04/06/2025, 17:00 — 18:00 — Online
Jani Lukkarinen, University of Helsinki
Generation of chaos in the cumulant hierarchy of the stochastic Kac model
Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution and degree of such independence, i.e., the degree of chaos in the system. In this talk, I will discuss our analysis of the cumulant hierarchy of the stochastic Kac model in the preprint [arxiv.org:2407.17068], a joint work with Aleksis Vuoksenmaa. We control generation of chaos via the magnitude of finite order cumulants of kinetic energies for arbitrary symmetric initial data, with the usual restriction of a fixed energy density. This allows estimating the accuracy of kinetic theory, uniformly in time and for any system with sufficiently large number of particles, N. We prove that the evolution of the system can be divided into three regimes: an initial regime of the length of at most O(ln N) in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime.
