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. . The smallest energies values $E_{\alpha }$ among the $M=2^{N}$, those with energy density $\epsilon =-{\sqrt {\log 2}}$, can be assumed to be distributed as $P_{N}(E)=C_{N}e^{{\frac {1}{\sqrt {2\log 2}}}\left(E-N{\sqrt {2\log 2}}\right)}$. Justify the form of this distribution (Hint: recall the discussion on extreme value statistics in Lecture 1!). Consider now one of these deep configurations of very small energy, which we will call <em< traps : what is the minimal energy density among the neighbouring configurations? Does it depend on $E_{\alpha }$? Why is this consistent with the results on the entropy of the REM that we computed in Problem 1.1? In which sense the energy landscape of the REM can be said to have a "golf course structure"?


2. The Arrhenius law states that the average time needed for the system to escape from a trap of energy $E$ and reach a configuration of zero energy under thermal dynamics is $\tau \sim e^{-\beta E}$. We call this a trapping time . Given the energy distribution of the energies of the deepest traps computed above, determine the distribution of trapping times. Show that there is a critical temperature below which the average trapping time diverges, and therefore the system needs infinite time to explore the whole configuration space.

The trap model is an effective model for the dynamics in complex landscapes. In this toy model, the configuration space is a collection of traps of randomly distributed depth, as 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.