Kinetic Regimes and Cross-Over Times in Many-Particle Reacting Systems

TL;DR

This paper analyzes the kinetics of single-species reactions in many-particle systems, identifying different regimes based on reactivity, dimension, and dynamical exponent, and provides a detailed theoretical framework without decoupling approximations.

Contribution

It introduces a comprehensive analysis of reaction kinetics across various regimes, incorporating general local reactivity and dynamical exponents, avoiding common decoupling approximations.

Findings

Mean field behavior above critical dimension d_c=z.

Identification of a diffusion-controlled regime for high reactivity.

Persistence of mean field kinetics for low reactivity Q<Qstar.

Abstract

We study kinetics of single species reactions ("A+A -> 0") for general local reactivity Q and dynamical exponent z (rms displacement x_t ~ t^{1/z}.) For small molecules z=2, whilst z=4,8 for certain polymer systems. For dimensions d above the critical value d_c=z, kinetics are always mean field (MF). Below d_c, the density n_t initially follows MF decay, n_0 - n_t ~ n_0^2 Q t. A 2-body diffusion-controlled regime follows for strongly reactive systems (Q>Qstar ~ n_0^{(z-d)/d}) with n_0 - n_t ~ n_0^2 x_t^d. For Q<Qstar, MF kinetics persist, with n_t ~ 1/Qt. In all cases n_t ~ 1/x_t^d at the longest times. Our analysis avoids decoupling approximations by instead postulating weak physically motivated bounds on correlation functions.