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The geometric approach to nonequilibrium thermodynamics is promising forunderstanding the role of dissipation and thermodynamic trade-off relations.This paper proposes a geometric framework for studying the nonequilibriumthermodynamics of reaction-diffusion systems. Based on this framework, weobtain several decompositions of the entropy production rate with respect toconservativeness, spatial structure, and wavenumber. We also generalize optimaltransport theory to reaction-diffusion systems and derive several trade-offrelations, including thermodynamic speed limits and thermodynamic uncertaintyrelations. The thermodynamic trade-off relations obtained in this paper shedlight on thermodynamic dissipation in pattern formation. We numericallydemonstrate our results using theFisher$\unicode{x2013}$Kolmogorov$\unicode{x2013}$Petrovsky$\unicode{x2013}$Piskunovequation and the Brusselator model and discuss how the spatial pattern affectsunavoidable dissipation.

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