Critical time scales for advection-diffusion-reaction processes.

Area of Science:

• Physical Chemistry
• Chemical Physics
• Mathematical Biology

Background:

• Reaction-diffusion equations model transient processes approaching steady states.
• Local Accumulation Time (LAT) was previously introduced as a critical time measure.
• Mean Action Time (MAT) was developed for analyzing particle lifetimes and diffusion.

Purpose of the Study:

• To demonstrate the identity between Local Accumulation Time (LAT) and Mean Action Time (MAT).
• To derive expressions for MAT in one-dimensional advection-diffusion-reaction systems.
• To explore the relationship between MAT and Mean Particle Lifetime (MPLT).

Main Methods:

• Derivation of Mean Action Time (MAT) for general 1D linear advection-diffusion-reaction problems.
• Application of both continuum and discrete mathematical approaches.
• Analysis of transitions between uniform states and more general cases.

Main Results:

• Local Accumulation Time (LAT) is shown to be equivalent to Mean Action Time (MAT).
• MAT and Mean Particle Lifetime (MPLT) are equivalent for specific uniform-to-uniform transitions.
• MAT can be evaluated without solving the governing partial differential equation (PDE).
• Accurate approximations for nonlinear kinetic processes can be derived using MAT.

Conclusions:

• MAT offers a practical and simple critical time measure for reaction-diffusion systems.
• The equivalence of MAT and MPLT provides a link between microscopic stochasticity and macroscopic timescales.
• MAT facilitates direct approximation of system dynamics without complex PDE solutions.