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The Sinai model of a tracer diffusing in a quenched Brownian potential is amuch studied problem exhibiting a logarithmically slow anomalous diffusion dueto the growth of energy barriers with the system size. However, if thepotential is random but periodic, the regime of anomalous diffusion crossesover to one of normal diffusion once a tracer has diffused over a few periodsof the system. Here we consider a system in which the potential is given by aBrownian Bridge on a finite interval $(0,L)$ and then periodically repeatedover the whole real line, and study the power spectrum $S(f)$ of the diffusiveprocess $x(t)$ in such a potential. We show that for most of realizations of$x(t)$ in a given realization of the potential, the low-frequency behavior is$S(f) \sim {\cal A}/f^2$, i.e., the same as for standard Brownian motion, andthe amplitude ${\cal A}$ is a disorder-dependent random variable with a finitesupport. Focusing on the statistical properties of this random variable, wedetermine the moments of ${\cal A}$ of arbitrary, negative or positive order$k$, and demonstrate that they exhibit a multi-fractal dependence on $k$, and arather unusual dependence on the temperature and on the periodicity $L$, whichare supported by atypical realizations of the periodic disorder. We finallyshow that the distribution of ${\cal A}$ has a log-normal left tail, andexhibits an essential singularity close to the right edge of the support, whichis related to the Lifshitz singularity. Our findings are based both on analyticresults and on extensive numerical simulations of the process $x(t)$.

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