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On the Einstein–Smoluchowski relation in the framework of generalized statistical mechanics

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  • Evangelista, L.R.
  • Lenzi, E.K.
  • Barbero, G.
  • Scarfone, A.M.

Abstract

Anomalous statistical distributions that exhibit asymptotic behavior different from the exponential Boltzmann–Gibbs tail are typical of complex systems constrained by long-range interactions or time-persistent memory effects at the stationary non-equilibrium or meta-equilibrium. In this framework, a nonlinear Smoluchowski equation, which models the system’s time evolution towards its steady state, is obtained using the gradient flow method based on a free-energy potential related to a given generalized entropic form. Comparison of the stationary distribution resulting from the maximization of entropy for a canonical ensemble with the steady state distribution resulting from the Smoluchowski equation gives an Einstein-Smoluchowski-like relation. Despite this relationship between the mobility of particle μ and the diffusion coefficient D retains its original expression: μ=βD, appropriate considerations, physically motivated, force us an interpretation of the parameter β different from the traditional meaning of inverse temperature.

Suggested Citation

  • Evangelista, L.R. & Lenzi, E.K. & Barbero, G. & Scarfone, A.M., 2024. "On the Einstein–Smoluchowski relation in the framework of generalized statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 635(C).
    • Handle: RePEc:eee:phsmap:v:635:y:2024:i:c:s0378437123010464
    DOI: 10.1016/j.physa.2023.129491
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    References listed on IDEAS

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    1. Abe, Sumiyoshi, 2001. "Heat and entropy in nonextensive thermodynamics: transmutation from Tsallis theory to Rényi-entropy-based theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(3), pages 417-423.
    2. T. Wada, 2010. "A nonlinear drift which leads to κ-generalized distributions," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 73(2), pages 287-291, January.
    3. Frank, T.D. & Daffertshofer, A., 2000. "Exact time-dependent solutions of the Renyi Fokker–Planck equation and the Fokker–Planck equations related to the entropies proposed by Sharma and Mittal," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 285(3), pages 351-366.
    4. Yanovsky, V.V. & Chechkin, A.V. & Schertzer, D. & Tur, A.V., 2000. "Lévy anomalous diffusion and fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 282(1), pages 13-34.
    5. T. Wada & A. M. Scarfone, 2009. "Asymptotic solutions of a nonlinear diffusive equation in the framework of κ-generalized statistical mechanics," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 70(1), pages 65-71, July.
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