Geometric thermodynamics of reaction-diffusion systems: Thermodynamic trade-off relations and optimal transport for pattern formation

Abstract

We establish universal relations between pattern formation and dissipation with a geometric approach to nonequilibrium thermodynamics of deterministic reaction-diffusion systems. We first provide a way to systematically decompose the entropy production rate (EPR) based on the orthogonality of thermodynamic forces, in this way identifying the amount of dissipation caused by each factor. This enables us to extract the excess EPR that genuinely contributes to the time evolution of patterns. We also show that a similar geometric method further decomposes the EPR into detailed contributions, e.g., the dissipation from each point in real or wavenumber space. Second, we relate the excess EPR to the details of the change in patterns through two types of thermodynamic trade-off relations for reaction-diffusion systems: thermodynamic speed limits and thermodynamic uncertainty relations. The former relates dissipation and the speed of pattern formation, and the latter bounds the excess EPR with partial information on patterns, such as specific Fourier components of concentration distributions. In connection with the derivation of the thermodynamic speed limits, we also extend optimal transport theory to reaction-diffusion systems, which enables us to measure the speed of the time evolution. This extension of optimal transport also solves the minimization problem of the dissipation associated with the transition between two patterns, and it constructs energetically efficient protocols for pattern formation. We numerically demonstrate our results using chemical traveling waves in the Fisher-Kolmogorov-Petrovsky-Piskunov equation and changes in symmetry in the Brusselator model. Our results apply to general reaction-diffusion systems and contribute to understanding the relations between pattern formation and unavoidable dissipation.