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Universal limiting behavior of reaction-diffusion systems with conservation laws

Joshua F. Robinson1,2,*, Thomas Machon2, and Thomas Speck3

  • *Contact author: joshua.robinson@stfc.ac.uk
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Phys. Rev. E 111, 065417 – Published 20 June, 2025

DOI: https://doi.org/10.1103/1bdc-1bjb

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 spatiotemporal 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 nonconserved 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.

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Phys. Rev. E 111, 065417– Published 20 June, 2025
  • Received 20 December 2024
  • Accepted 28 May 2025
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