Non-Markovian random walks and nonlinear reactions: subdiffusion and propagating fronts

Area of Science:

• Physics
• Physical Chemistry
• Chemical Engineering

Background:

• Non-Markovian transport phenomena are crucial in complex systems.
• Continuous Time Random Walk (CTRW) models describe anomalous diffusion with non-exponential waiting times.
• Incorporating nonlinear kinetics into these models is essential for accurate reaction-transport system analysis.

Purpose of the Study:

• To incorporate nonlinear kinetic terms into non-Markovian transport equations using CTRW.
• To derive nonlinear Master equations for mesoscopic densities in reacting particle systems.
• To apply these equations to analyze front propagation in reaction-transport systems with anomalous diffusion.

Main Methods:

• Development of three distinct CTRW models incorporating reactions.
• Derivation of nonlinear Master equations applicable to arbitrary jump and waiting time distributions.
• Application of derived equations to reaction-transport systems exhibiting Kolmogorov-Petrovskii-Piskunov (KPP) kinetics and anomalous diffusion.

Main Results:

• Successful derivation of nonlinear Master equations for mesoscopic densities.
• Explicit expression for the speed of a propagating front in subdiffusive transport scenarios.
• Demonstration of the applicability of CTRW models to complex reaction-transport dynamics.

Conclusions:

• The study provides a robust framework for modeling nonlinear reaction-transport systems with anomalous diffusion.
• The derived nonlinear Master equations offer a powerful tool for understanding mesoscopic phenomena.
• The explicit solution for front propagation speed advances the understanding of pattern formation in complex media.