title: Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model.
abstract: 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(f) of the diffusive process 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)∼A/f^{2}, i.e., the same as for standard Brownian motion, and the amplitude 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 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 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(t).
authors.affiliation: Université Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, F-33400 Talence, France.
authors.forename: David S
authors.initials: DS
authors.lastname: Dean
authors.affiliation: Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, I-00185 Roma, Italy.|Dipartimento di Matematica e Fisica, Università Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy.
authors.forename: Antonio
authors.initials: A
authors.lastname: Iorio
authors.affiliation: Dipartimento di Fisica, Sapienza Università di Roma, P.le A. Moro 2, I-00185 Roma, Italy.|INFN, Sezione di Roma 1 and Nanotech-CNR, UOS di Roma, P.le A. Moro 2, I-00185 Roma, Italy.
authors.forename: Enzo
authors.initials: E
authors.lastname: Marinari
authors.affiliation: Sorbonne Universités, UPMC Univ Paris 06, UMR 7600, LPTMC, F-75005, Paris, France.|CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005, Paris, France.
authors.forename: Gleb
authors.initials: G
authors.lastname: Oshanin