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Explore critical dynamical fluctuations in one-dimensional microscopic reaction-diffusion processes through this 49-minute mathematical lecture. Examine a system combining Glauber and Kawasaki dynamics where particle density evolves according to a reaction-diffusion equation with fixed points corresponding to specific total mass values. Investigate density fluctuations around fixed points during dynamical phase transitions, focusing on models with non-linear reaction terms where stable fixed points lose stability as parameters vary. Discover how critical density fluctuations at the exact critical point exhibit non-Gaussian behavior, evolve on extended time-scales, and follow specific scaling limits. Learn about the identification of a single slow observable - the total number of particles - that follows a one-dimensional non-linear stochastic differential equation under non-Gaussian space rescaling, while other Fourier modes remain fast and Gaussian. Understand the mathematical proof techniques involving decoupling of slow and fast modes through relative entropy arguments, addressing major technical challenges including the absence of local equilibrium due to non-linearity and the need for replacement estimates on diverging time intervals caused by critical slowdown. Gain insights into advanced mathematical concepts in probability theory, statistical mechanics, and dynamical systems through research presented at the Centre International de Rencontres Mathématiques during the thematic meeting on "Interacting particle systems and related fields."
