Activated dynamics of the Ising p-spin disordered model with finite number of variables

Abstract

We study the dynamic and metastable properties of the fully connected Ising p-spin model with finite number of spins, with a focus on activated dynamics and trap-like characteristics. We propose a definition of trapping regions based on purely dynamical criteria. We compute trapping energies, trapping times and self correlation functions and we analyse their statistical properties in comparison to the predictions of the well-known Bouchaud trap model.

Dynamical instantons and activated processes in mean-field glass models

“…This allows us to get some insight on the kind of dynamics one should expect in the activated regime, at least for the particular model we are considering: indeed, it is natural to expect that these saddles will matter in the earlier times of the dynamics, and that escaping through them the system would undergo a back and forth motion with frequent returns to the original minimum, given that the energy barrier associated to going back to the reference minimum is extensively lower. Such frequent returns have been recently observed in numerical simulations of the low-temperature dynamics of the Ising p-spin model of finite-size [13,47]. In [47] in particular it is shown that most of the stable configurations (the analogous of local minima in a discrete setting) that the system visits consecutively in its activated dynamics have a large overlap with each others; moreover, it appears that the system has to climb higher in the energy landscape in order to reach stable configurations that are less correlated with the previous one, consistently with our finding that the minima at smaller overlap with the reference one are connected to it by saddles at higher energy density (at least when focusing on the saddles at larger overlap and zero complexity, see Fig.…”

“…An activated regime in which the system jumps over increasingly larger barriers and is able to explore fully the energy landscape. To observe such activated processes in mean-field models one has to probe the dynamics on time-scales which are exponentially large in N since the barriers between low energy metastable states scale as N [10][11][12][13]. Whereas a theory of the first dynamical regime has been progressively developed in the last twenty years, constructing a theoretical framework to understand the second one remains an open problem-a central one in many of the contexts in which rough energy landscapes play a role.…”

Effective traplike activated dynamics in a continuous landscape

“…Barrier crossing is, in fact, also possible in the mean-field approximation, provided that the system size N is kept finite [4,5]. Keeping N finite makes calculations especially hard, so, save for some exceptions [6][7][8], most work on activated dynamics consists of numerical simulations [9][10][11].…”

“…The first of these two issues was successfully addressed by showing that some problems from number theory can be reformulated as physics problems on a lattice, which behave like the TM [19,20]. On the other side, the TM paradigm of activated dynamics is probably not suitable for the description of most systems with strong enough correlations [11,21,22]. One must therefore try to understand the limits of the applicability of the TM paradigm, and whether it is possible to create a connection between the TM and other systems such as sphere packings.…”

“…This concept of threshold energy is related to its definition in the TM, as the energy to reach to jump barriers and have renewal dynamics[13]. Operative methods for the detection of the threshold energy based on the behavior of p-spin-like models (e.g., those used in Refs [11,31]…”