After recalling the many situations that can be described by reaction-diffusion equations (chemistry, physics, biology, ecology, sociology, genealogies) and the different types of theoretical approaches used (dynamical systems, partial differential equations, stochastic equations, mean-field approximations, exact solutions, simulations, renormalization), this lecture began by presenting a few examples : chemical reactions with Gulberg and Waage's mass action law (1864), evolutionary models in ecology such as the Lotka-Volterra equations (1910) for a predator-prey system, infection models in biology.
For reaction-diffusion problems, two important aspects that mean-field descriptions neglect are fluctuations and spatial dependencies. The lecture continued with a more detailed discussion of reactions of the type A + A → A which appears in problems of polymerization, coagulation, genealogies, A + A ⇌ ∅ which describes the adsorption and desorption of diatomic molecules on the surface of a metal, A + B → 2Awhich models the spread of an infection or opinion in a population, A + B → ∅ which represents reactions in a medium composed of matter and antimatter. For all these models, the mass action laws, obtained by making a mean-field approximation, correctly predict behavior at long times only above an upper critical dimension dc. Below this critical dimension (dc = 2 for A + A → A or A + A → ∅ and dc = 4 for A + B → ∅), the role of fluctuations can no longer be neglected. However, a simple argument allows us to predict asymptotic behavior for dimensions below the critical dimension. The lecture ended by showing how microscopic reaction-diffusion models can be accurately described by stochastic partial differential equations, with in cases like A + A → ∅ imaginary white Gaussian noise.
