# On Non-ideal Chemical-Reaction Networks and Phase Separation

Much of the theory on chemical-reaction networks (CRNs) has been developed in the ideal-solution limit, where interactions between the solutes are negligible. However, there is a large variety of phenomena in biological cells and soft-matter physics which appear to deviate from the ideal-solution behaviour. Particularly striking is the case of liquid-liquid phase separation, which is typically caused by inter-particle interactions. Here, we revisit a number of known results in the domain of ideal CRNs, and we generalise and adapt them to arbitrary interactions between the solutes which stem from a given free energy. We start by reviewing the theory of chemical reaction networks, linking it to concepts in statistical physics. Then we obtain a number of new results for non-ideal complex-balanced networks, where the creation and annihilation rates are equal for all chemical complexes which appear as reactants or products in the CRN. Among these is the form of the steady-state probability distribution and Lyapunov functions for such networks. Finally, this allows us to draw a phase diagram for complex-balanced reaction-diffusion systems based on the minimisation of such Lyapunov function, with a rationale similar to that of equilibrium thermodynamics but for systems that may sustain non-equilibrium chemical currents at steady state. In addition, we show that for complex-balanced networks at steady-state, there are no diffusion currents.