Autonomous evolution of electron speeds in a thermostatted system: exact results

We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $E\to 0$ and $t\to t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere. This verifies previously conjectured behavior. Our results are based on splitting the system's evolution into a "slow" process and an independent "noise". We show that the noise, suitably rescaled, converges a Brownian motion, enhanced in the sense of rough paths. Then we employ the It\^o-Lyons continuity theorem to identify the limit of the slow process.