T-6

Goal: The goal of these problems is to understand some crucial features of glassy dynamics (power laws, aging) in a simplified single particle description, the so called trap model.

Langevin dynamics, energy barriers, power-law decay, aging

Noise and Langevin dynamics. In problems 5 we have characterized the energy landscape of the spherical p-spin, and showed that it is Madde by plenty of stationary points where gradient descent gets stuck. How to modify the dynamics to escape from a given local minimum and explore other regions of the energy landscape? One possibility is to add some stochasticity (or noise), i.e. some random terms that kick the systems in random directions in configuration space, towards which maybe the energy increases instead of decreasing: ${\frac {d{\vec {\sigma }}(t)}{dt}}=-\nabla E({\vec {\sigma }})+{\vec {\eta }}(t)$
The simplest choice is ${\vec {\eta }}(t)$ a Gaussian vector at each time $t$ , uncorrelated from the vectors at other times $t'\neq t$ , with zero average and some constant variance. This variance, which measures the strength of the noisy kicks, can be interpreted as a temperature: the resulting dynamics is known as Langevin dynamics .

Activation and Arrhenius law. When the noise in the Langevin dynamics is weak (temperature is small), the dynamics does not get stuck in local minima forever, but for very large time. This time depends crucially on the energy barrier which separate the minimum from the other configurations (see sketch). The Arrhenius law states that the typical timescale $\tau$ required to escape from a local minimum through a barrier of height $\Delta E$ with thermal dynamics with inverse temperature $\beta$ scales as $\tau \sim \tau _{0}e^{-\beta \,\Delta E}$ . A dynamics made of jumps from minimum to minimum through the crossing of energy barriers is called activated .

Fast and slow dynamics. An important quantity to characterize the dynamics is the correlation function
$C(t_{w}+t,t_{w})={\frac {1}{N}}\sum _{i=1}^{N}S_{i}(t_{w})S_{i}(t_{w}+t).$

When the dynamics is fast, this decays exponentially to zero. This is what happens during the dynamical evolution of glassy models in the high-temperature phase. Slow dynamics corresponds to the fact that the correlation is decaying much slower, as a power law in time, towards an asymptotic value. This is what happens in the glassy phase. When ergodicity breaking occurs, the asymptotic value is $q_{EA}$ .
Aging. Aging is a property of the dynamics, which characterized the dynamics of glasses. It means that the relaxation timescale of a system (how slow the system evolves) depends on the age of the system itself (on how long the system has evolved so far). Aging can be seen in the behaviour of correlation function...

Problem 6.1: a simple model for aging

Traps in the trap model.

The trap model is an abstract model for the dynamics in complex landscapes introduced in ^[1]. The configuration space is a collection of $M\gg 1$ 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 $dt/\tau _{\alpha }$ .). When the system exits the trap, it jumps into another one randomly chosen among the $M$ . The average times are distributed as

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

where $m$ is a parameter. In this exercise, we aim at understanding the main features of this dynamics.

Aging. Compute the average trapping time (between the various traps) 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, assuming that the system has spent time $\tau _{\alpha }$ in each visited trap $\alpha$ . 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?

Correlation functions and slow dynamics. 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:
$\Pi (t,t_{w})={\frac {\sin(\pi m)}{\pi }}\int _{\frac {t}{t+t_{w}}}^{1}du(1-u)^{m-1}u^{-m}.$
Why is this, again, an indication of aging? Study the asymptotic behaviour of the correlation function for $t\ll t_{w}$ and $t\gg t_{w}$ and show that the dynamics is slow, characterized by power laws. Show that
$\lim _{t\to \infty }C(t_{w}+t,t_{w})=q_{0}\quad {\text{ for finite }}t_{w},\quad \quad \lim _{t_{w}\to \infty }C(t_{w}+t,t_{w})=q_{EA}\quad {\text{ for finite }}t$ When $q_{0}=0$ , this behaviour is called "weak ergodicity breaking".

Problem 6.2: from traps to landscapes

In this exercise, we aim at understanding why the trap model is a good effective model for the exploration of the energy landscape of two models that we have studied so far: the REM and spherical p-spin model. While for the p-spin we think about Langevin dynamics, for the REM we can not due to the discreteness of configurations. We will consider Monte Carlo dynamics: at each time step the system in a given configuration ${\vec {S}}$ with energy $E_{1}$ tries to transition to another configuration that differs with respect to the previous one by a single spin flip; let the energy of this second configuration be $E_{2}$ . The transition occurs with probability one if $E_{2}<E_{1}$ , and with probability $e^{-\beta (E_{2}-E_{1})}$ otherwise.

REM: the golf course landscape. In the REM, the smallest energies values $E_{\alpha }$ among the $M=2^{N}$ can be assumed to be distributed as
$P_{N}^{\text{extrm}}(E)=C_{N}{\text{exp}}\left[2{\sqrt {\log 2}}(E+N{\sqrt {\log 2}})\right],\quad \quad E<0,\quad \quad C_{N}{\text{ normalization}}$
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 energy close to that of the ground state: what is the minimal energy among the $N$ neighbouring configurations, which differs from the previous one by a spin flip? Does it depend on the energy of the original one? Why is this consistent with the results on the entropy of the REM that we computed in Problem 1.1?

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 density $\epsilon <0$ and reach a configuration of zero energy density is $\tau \sim e^{-\beta N\epsilon }$ . This is a trapping time. Given the energy distribution $P_{N}(E)$ , determine the distribution of trapping times $P_{m}(\tau )$ : what plays the role of $m$ ? DDO you recognise a critical temperature?

p-spin: the “trap” picture and the assumptions behind. In Problems 5, we have seen that the energy landscape of the spherical p-spin is characterized by the threshold energy, below which plenty of minima appear. Explain why the tap model corresponds to the following picture for the dynamics: the system is trapped into minima below the threshold for exponentially large times, and then jumps from minimum to minimum passing through the threshold energy. Do you see any assumption behind the trap description that is not straightforwardly justified in the p-spin case?