Universal limiting behaviour of reaction-diffusion systems with conservation laws

TL;DR

This paper demonstrates that reaction-diffusion systems with multiple species and conservation laws can be effectively described by a Cahn-Hilliard-like equation, simplifying the analysis of complex pattern formation and phase separation phenomena.

Contribution

It introduces a geometric framework based on nullclines to derive coarse-grained equations for multi-species reaction-diffusion systems with conservation laws.

Findings

Conservation laws constrain dynamics to a Cahn-Hilliard-like equation.

A geometric framework captures effects of eliminating fast degrees of freedom.

The theory aids understanding of biomolecular condensates.

Abstract

Making sense of complex inhomogeneous systems composed of many interacting species is a grand challenge that pervades basically all natural sciences. Phase separation and pattern formation in reaction-diffusion systems have been largely studied as two separate paradigms. Here we show that in reaction-diffusion systems composed of many species, the presence of a conservation law constrains the evolution of the conserved quantity to be governed by a Cahn-Hilliard-like equation. This establishes a direct link with the paradigm of coexistence and recent "active" field theories. Hence, even for complex many-species systems a dramatically simplified but accurate description emerges over coarse spatio-temporal scales. Using the nullcline (the line of homogeneous steady states) as the central motif, we develop a geometrical framework which endows chemical space with a basis and suitable coordinates. This framework allows us to capture and understand the effect of eliminating fast non-conserved degrees of freedom, and to explicitly construct coefficients of the coarse field theory. We expect that the theory we develop here will be particularly relevant to advance our understanding of biomolecular condensates.