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.Langevin dynamics.

-Arrhenius law, trapping and activation.

-aging

The energy landscape of the REM. 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.

Problem 6.1: the trap model, and its foundations

The trap model is an abstract, effective model for the dynamics in complex landscapes. The configuration space is described as a collection of $M=2^{N}$ traps labeled by $\alpha$ having random depths/energies, see sketch. The dynamics is a sequence of jumps between the traps: the system spends in a trap $\alpha$ an exponentially large time with average $\tau _{\alpha }$ (the probability to jump out of the trap in time $[t,t+dt]$ is $dt/\tau _{\alpha }$.). The average times are distributed as

$P_{m}(\tau )={\frac {m\tau _{0}^{m}}{\tau ^{1+m}}}$

where $m$ is a parameter. When the system exits the trap, it jumps into another one randomly chosen among the $M=2^{N}$. In this exercise, we aim at understanding the main features of this dynamics and at connecting it to models discussed before, like the REM and the $p-$spin.

1. Aging. Compute the average trapping time and show that there is a critical value of $m$ below which it diverges, signalling a non-ergodic phase (the system needs infinite time to explore the whole configuration space). Consider a dynamics running from time $t_{w}$ to some later time $t_{w}+t$: compute the typical value of the maximal trapping time $\tau _{\text{max}}(t)$ encountered in this time interval, an show that in the non-ergodic phase $\tau _{\text{max}}(t)\sim t$. Why is this interpretable as a condensation phenomenon, as the ones discussed in Problems 1? Why is this an indication of aging?


2. Correlation functions and weak ergodicity breaking. Assume now that the trap represent a collection of microscopic configurations having self overlap $q_{EA}$. Assume that the overlap between configurations of different traps is $q_{0}$. Justify why the correlation function can be written as $C(t_{w}+t,t_{w})=q_{EA}\Pi (t,t_{w})+q_{0}\left(1-\Pi (t,t_{w})\right),\quad \quad \Pi (t,t_{w})={\text{probability that systems has not jumped in }}[t_{w},t_{w}+t].$

   In the non-ergodic regime, one finds:

   $\Pi (t,t_{w})={\frac {\sin(\pi m)}{\pi (1-m)}}\int _{\frac {t}{t+t_{w}}}^{1}du(1-u)^{m-1}u^{-m}.$

   Study the asymptotic behaviour of the correlation function for $t\ll t_{w}$ and $t\gg t_{w}$. Show that

   $\lim _{t\to \infty }C(t_{w}+t,t_{w})=0\quad {\text{ for finite }}t_{w},\quad \quad \lim _{t_{w}\to \infty }C(t_{w}+t,t_{w})=q_{EA}\quad {\text{ for finite }}t$

   This is called "weak ergodicity breaking".
   

3. REM: the golf course landscape. We now want to argue that the trap model is a good representation for the dynamics of the REM. In the REM, the smallest energies values $E_{\alpha }$ among the $M=2^{N}$ can be assumed to be distributed as $P_{N}(E)\approx C_{N}{\text{exp}}\left[2{\sqrt {\log 2}}(E+N{\sqrt {\log 2}})\right],\quad \quad E<0$ 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 (or trap) of very small energy: what is the minimal energy density among the $N$ neighbouring configurations, which differs from the previous one by a spin flip? Does it depend on the enery of the trap itself? Why is this consistent with the results on the entropy of the REM that we computed in Problem 1.1?


4. REM: 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 $\tau \sim e^{\beta |E|}$. This is a trapping time. Given the energy distribution $P_{N}(E)$, determine the distribution of trapping times $P_{m}(\tau )$: what is $m$? What is the critical temperature?
5. p-spin: the “trap” picture of long time dynamics.

the system has spent almost all the time up to $t$ in only one trap, the deepest one.

the age of the system is also the typical timescale for its evolution