A simple analytical description of the non-stationary dynamics in Ising spin systems

Abstract

The analytical description of the dynamics in models with discrete variables (e.g. Ising spins) is a notoriously difficult problem, that can be tackled only under some approximation. propose to use the same approximation based on the cluster variational method also for the non-stationary regime, which has not been considered up to now within this framework. We check the validity of this approximation in describing the non-stationary dynamical regime of several Ising models defined on Erdos-Rényi random graphs: we study ferromagnetic models with symmetric and partially asymmetric couplings, models with random fields and also spin glass models. A comparison with the actual Glauber dynamics, solved numerically, shows that one of the two studied approximations (the so-called 'diamond' approximation) provides very accurate results in all the systems studied. Only for the spin glass models we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states. Given the simplicity of the equations to be solved, we believe the diamond approximation should be considered as the 'minimal standard' in the description of the non-stationary regime of Ising-like models: any new method pretending to provide a better approximate description to the dynamics of Ising-like models should perform at least as good as the diamond approximation.2

Cluster Variational Approximations for Structure Learning of Continuous-Time Bayesian Networks from Incomplete Data

“…where the bound F ≡ −D KL (Q(X)||P (X)P (Y | X)) is also known as the Kikuchi functional [10], or the Kikuchi variational energy. The Kikuchi functional has recently found heavy use in variational approximations for probabilistic models [21,20,16], because of the freedom it provides for choosing clusters in space and time. We will now make use of this feature.…”

“…Indeed, if we replace our star-shape cluster by the completely local one A mf j (t), we recover exactly their previous result, demonstrating the generality of our method (see Appendix B.3). In principle, higher-order clusters can be considered [20,16]. Lastly, we enforce continuity by (3) fulfilling the constraint.…”

“…Our method combines two key ideas. We extend the framework of variational inference for multi-component Markov chains by borrowing results from statistical physics on cluster-variational-methods [21,20,16]. Here the previous result in [5] is recovered as a special case.…”

The matrix product approximation for the dynamic cavity method

“…While this can be expected to work well for stationary states at high temperatures, such approximations are usually severe for short to intermediate times or low temperatures. Other approximative approaches are the cluster variational method [28][29][30] (applicable for short-range spatio-temporal correlations) or perturbative schemes [25,31,32], generating functional analysis [33][34][35][36][37], and the generalized mean field approximation [38].…”

Matrix product algorithm for stochastic dynamics on networks applied to nonequilibrium Glauber dynamics

“…While this can be expected to work well for stationary states at high temperatures, such approximations are usually quite severe for short to intermediate times or low temperatures. Also, for dense networks, where the cavity method is not applicable, approximation schemes like the cluster variational method [18][19][20] or perturbative schemes [15,21,22] have been developed.…”