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We prove approach to thermal equilibrium for the fully Hamiltonian dynamicsof a dynamical Lorentz gas, by which we mean an ensemble of particles movingthrough a $d$-dimensional array of fixed soft scatterers that each possess aninternal harmonic or anharmonic degree of freedom to which moving particleslocally couple. We establish that the momentum distribution of the movingparticles approaches a Maxwell-Boltzmann distribution at a certain temperature$T$, provided that they are initially fast and the scatterers are in asufficiently energetic but otherwise arbitrary stationary state of their freedynamics--they need not be in a state of thermal equilibrium. The temperature$T$ to which the particles equilibrate obeys a generalized equipartitionrelation, in which the associated thermal energy $k_{\mathrm B}T$ is equal toan appropriately defined average of the scatterers' kinetic energy. In theequilibrated state, particle motion is diffusive.

Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases