# Population dynamics of driven autocatalytic reactive mixtures

## TL;DR

This paper uses the Fokker-Planck equation to explore how autocatalysis and reaction kinetics influence phase separation and population dynamics in driven reactive mixtures, with implications for batteries and materials science.

## Contribution

It introduces a novel application of the Fokker-Planck framework to analyze how autocatalysis affects phase behavior beyond thermodynamic predictions.

## Key findings

- Autocatalysis can induce phase separation in single-phase thermodynamic systems.
- Autoinhibitory reactions suppress phase separation.
- Asymmetric kinetics lead to different population dynamics upon reaction reversal.

## Abstract

Motivated by the theory of reaction kinetics based on nonequilibrium thermodynamics and the linear stability of driven reaction-diffusion, we apply the Fokker-Planck equation to describe the population dynamics of an ensemble of reactive particles in contact with a chemical reservoir. We illustrate the effect of autocatalysis on the population dynamics by comparing systems with identical thermodynamics yet different reaction kinetics. The dynamic phase behavior of the system may be entirely different from what its thermodynamics may suggest. By defining phase separation for a particle ensemble to be when the probability distribution is bimodal, we find that thermodynamic phase separation may be suppressed by autoinhibitory reactions, while autocatalysis enhances phase separation and in some cases induce the ensemble that consists of thermodynamically single-phase systems to segregate into two distinct populations, which we term fictitious phase separation. Asymmetric reaction kinetics also results in qualitatively different population dynamics upon reversing the reaction direction. In the limit of negligible fluctuations, we use method of characteristics and linearization to study the evolution of the standard deviation of concentration as well as the condition for phase separation, in good agreement with the full numerical solution. Applications are discussed to Li-ion batteries and {\it in situ} x-ray diffraction.