Abstract
The temporal Fokker–Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation–dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker–Planck equation for the first passage distribution function $$f_j(r,t)$$ of a particle moving on a substrate with time delays $$\tau _j$$ . Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability $$P_j$$ is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, $$P_j \propto f_j^{\nu - 1}$$ , the generalized Fokker–Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, $$P_j \propto \tau _j^{-1-\alpha }$$ (with $$0< \alpha < 2$$ ), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.