Crossover in diffusion equation: Anomalous and normal behaviors

Ubiquitous phenomena exist in nature where, as time goes on, a crossover is observed between different diffusion regimes (e.g., anomalous diffusion at early times which becomes normal diffusion at long times, or the other way around). In order to focus on such situations we have analyzed particular relevant cases of the generalized Fokker-Planck equation ${∫}d{{γ}}^{{'}}{τ}({{γ}}^{{'}})[{{∂}}^{{{γ}}^{{'}}}{ρ}(x,t)]/{∂}{t}^{{{γ}}^{{'}}}={∫}d{{μ}}^{{'}}d{{ν}}^{{'}}D({{μ}}^{{'}},{{ν}}^{{'}})[{{∂}}^{{{μ}}^{{'}}}[{ρ}(x,t){]}^{{{ν}}^{{'}}}]/{∂}{x}^{{{μ}}^{{'}}},$ where ${τ}({{γ}}^{{'}})$ and $D({{μ}}^{{'}},{{ν}}^{{'}})$ are kernels to be chosen; the choice ${τ}({{γ}}^{{'}})={δ}({{γ}}^{{'}}{-}1)$ and $D({{μ}}^{{'}},{{ν}}^{{'}})={δ}({{μ}}^{{'}}{-}2){δ}({{ν}}^{{'}}{-}1)$ recovers the normal diffusion equation. We discuss in detail the following cases: (i) a mixture of the porous medium equation, which is connected with nonextensive statistical mechanics, with the normal diffusion equation; (ii) a mixture of the fractional time derivative and normal diffusion equations; (iii) a mixture of the fractional space derivative, which is related with L\'evy flights, and normal diffusion equations. In all three cases a crossover is obtained between anomalous and normal diffusions. In cases (i) and (iii), the less diffusive regime occurs for short times, while at long times the more diffusive regime emerges. The opposite occurs in case (ii). The present results could be easily extended to more complex situations (e.g., crossover between two, or even more, different anomalous regimes), and are expected to be useful in the analysis of phenomena where nonlinear and fractional diffusion equations play an important role. Such appears to be the case for isolated long-ranged interaction Hamiltonians, which along time can exhibit a crossover from a longstanding metastable anomalous state to the usual Boltzmann-Gibbs equilibrium one. Another illustration of such crossover occurs in active intracellular transport.