T-5

Goal: So far we have discussed the equilibrium properties of disordered systems, that are encoded in their partition function and free energy. In this set of problems, we characterize the energy landscape of a prototypical model, the spherical <math>p${\displaystyle }$-spin.

Key concepts: Langevin dynamics, gradient descent, oout-of-equilibrium dynamics, metastable states, Hessian matrices, random matrix theory.

Metastability and optimisation

- Energy landscapes. Consider p-spin. The function E defines a random energy landscape. The global minimum is the GS. We have seen that we can compute its energy density doing equilibrium calculation in the limit of zero T. There can be local minima of the energy landscape, see picture. In particular, glassy systems are characterized by the fact that there are plenty.

- consider situation in which we want the system to reach the global minimum in energy. start from an arbitrary configuration. want to define a dynamics such that after sufficiently long time the system is in GS. one possibility is gradient descent.

- moves towards configs where energy is smaller. dynamics is stuck when reach a conf stationary point (local minima, max, saddles). If the landscape is convex, then go to GS. If not, can end up in other stationary point, in particular local minima.

- can add a bit of noise: Langevin dynamics.