Cluster size distributions in particle systems with asymmetric dynamics

Area of Science:

• Statistical Mechanics
• Probability Theory
• Complex Systems

Background:

• The totally asymmetric exclusion process (TASEP) is a fundamental model in statistical mechanics.
• Understanding cluster formation and growth is crucial in various physical and biological systems.
• Previous studies have explored TASEP dynamics, but detailed analysis of largest cluster behavior under generalized conditions is less common.

Purpose of the Study:

• To derive exact and asymptotic results for the largest cluster size in one-dimensional TASEP.
• To analyze two distinct TASEP dynamics: ordinary and generalized.
• To investigate the connection between TASEP cluster statistics and extreme order statistics.

Main Methods:

• Analytical derivation of exact and asymptotic formulas.
• Mathematical analysis of cluster size distributions.
• Comparison with established theories of extreme order statistics.
• Modeling of a related interface growth process.

Main Results:

• The expected largest cluster length diverges logarithmically with system size for ordinary TASEP.
• For generalized TASEP dynamics, the largest cluster length diverges as a logarithm divided by a double logarithm.
• A clear link is established between TASEP cluster statistics and extreme value theory.
• The study also examines an interface growth model with particle relaxation.

Conclusions:

• The growth rate of the largest cluster in TASEP is sensitive to the specific dynamics employed.
• Generalized TASEP dynamics exhibit slower cluster growth compared to ordinary dynamics.
• The findings provide insights into large-scale structures in driven particle systems and their relation to extreme value phenomena.