T-7

Goal: The goal of these problems is to understand some features of glassy dynamics (power laws, aging) in a simplified description, the so called trap model.
Techniques: extreme value statistics, asymptotic analysis.

A dynamical dictionary: out-of-equilibrium, aging

Fig. 7 - Behaviour of the correlation function in a system displaying aging.

• Equilibrating dynamics. A system evolving with thermal dynamics (e.g. Langevin dynamics) equilibrates dynamically if there is a timescale ${\tau }_{\text{eq}}$ beyond which the dynamical trajectories sample the configurations of the system $\overrightarrow{\sigma }$ with the frequency that is prescribed by the Gibbs Boltzmann measure, $\sim e^{-\beta E(\overrightarrow{\sigma })}$, where $\beta$ is the inverse temperature associated to the noise. At equilibrium, one-point functions in time, like the energy of the system, reach a stationary value (the equilibrium value predicted by thermodynamics at that temperature), while two-point functions like the correlation function $$C(t_w+t,t_w)=\frac{1}{N}\sum _{i=1}^N{\sigma }_i(t_w){\sigma }_i(t_w+t)$$ are time-translation invariant, meaning that $C(t_w+t,t_w)\sim c(t)$ is only a function of the difference between the two times, and does not depend on $t_w$.


• Out-of-equilibrium and aging. In some systems the equilibration timescale ${\tau }_{\text{eq}}$ is extremely large/diverging with some parameter of the model (like $N$), and for very large time-scales the dynamics is out-of-equilibrium . In glassy systems, out-of-equilibrium dynamics is often characterized by aging: 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, see Fig 7: the timescale that the system needs to leave the plateau increases with the age of the system $t_w$, meaning that the system is becoming more and more slow as it gets more and more old.



Problems

Problem 7: a simple model for aging



Fig 6.3 - Traps in the trap model.

The trap model is an abstract model for the dynamics in complex landscapes studied 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 $$P_{\mu }(\tau )=\frac{\mu {\tau }_0^{\mu }}{{\tau }^{1+\mu }}\tau \ge {\tau }_0$$ where $\mu$ is a parameter. In this exercise, we aim at understanding the main features of this dynamics.





1. Ergodicity breaking and condensation. Compute the average trapping time (averaging between the traps) and show that there is a critical value of $\mu$ 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 exactly a 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?


2. Aging 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 $$C(t_w+t,t_w)=q_{EA}\mathrm{\Pi }(t,t_w)+q_0(1-\mathrm{\Pi }(t,t_w)),\mathrm{\Pi }(t,t_w)=\text{probability that systems has not jumped in }[t_w,t_w+t].$$ In the non-ergodic regime, one finds: $$\mathrm{\Pi }(t,t_w)=\frac{\sin (\pi \mu )}{\pi }\underset{\frac{t}{t+t_w}}{\overset{1}{\int }}du(1-u{)}^{\mu -1}{u}^{-\mu }.$$ Why is this an indication of aging? Show that $$\underset{t\to \mathrm{\infty }}{\lim }C(t_w+t,t_w)=q_0\text{ for finite }{t}_w,\underset{t_w\to \mathrm{\infty }}{\lim }C(t_w+t,t_w)=q_{EA}\text{ for finite }t$$ When $q_0=0$, this behaviour is called "weak ergodicity breaking".


3. Extra. Power laws. Study the asymptotic behavior 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.







Check out: key concepts and exercises

Key concepts: aging, activation, time-translation invariance, out-of equilibrium dynamics, power laws, decorrelation, condensation, extreme values.


In Exercise 13 , you will see in which sense the trap model is a good effective model to describe a dynamics exploring a complicated energy landscape with many metastable states, focusing on the REM landscape as an example.

To know more

• Bouchaud. Weak ergodicity breaking and aging in disordered systems [1]
• Biroli. A crash course on aging [2]
• Kurchan. Six out-of-equilibrium lectures [3]