The exact distribution of the oscillation period in the underdamped one-dimensional Sinai model

2001/10/24 by David S. Dean, Satya N. Majumdar · 3 citations
Mathematics · Physics and Astronomy · #Statistical Mechanics and Entropy #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech

paper · pdf · doi:10.1088/0305-4470/34/49/102

published as J. Phys. A 34, L697 (2001) · 9 pages LateX, 2 .eps figures

arxiv created 2001/10/24 · openalex publication_date 2001/12/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We consider the Newtonian dynamics of a massive particle in a one-dimensional random potential which is a Brownian motion in space. This is the zero-temperature nondamped Sinai model. As there is no dissipation the particle oscillates between two turning points where its kinetic energy becomes zero. The period of oscillation is a random variable fluctuating from sample to sample of the random potential. We compute the probability distribution of this period exactly and show that it has a power law tail for large period, P ( T ) ~ T -5/3 , and an essential singularity P ( T ) ~ exp (-1/ T ) as T →0. Our exact results are confirmed by numerical simulations and also via a simple scaling argument.

Cited by

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• Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules