T-6

Goal: Trap model. Captures aging in a simplified single particle description.

Key concepts: gradient descent, rugged landscapes, metastable states, Hessian matrices, random matrix theory, landscape’s complexity.

Langevin, Activation

- Monte Carlo dynamics. Consider the REM discussed in Problems 1. Assume that the $2^{N}$ configurations are organised in an hypercube of connectivity $N$: each configuration has $N$ neighbours, that are obtained flipping one spin of the first configuration. -Langevin dynamics -Arrhenius law, trapping and activation.

The energy landscape of the REM.

Problem 6.1: from the REM to the trap model

1. . Consider the energy landscape of the REM. Justify why the smallest energies values $E_{\alpha }$ among the $2^{N}$ ones, those with energy density $\epsilon \sim -{\sqrt {\log 2}}$, are distributed according to $P(E)$

   - Consider the smallest energy values $E_{\alpha }$ among the $2^{N}$ ones, those with energy density $\epsilon \sim -{\sqrt {\log 2}}$: what is their distribution? (Hint: recall extreme value statistics discussed in Lecture 1)

   - Assume to be in a configuration of given (very small) energy $E_{\alpha }$: what is the minimal energy density among the neighbouring configurations? Does it depend on $E_{\alpha }$? In which sense the energy landscape of the REM has a golf course structure?
   

   
   The trap model is an effective model for the dynamics, which mimics the exploration of energy landscapes in which there are plenty of minima separated by high energy barriers. ....

   -Compute the distribution of trapping times: when is it fat-tailed?

   - consider now a dynamics from time $t=0$ to time $t=t_{w}$. Assume that in this time interval the system has visited $N(t_{w})$ distinct traps. Show that the system has spent almost all the time up to $t_{w}$ in only one trap, the deepest one.