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The kinetics of chemical reactions are determined by the law of mass action, which has been successfully applied to homogeneous, dilute mixtures. At nondilute conditions, interactions among the components can give rise to coexisting phases, which can significantly alter the kinetics of chemical reactions. Here, we derive a theory for chemical reactions in coexisting phases at phase equilibrium. We show that phase equilibrium couples the rates of chemical reactions of components with their diffusive exchanges between the phases. Strikingly, the chemical relaxation kinetics can be represented as a flow along the phase equilibrium line in the phase diagram. A key finding of our theory is that differences in reaction rates between coexisting phases stem solely from phase-dependent reaction rate coefficients. Our theory is key to interpreting how concentration levels of reactive components in condensed phases control chemical reaction rates in synthetic and biological systems.

Chemically active droplets provide simple models for cell-like systems that can grow and divide. Such active droplet systems are driven away from thermodynamic equilibrium and turn over chemically, which corresponds to a simple metabolism. Two scenarios of nonequilibrium driving are considered. First, droplets are driven via the system boundaries by external reservoirs that supply nutrient and remove waste (boundary-driven). Second, droplets are driven by a chemical energy provided by a fuel in the bulk (bulk-driven). For both scenarios, the conservation of energy and matter as well as the balance of entropy are discussed. Conserved and nonconserved fields are used to analyse the nonequilibrium steady states of active droplets. Using an effective droplet model, droplet stability and instabilities leading to droplet division are explored. This work reveals that droplet division occurs quite generally in active droplet systems. The results suggest that life-like processes such as metabolism and division can emerge in simple nonequilibrium systems that combine the physics of phase separation and chemical reactions.

We generalize the Susceptible-Infected-Removed model for epidemics to take into account generic effects of heterogeneity in the degree of susceptibility to infection in the population. We introduce a single new parameter corresponding to a power-law exponent of the susceptibility distribution that characterizes the population heterogeneity. We show that our generalized model is as simple as the original model which is contained as a limiting case. Because of this simplicity, numerical solutions can be generated easily and key properties of the epidemic wave can still be obtained exactly. In particular, we present exact expressions for the herd immunity level, the final size of the epidemic, as well as for the shape of the wave and for observables that can be quantified during an epidemic. We find that in strongly heterogeneous populations the epidemic reaches only a small fraction of the population. This implies that the herd immunity level can be much lower than in commonly used models with homogeneous populations. Using our model to analyze data for the SARS-CoV-2 epidemic in Germany shows that the reported time course is consistent with several scenarios characterized by different levels of immunity. These scenarios differ in population heterogeneity and in the time course of the infection rate, for example due to mitigation efforts or seasonality. Our analysis reveals that quantifying the effects of mitigation requires knowledge on the degree of heterogeneity in the population. Our work shows that key effects of population heterogeneity can be captured without increasing the complexity of the model. We show that information about population heterogeneity will be key to understand how far an epidemic has progressed and what can be expected for its future course., Comment: 34 pages, 8 figures