Linearization principle: The geometric origin of nonlinear Fokker-Planck equations

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.