Linearization Principle: The Geometric Origin of Nonlinear Fokker-Planck Equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910247185022976 |
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| author | Suyari, Hiroki |
| author_facet | Suyari, Hiroki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01278 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linearization Principle: The Geometric Origin of Nonlinear Fokker-Planck Equations Suyari, Hiroki Statistical Mechanics Mathematical Physics 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. |
| title | Linearization Principle: The Geometric Origin of Nonlinear Fokker-Planck Equations |
| topic | Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2603.01278 |