Analytically solvable model of a driven system with quenched dichotomous disorder

Abstract

We perform a time-dependent study of the driven dynamics of overdamped particles which are placed in a one-dimensional, piecewise linear random potential. This set-up of spatially quenched disorder then exerts a dichotomous varying random force on the particles. We derive the path integral representation of the resulting probability density function for the position of the particles and transform this quantity of interest into the form of a Fourier integral. In doing so, the evolution of the probability density can be investigated analytically for finite times. It is demonstrated that the probability density contains both a δ-singular contribution and a regular part. While the former part plays a dominant role at short times, the latter rules the behavior at large evolution times. The slow approach of the probability density to a limiting Gaussian form as time tends to infinity is elucidated in detail.

Arrival time distribution for a driven system containing quenched dichotomous disorder

“…This result is corroborated by the fact that the long-time asymptotic of the average particle velocity equals (f 2 − g 2 )/f [22].…”

“…Next, introducing the quantities S n (x) = Ωn(x) n j=1 p(s j )ds j , we find the representations W 0 (x) = 1 − S 1 (x) and W n (x) = S n (x) − S n+1 (x), see also [22]. Finally, taking into account that S ∞ (x) = 0 and ∞ n=1 W n (x) = S 1 (x), we assure that the normalization condition holds true for an arbitrary p(s).…”

“…(3.6) into a form with separate integrations over the variables s j . To this end, we use an approach [22] based on the integral representation of the step function…”

“…In [22] we derived the probability density of the solution of Eq. (1.1) and investigated explicitly its time evolution.…”

“…Although temporal noise terms are absent, there are only very few exact results available. Therefore, in order to fill this gap, we have examined the following dimensionless equation of motion for an overdamped particle [22]:Ẋ t = f + g(X t ).…”