Solving the equilibrium dynamics of particle systems in infinite dimensions

In the last years, a general framework to study the dynamics of particle systems in infinite dimensions has been developed. This theory can be applied to a wide class of physical cases both in and out of equilibrium, including the physics of glasses, colloids and active matter. On the other hand, even for the equilibrium - simplest - case, an analytical solution of the dynamical equations is out of reach. I will show how a numerical solution can be found, leading to the emergence of a dynamical transition in the case of short-ranged repulsive potentials.
The numerical analysis for long times matches the predictions obtained by thermodynamical arguments at equilibrium, and provides new observations such as the behavior in time of the memory kernel and the critical scaling of the diffusivity when approaching the dynamically arrested phase. The consistency of the results at equilibrium is the first step to obtain new insights into non-equilibrium physical phenomena, which will represent the future direction of the research.