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 golf course energy landscape. 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 $P_{N}(E)\approx C_{N}{\text{exp}}\left[{\frac {(E-N{\sqrt {2\log 2}})}{\sqrt {2\log 2}}}\right]$, where $C_{N}$ is a normalisation factor. 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 of the trap? Does it depend on the entry of the trap itself? Why is this consistent with the results on the entropy of the REM that we computed in Problem 1.1?


2. Trapping times. The results above show that the energy landscape of the REM has a "golf course" structure: configuration with deep energy are isolated, surrounded by configurations of much higher energy (zero energy density). The Arrhenius law states that the time needed for the system to escape from a trap of energy $E$ and reach a configuration of zero energy is $\tau \sim e^{-\beta E}$. We call this a trapping time . Given the energy distribution $P_{N}(E)$, 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. Do you recognise this temperature?

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.