Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model

The Sinai model of a tracer diffusing in a quenched Brownian potential is a much-studied problem exhibiting a logarithmically slow anomalous diffusion due to the growth of energy barriers with the system size. However, if the potential is random but periodic, the regime of anomalous diffusion crosses over to one of normal diffusion once a tracer has diffused over a few periods of the system. Here we consider a system in which the potential is given by a Brownian bridge on a finite interval $(0,L)$ and then periodically repeated over the whole real line and study the power spectrum $S\left(f\right)$ of the diffusive process $x\left(t\right)$ in such a potential. We show that for most of realizations of $x\left(t\right)$ in a given realization of the potential, the low-frequency behavior is $S\left(f\right)\sim \mathcal{A}/f^2$, i.e., the same as for standard Brownian motion, and the amplitude $\mathcal{A}$ is a disorder-dependent random variable with a finite support. Focusing on the statistical properties of this random variable, we determine the moments of $\mathcal{A}$ of arbitrary, negative, or positive order $k$ and demonstrate that they exhibit a multifractal dependence on $k$ and a rather unusual dependence on the temperature and on the periodicity $L$, which are supported by atypical realizations of the periodic disorder. We finally show that the distribution of $\mathcal{A}$ has a log-normal left tail and exhibits an essential singularity close to the right edge of the support, which is related to the Lifshitz singularity. Our findings are based both on analytic results and on extensive numerical simulations of the process $x\left(t\right)$.

Figure 1

Potential $V\left(x\right)$ as a periodically extended Brownian bridge with $V(x=0)=V(x=L)=0$.

Figure 2

Main: $\mathbb{E}(\overline{x^2\left(t\right)})$ in a periodic Sinai model (with a periodic BB), numerical data shown as points. Also shown by the dashed line is the fit $c_1{ln}^4\left(t\right)$ for the short-time Sinai regime along with the solid line late-time fit $c_{2}t$. Inset: As in the main figure but for an unconstrained periodic Sinai potential.

Figure 3

The second moment of the fitted amplitudes of the power spectrum in a BB potential. Numerical results are shown as circles along with the fit of the data by $aexp(-L^b)$, shown as a solid line, yielding the fitted value $b=0.37$.

Figure 4

The exponent $b^{\left(1\right)}(L,2L)$ in Eq. (38) as a function of $L$.

Figure 5

Distribution $P\left(\mathcal{A}\right)$ of the amplitudes $\mathcal{A}$ for a BB potential, plotted with circles (numerical results). In the main figure $L=64$ while the inset shows the results for $L=32$. The log-normal fit corresponding to Eq. (39) is shown for small $\mathcal{A}$ as a solid green curve while the prediction of Eq. (47) for the right tail is shown for large values of $\mathcal{A}$ by the dashed blue line.