A Kac Model with Exclusion

We consider a one dimension Kac model with conservation of energy and an exclusion rule: Fix a number of particles $n$, and an energy $E>0$. Let each of the particles have an energy $x_j \geq 0$, with $\sum_{j=1}^n x_j = E$. For some $\epsilon$, the allowed configurations $(x_1,\dots,x_n)$ are those that satisfy $|x_i - x_j| \geq \epsilon$ for all $i\neq j$. At each step of the process, a pair $(i,j)$ of particles is selected uniformly at random, and then they "collide", and there is a repartition of their total energy $x_i + x_j$ between them producing new energies $x^*_i$ and $x^*_j$ with $x^*_i + x^*_j = x_i + x_j$, but with the restriction that exclusion rule is still observed for the new pair of energies. This process bears some resemblance to Kac models for Fermions in which the exclusion represents the effects of the Pauli exclusion principle. However, the "non-quantized" exclusion rule here, with only a lower bound on the gaps, introduces interesting novel features, and a detailed notion of Kac's chaos is required to derive an evolution equation for the evolution of rescaled empirical measures for the process, as we show here.