Liquid to supercooled-liquid crossover from a Boltzmann transport approach to escape and diffusion.

We develop a model describing the motion of a non-Brownian particle in a periodic potential, which we then use to predict the temperature dependence of the diffusivity of a glass-former. In the model, the velocity of the particle is drawn for the equilibrium distribution at rate $1/t_c$, where $t_c$ is the intercollision time in the relaxation time approximation. Solutions within a Boltzmann transport approach show that the diffusivity crossovers from a low-$t_c$ regime in which the particle at most crosses a single barrier in between two successive collisions, to a high-$t_c$ regime in which the particle may cross several barriers. We then use our model to predict the temperature dependence of the diffusion coefficient of a system of harmonic-spheres, whose energy landscape has features resembling those of the potential considered in our model. We successfully recover a crossover in the temperature dependence of the diffusion coefficient observed through numerical dynamics simulations, as well as the relationship of the diffusivity on the temperature in the high-temperature limit.