Decay of Information for the Kac Evolution

We consider a system of M particles in contact with a heat reservoir of $$N\gg M$$ particles. The evolution in the system and the reservoir, together with their interaction, is modeled via the Kac’s master equation. We chose the initial distribution with total energy $$N+M$$ and show that if the reservoir is initially in equilibrium, that is, if the initial distribution depends only on the energy of the particle in the reservoir, then the entropy of the system decays exponentially to a very small value. We base our proof on a similar property for the Information. A similar argument allows us to greatly simplify the proof of the main result in Bonetto et al. (Commun Math Phys 363(3):847–875, 2018).