Abstract:Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by introducing the Linearization Principle directly at the dynamical stage. By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. Within this framework, we show that the $q$-deformed geometry, corresponding to Tsallis statistics, exhibits a fundamental duality between the dynamic index $q$ and the thermodynamic index $2-q$: the stationary state is a $q$-Gaussian distribution that minimizes a free energy functional defined by a generalized entropy of index $2-q$. We prove the $H$-theorem for the derived equation and demonstrate its application to the harmonic oscillator and the free particle. This framework describes anomalous diffusion without relying on ad-hoc constraints or phenomenological nonlinear drift forces.
| Comments: | 5 pages, 1 figure. Accepted for publication in Physical Review E |
| Subjects: | Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph) |
| Cite as: | arXiv:2603.01278 [cond-mat.stat-mech] |
| (or arXiv:2603.01278v3 [cond-mat.stat-mech] for this version) | |
| https://doi.org/10.48550/arXiv.2603.01278 arXiv-issued DOI via DataCite |
From: Hiroki Suyari [view email]
[v1]
Sun, 1 Mar 2026 21:31:32 UTC (25 KB)
[v2]
Mon, 6 Apr 2026 07:24:44 UTC (26 KB)
[v3]
Fri, 22 May 2026 06:39:11 UTC (26 KB)
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