Generalized empty-interval method applied to a class of one-dimensional stochastic models.

Area of Science:

• Statistical Mechanics
• Mathematical Physics
• Computational Science

Background:

• Reaction-diffusion models are crucial for understanding complex systems.
• Many models are analytically tractable only if they map to free-fermion models.
• One-dimensional systems present unique challenges due to fluctuations.

Purpose of the Study:

• To develop a method for analyzing one-dimensional reaction-diffusion models intractable by conventional techniques.
• To investigate a generalized voter/epidemic model and a diffusion-coagulation model.
• To examine wave front propagation in microscopic models.

Main Methods:

• Extension of the empty-interval method (interparticle distribution function method) using a string function.
• Analysis of specific reaction-diffusion models, including generalizations of the voter and epidemic models.
• Application of similarity transformations to map models.

Main Results:

• A novel string function approach successfully analyzes models not solvable by the conventional interparticle distribution function method.
• The study identifies reaction-diffusion models mappable to a reversible diffusion-coagulation model.
• Mean-field predictions from Fisher's equation are shown to be invalid for these one-dimensional microscopic models.

Conclusions:

• The string function method provides a powerful tool for studying complex reaction-diffusion systems.
• The behavior of one-dimensional microscopic models can deviate significantly from macroscopic mean-field approximations.
• This work offers new analytical pathways for complex systems modeling.