On the Einstein-Smoluchowski relation in the framework of generalized statistical mechanics. (English) Zbl 07820909
Summary: 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 \(\mu\) and the diffusion coefficient \(D\) retains its original expression: \(\mu = \beta D\), appropriate considerations, physically motivated, force us an interpretation of the parameter \(\beta\) different from the traditional meaning of inverse temperature.
MSC:
| 82-XX | Statistical mechanics, structure of matter |
Keywords:
Einstein-Smoluchowski relation; maximal entropy principle; zeroth principle of thermodynamics; physical temperatureReferences:
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