Abstract
We discuss the dynamics of a Brownian particle under the influence of a spatially periodic noise strength in one dimension using analytical theory and computer simulations. In the absence of a deterministic force, the Langevin equation can be integrated formally exactly. We determine the short- and long-time behaviour of the mean displacement (MD) and mean-squared displacement (MSD). In particular, we find a very slow dynamics for the mean displacement, scaling as t^{-1/2} with time t. Placed under an additional external periodic force near the critical tilt value we compute the stationary current obtained from the corresponding Fokker–Planck equation and identify an essential singularity if the minimum of the noise strength is zero. Finally, in order to further elucidate the effect of the random periodic driving on the diffusion process, we introduce a phase factor in the spatial noise with respect to the external periodic force and identify the value of the phase shift for which the random force exerts its strongest effect on the long-time drift velocity and diffusion coefficientGraphical
Highlights
Dating back to the important paper by Einstein in the annus mirabilis 1905 [1], the dynamics of Brownian particles has been in the focus of statistical physics for more than 100 years [2]
While the case where D is only an explicit function of time t is well studied, for example in the context of Brownian ratchets [11–15] and heat engines [16–20], in this work we focus on the case where we have a spatially dependent noise strength [21–24] modelled by a positive function D(x), i.e. a space-dependent diffusion coefficient, such that the most basic model for such processes is given by the Langevin equation x (t) = D(x(t))η(t)
The paper is organized as follows: in the beginning we focus on the free case, for which we study the shortand long-time behaviour of mean displacement (MD) and mean-squared displacement (MSD), we proceed with the full model, including the tilted potential, for which we study the stationary distribution and the dependence of long time diffusion and drift on φ and ν
Summary
Dating back to the important paper by Einstein in the annus mirabilis 1905 [1], the dynamics of Brownian particles has been in the focus of statistical physics for more than 100 years [2]. There is a variety of excellent realizations of Brownian particles including mesoscopic colloidal particles in suspension [3], random walkers in the macroscopic world (such as [4]) and in the microscopic biological context [5], and even elements of the stock exchange market [6]. This facilitates a direct comparison of the stochastic averages between the stochastic modelling and real experimental data.
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