Large Deviations of Kac’s Elastic Particle System
A talk in the Bielefeld Stochastic Afternoon series by
Daniel Heydecker from University of Cambridge
| Abstract: | We consider Kac’s stochastic model for a $N$-particle gas, with particles interacting by elastic collisions, and study the large deviations in the limit $N\to\infty$ in the weak topology. There is a very natural candidate rate function, first proposed by Léonard, which amounts to the dynamic entropy of a flux measure plus an initial cost; with this rate function, we sketch the proof of an upper bound, and a lower bound restricted to a class of `sufficiently good’ paths. Perhaps surprisingly, we will see that the proposed rate function does not capture all of the possible large deviation behaviour. Lu and Wennberg showed that, even though the microscopic collisions preserve energy, there are solutions to the Boltzmann equation for which the energy is increasing. We will show that such paths can arise as large deviation limits with finite exponential cost, so cannot be excluded from large deviation analysis, but occur strictly more rarely than predicted by the rate function. At the level of the particle system, this occurs when a macroscopic proportion of the energy concentrates in $o(N)$ particles, which can happen with probability $e^{-O(N)}$. Please contact stochana@math.uni-bielefeld.de for Meeting-ID and Password. (New meeting details since April 1!) |