8 problems

Let $\mathcal{L}_N$ be Kac's master-equation operator on $L^2\left(\mathbb{S}^{N-1}\left(\sqrt{N}\right),d\sigma_N\right)$, and let … be its spectral gap. Kac's uniform spectral-ga…

Let $F_N$ range over probability densities on Kac's sphere, and define the entropy-entropy production ratio by … where $H_N\left(F_N\right)$ is the relative entropy and…

Detailed-chaos propagation conjecture. The Kac process on $S_{n,\epsilon_n}$ propagates detailed chaos according to Definition$2$.

Exponential-gap propagation conjecture. The process defined here propagates chaos with exponential gap distribution.

Let $44$ be a Borel probability measure on $44$ invariant under permutation of coordinates, with finite second moment and zero mean. Let $d_2$ be the GTW metric, let $R_4$ be its a…

Let $_N$ denote the spectral gap for the Kac model with $N$ particles and unit energy per particle. Let $_N$ be the variational quantity obtained by restricting the gap problem to…

Let $\Gamma_{N}$ denote the entropy production for symmetric probability densities $\psi_N$ on $\mathbb{S}^{N-1}(\sqrt{N})$, defined by … where the infimum is over all such probabi…

Villani's entropy-production conjecture. The bounds are essentially sharp, in the sense that