Non-normalizable quasiequilibrium states under fractional dynamics

Lucianno Defaveri, Maike A.F. Dos Santos, David A. Kessler, Eli Barkai, Celia Anteneodo

Research output: Contribution to journalArticlepeer-review


Particles anomalously diffusing in contact with a thermal bath are initially released from an asymptotically flat potential well. For temperatures that are sufficiently low compared to the potential depth, the dynamical and thermodynamical observables of the system remain almost constant for long times. We show how these stagnated states are characterized as non-normalizable quasiequilibrium (NNQE) states. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate ensemble averages. We obtain analytical estimates of the durations of NNQE states, 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 timescales for transient stagnation depend exponentially on the value of the depth of the potential well, in units of temperature, multiplied by a function of the fractional exponent.

Original languageEnglish
Article number024133
JournalPhysical Review E
Issue number2
StatePublished - Aug 2023

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Statistics and Probability


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