Effective diffusion coefficients in reaction-diffusion systems with anomalous transport.

Area of Science:

• Chemical kinetics
• Non-equilibrium systems
• Statistical physics

Background:

• Subdiffusion and anomalous transport are common in complex systems.
• Turing patterns arise from reaction-diffusion processes, but modeling subdiffusive cases is challenging.
• Cross-diffusion offers a potential framework for understanding anomalous transport phenomena.

Purpose of the Study:

• To demonstrate that Turing patterns in subdiffusion systems can be effectively replicated by a Markovian cross-diffusion system.
• To establish an effective system that shares the same Turing instability and patterns as the original subdiffusive system.
• To define effective diffusion coefficients for subdiffusion systems using the cross-diffusion framework.

Main Methods:

• Developing an effective Markovian cross-diffusion system to model subdiffusion.
• Analyzing Turing instability in both the original and effective systems.
• Calculating effective diffusion coefficients from the cross-diffusion model.
• Numerically integrating fractional reaction-diffusion equations and their normally diffusing counterparts.

Main Results:

• The effective cross-diffusion system replicates Turing patterns and instability from subdiffusion systems.
• Transient dynamics are accurately captured when particles are short-lived.
• Effective diffusion coefficients were defined and shown to accurately describe Turing instability.
• The mean-squared displacement of subdiffusing particles exhibits linear growth, consistent with calculations.

Conclusions:

• Cross-diffusion provides an effective framework for studying Turing patterns in subdiffusion systems.
• Anomalous transport can lead to emergent cross-diffusive behavior.
• The defined effective diffusion coefficients simplify the analysis of Turing instability in complex systems.