Population heterogeneity in the fractional master equation, ensemble self-reinforcement and strong memory effects

Abstract

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

References in corpus (14)

• Lévy walks
• Assessing non-Markovian dynamics
• "Diffusing diffusivity": A model for anomalous and "anomalous yet Brownian" diffusion
• Non-Markovian quantum jumps
• Diffusion-limited reactions in dynamic heterogeneous media
• Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces
• Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving
• Paradoxes of Subdiffusive Infiltration in Disordered Systems
• Aging power spectrum of membrane protein transport and other subordinated random walks
• Optimal non-Markovian search strategies with n-step memory
• Self-reinforcing directionality generates Lévy walks without the power-law assumption
• Inhomogeneous diffusion and ergodicity breaking induced by global memory effects
• Superdiffusion in self-reinforcing run-and-tumble model with rests
• Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution

Cited by in corpus (2)

• Fractional Brownian motion with fluctuating diffusivities
• Heterogeneous model for superdiffusive movement of dense-core vesicles in C. elegans