Non-normalizable quasi-equilibrium states under fractional dynamics

We study non-normalizable quasi-equilibrium states (NNQE) arising from anomalous diffusion. Initially, particles in contact with a thermal bath are released from an asymptotically flat potential well, with dynamics that is described by fractional calculus. For temperatures that are sufficiently low...

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Bibliographic Details
Published in:arXiv.org 2023-04
Main Authors: Defaveri, Lucianno, Maike A F dos Santos, Kessler, David A, Barkai, Eli, Anteneodo, Celia
Format: Article
Language:English
Subjects:
Diffusion
Fokker-Planck equation
Fractional calculus
Mathematical analysis
Random walk
Thermal baths
Online Access:Get full text
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Summary:We study non-normalizable quasi-equilibrium states (NNQE) arising from anomalous diffusion. Initially, particles in contact with a thermal bath are released from an asymptotically flat potential well, with dynamics that is described by fractional calculus. For temperatures that are sufficiently low compared to the potential depth, the properties of the system remain almost constant in time. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate the ensemble averages of observables. We obtain analytical estimates of the duration of NNQE, depending on the fractional order, from approximate theoretical solutions of the FTFPE. We study and compare two types of observables, the mean square displacement typically used to characterize diffusion, and the thermodynamic energy. We show that the typical time scales for stagnation depend exponentially on the activation energy in units of temperature multiplied by a function of the fractional exponent.
ISSN:2331-8422
DOI:10.48550/arxiv.2304.08834