Local equilibrium properties of ultraslow diffusion in the Sinai model

Amin Padash, Erez Aghion, Alexander Schulz, Eli Barkai, Aleksei V. Chechkin, Ralf Metzler, Holger Kantz

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We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 1017 steps and thereby also study finite-time crossover phenomena.

Original languageEnglish
Article number073026
JournalNew Journal of Physics
Issue number7
StatePublished - 1 Jul 2022

Bibliographical note

Funding Information:
AC acknowledges support of the Polish National Agency for Academic Exchange (NAWA). The support of Israel Science Foundation’s Grant 1614/21 is acknowledged (EB). RM acknowledges the German Science Foundation (DFG, Grant No. ME 1535/12-1) and the Foundation for Polish Science (Fundacja na rzecz Nauki Polskiej, Humboldt Polish Honorary Research Scholarship) for support.

Publisher Copyright:
© 2022 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.


  • Sinai diffusion
  • clustering
  • local equilibrium


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