Subdiffusion via dynamical localization induced by thermal equilibrium fluctuations

Abstract

We reveal the mechanism of subdiffusion which emerges in a straightforward, one dimensional classical nonequilibrium dynamics of a Brownian ratchet driven by both a time-periodic force and Gaussian white noise. In a tailored parameter set for which the deterministic counterpart is in a non-chaotic regime, subdiffusion is a long-living transient whose lifetime can be many, many orders of magnitude larger than characteristic time scales of the setup thus being amenable to experimental observations. As a reason for this subdiffusive behaviour in the coordinate space we identify thermal noise induced dynamical localization in the velocity (momentum) space. This novel idea is distinct from existing knowledge and has never been reported for any classical or quantum system. It suggests reconsideration of generally accepted opinion that subdiffusion is due to broad distributions or strong correlations which reflect disorder, trapping, viscoelasticity of the medium or geometrical constraints.

Figure 1

Diffusion anomalies of an inertial Brownian particle moving in a periodic potential and driven by a unbiased time-periodic force. (a) The time dependence of the diffusion coefficient D(t). (b) The evolution of variance of the period averaged velocity ${\sigma }_{\mathbf{v}}^2(t)$ is presented for three values of thermal noise intensity θ proportional to temperature. The region corresponding to the subdiffusive behaviour for θ = 0.0007 is indicated with the cyan colour. Parameters are m = 6, a = 1.899, ω = 0.403. At zero temperature θ = 0, the system is non-chaotic.

Figure 2

(a) The scaling index α vs temperature θ fitted to the asymptotic parts of ${\sigma }_x^2(t)$. The temperature interval corresponding to subdiffusion is indicated with the cyan colour. Superdiffusion is for lower temperatures (α > 1) and normal diffusion - for higher temperatures (α = 1). (b) The period averaged velocity variance ${\sigma }_{\mathbf{v}}^2(t)$ in the long time regime. The vertical lines indicate three values of θ = 0.00016 (red), θ = 0.0007 (blue) and θ = 0.001175 (green). (c) The conditional probabilities in the three-states model: the minus v ₋ ≈ −0.4, the zero v ₀ ≈ 0 and the plus v ₊ ≈ 0.4 solution all presented as function of temperature θ of the system. Other parameters are the same as in Fig. 1.

Figure 3

A representative trajectory of the Brownian particle coordinate x(t) and the period averaged velocity v(t). Each red dot in panel (b) depicts the period averaged velocity v(t) for a given t. The dynamical localization of the latter in the state pointing to the positive direction v ₊ ≈ 0.4 is illustrated. In the inset of panel (a) we show the ensemble of 100 trajectories of the Brownian particle. Parameters are the same as in Fig. 1 with θ = 0.0004.