Mean field theory of the swap Monte Carlo algorithm

Abstract

The swap Monte Carlo algorithm combines the translational motion with the exchange of particle species, and is unprecedentedly efficient for some models of glass former. In order to clarify the physics underlying this acceleration, we study the problem within the mean field replica liquid theory. We extend the Gaussian ansatz so as to take into account the exchange of particles of different species, and we calculate analytically the dynamical glass transition points corresponding to the swap and standard Monte Carlo algorithms. We show that the system evolved with the standard Monte Carlo algorithm exhibits the dynamical transition before that of the swap Monte Carlo algorithm. We also test the result by performing computer simulations of a binary mixture of the Mari-Kurchan model, both with standard and swap Monte Carlo. This scenario provides a possible explanation for the efficiency of the swap Monte Carlo algorithm. Finally, we discuss how the thermodynamic theory of the glass transition should be modified based on our results.

Bypassing sluggishness: SWAP algorithm and glassiness in high dimensions

“…How can one explain the relatively rapid suppression of swap efficiency despite of the slow decay of the density gap (ϕ swap d − ϕ std d )/ϕ std d to a nonzero value? While the relative increase of ϕ d is qualitatively consistent with mean-field treatments in d = 3 [17,18], the saturation and the asymptotic behavior of the gap with d were not anticipated. Plugging this result into the critical scaling forms τ α = A(ϕ d − ϕ) −γ , we obtain an approximate expression for the efficiency ratio…”

Does the Adam-Gibbs relation hold in simulated supercooled liquids?

“…This is not a small accomplishment. One can take the alternative view that the deviations from the canonical exponent values should be taken as an indirect sign that thermodynamics only contributes some part of the slowing down, in addition to other physical factors [105][106][107][108][109][110]. This view is sometimes also invoked to rationalise the "modest" growth of static correlation lengthscale observed numerically and experimentally [89,111].…”

Accelerated relaxation and suppressed dynamic heterogeneity in a kinetically constrained (East) model with swaps

“…To see this, we consider the sequence of processes obtained by reading from left to right in Fig. 1): the first factor in (20) is the rate for the initial transition in Fig. 1, the second factor is the probability that the excitation is destroyed (second step) before the system reverts back to the first state.…”

“…This work has enabled new computational studies of glassy states at low temperatures [14,15,16]. Several theoretical works have proposed explanations of the effects of swap dynamics [17,18,19,20,21]. This strong dependence of relaxation time on dynamics is unexpected within thermodynamic theories [17], although explanations have been proposed, within an RFOT-like picture [20,19].…”

“…Several theoretical works have proposed explanations of the effects of swap dynamics [17,18,19,20,21]. This strong dependence of relaxation time on dynamics is unexpected within thermodynamic theories [17], although explanations have been proposed, within an RFOT-like picture [20,19]. In dynamical facilitation theory, it is natural to expect strong dependence of time scales on dynamics, but it is not clear why MC moves that swap particle sizes would have such a dramatic effect.…”