \nu }}}}[{\rm{q(t)]}}{{\rm{\xi }}_{\rm{\nu }}}({\rm{t}})\;{\rm{\mu = 1, }} \ldots ,\,{\rm{N; \nu = 1, }} \ldots {\rm{, M}} $$

(1.1)

where ξ(t) is not a white noise but has a finite correlation time (“colored noise”). The q dependence of g_µν gives its “multiplicative” character to the noise term. It is our purpose here to elucidate some phenomena appearing in nonequilibrium systems described by (1.1) with special emphasis on the effect of considering a finite correlation time as compared to the white noise case. The interest in this problem is not only purely mathematical. In fact, there are at least two important sources of these equations for a realistic description of a system. The first one is the elimination of fast variables from the equations of motion. A careful adiabatic elimination procedure [1] from a set of additive white noise Langevin equations leads in general to colored multiplicative noise.