Accelerated relaxation and suppressed dynamic heterogeneity in a kinetically constrained (East) model with swaps
AAccelerated relaxation and suppressed dynamicheterogeneity in a kinetically constrained (East)model with swaps
Ricardo Guti´errez
Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933M´ostoles, Madrid, Spain
Juan P. Garrahan
School of Physics and Astronomy and Centre for the Mathematics andTheoretical Physics of Quantum Non-equilibrium Systems, University ofNottingham, Nottingham NG7 2RD, UK
Robert L. Jack
Department of Applied Mathematics and Theoretical Physics, University ofCambridge, Wilberforce Road, Cambridge CB3 0WA, United KingdomDepartment of Chemistry, University of Cambridge, Lensfield Road, CambridgeCB2 1EW, United Kingdom
Abstract.
We introduce a kinetically constrained spin model with a localsoftness parameter, such that spin flips can violate the kinetic constraint withan (annealed) site-dependent rate. We show that adding MC swap moves tothis model can dramatically accelerate structural relaxation. We discuss theconnection of this observation with the fact that swap moves are also able toaccelerate relaxation in structural glasses. We analyse the rates of relaxation inthe model. We also show that the extent of dynamical heterogeneity is stronglysuppressed by the swap moves.
1. Introduction
The glass transition is a dynamical phenomenon [1, 2, 3]. In the supercooled regime,the structural relaxation time of a typical liquid behaves as τ α = τ exp (cid:18) A ( T ) k B T (cid:19) (1)where τ is a microscopic relaxation time, and A ( T ) is a function (with units of energy)that increases on cooling. It is natural to interpret A ( T ) as a free energy barrierassociated with structural relaxation, and a central aim of any theory of the glasstransition is to explain the temperature-dependence of this quantity. Some theories,including random first-order transition theory [2, 4] and other mean-field theoriesof the glass transition [5] propose that A ( T ) can be determined on thermodynamicgrounds – the idea is that different amorphous states of the glass are separated bylarge free energy barriers, which must necessarily be crossed, in order for structural a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p ccelerated relaxation in a kinetically constrained model with swaps A ( T ) may be expected to depend stronglyon the dynamical rules by which a glassy system evolves in time.For atomistic systems, it is known that many properties of supercooled liquidsare independent of microscopic details of the dynamics [1, 2, 3]. For example, systemswhere particles evolve by Newton’s equations have very similar properties to thosewith Monte Carlo (MC) dynamics, and to overdamped Langevin dynamics [7]. Thisobservation is consistent with thermodynamic pictures – the interpretation is that thesystems equilibrate quickly in local minima of the free energy, and the mechanisms bywhich they escape from these minima are controlled by thermodynamic barriers (whichare independent of dynamics). However, recent studies have shown that structuralrelaxation of some supercooled liquids can be accelerated by many orders of magnitude,by the simple addition of an extra dynamical process (MC move), in which particlesof different sizes can swap their locations [8, 9, 10, 11, 12]. This has allowed theequilibration (by computer simulation) of supercooled liquid states with extremelylarge viscosities, comparable with experimental glasses [13]. This work has enablednew computational studies of glassy states at low temperatures [14, 15, 16]. Severaltheoretical 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 withinthermodynamic theories [17], although explanations have been proposed, within anRFOT-like picture [20, 19]. In dynamical facilitation theory, it is natural to expectstrong dependence of time scales on dynamics, but it is not clear why MC moves thatswap particle sizes would have such a dramatic effect. The predictions of facilitationtheory are based on kinetically constrained models (KCMs) [22, 23], in which thedegrees of freedom occupy the sites of a lattice, such as Ising spins in the caseof facilitated spin models. In such models, spin i can only change its state if theneighbouring spins satisfy a constraint [22, 23]. In this article, we consider softenedKCMs [24, 25, 26], in which the local structure of the liquid is accounted for by anadditional variable on each site of the lattice, which we call the (local) softness. If thisquantity is large, there is a substantial probability that the system can relax locally, byviolating the kinetic constraint. We will show that even if the softness has a minimaleffect on the natural dynamics of the model, introducing MC moves that swap thevalues of the local softness can dramatically accelerate relaxation.The resulting models are simple ones, but we argue that the mechanism foracceleration by swaps may be general. In a deeply supercooled liquid, most regionsof the system are characterised by very large free energy barriers, but there are afew excitations that facilitate local motion [6, 27]. The swap mechanism acceleratesdynamics because a region with a large local barrier can become soft via a swap move,which then enables relaxation. In an atomistic system, one can imagine that a regionmight temporarily swap its particles for smaller ones; it can then relax, and then swapback the small particles for typically-sized ones. That is, the free energy barrier forlocal relaxation can be lowered by a fluctuation in the local structure, which triggers ccelerated relaxation in a kinetically constrained model with swaps
2. Model: Soft East KCM with swaps
We describe several variants of the East model, with a local softness variables, and weexplain how swaps are introduced as part of the set of rules that govern its dynamics.The local softness may be either a positive real number, or a binary variable. Bothmodels behave similarly at low temperatures: the variant with binary softness isextensively used in the numerical explorations of Section 4. Different variants of themodel require slightly different sets of parameters: these are summarised in Sec. 2.4,below. We emphasise that the main results of this work are robust across a broadrange of parameters.
The East model [28, 22, 23] consists of N binary spins n i ∈ { , } for i = 1 , , . . . , N .For simplicity, we consider the one-dimensional case with periodic boundaries, so weidentify spin 0 with spin N and spin N + 1 with spin 1. In contrast to the standardEast model [28, 22, 23], we consider a version where the kinetic constraint is “soft”,cf. [24, 25, 26, 29, 30]. The model has two controlling parameters, (cid:15) and c , with (cid:15) ≥ < c <
1. At site i we define the softened kinetic constraint C i = n i − + (cid:15) (2)If spin i has n i = 1 then it flips to state n i = 0 with rate C i ; if n i = 0 then spin i flips (to n i = 1) with rate C i c/ (1 − c ). The standard “hard” East model [28, 22, 23]occurs for (cid:15) = 0, in which case spin i is only updated when it is facilitated by its leftneighbour. For (cid:15) > (cid:15) softens the constraint. The physical idea is that facilitated spins flipwith a rate of order unity (“facilitated mechanism”) but all spins can also flip by anadditional “soft mechanism”, with a rate (cid:15) that is small at low temperatures.In general, the parameters c, (cid:15) may both depend on temperature. We take c = 11 + e J/T (3)so that the system obeys detailed balance with respect to a Boltzmann distributionwith energy E = J (cid:80) i n i (this fact is independent of (cid:15) ). This Boltzmann distributioncorresponds to independent spins with (cid:104) n i (cid:105) = c . (4) ccelerated relaxation in a kinetically constrained model with swaps (cid:15) with an energy barrier U that may alsodepend on temperature, cf. [25, 29, 30]: (cid:15) = e − U ( T ) /T . (5)The relaxation time of these models can be defined by considering the spin-spincorrelation function (cid:104) n i ( t ) n i (0) (cid:105) − (cid:104) n i (cid:105) , or the spectral gap of the generator. For lowtemperatures (small c ), the relaxation time of the hard ( (cid:15) = 0) East model is [31, 32] τ East ∼ exp (cid:18) J T ln 2 + bJT (cid:19) (6)where b is a constant (of order unity). The dominant contribution to the relaxationtime is the super-Arrhenius term so the value of b is relatively unimportant in thefollowing. We note at this point that this super-Arrhenius scaling appears becausefor an East model with (typical) excitation density c , the (free)-energy barrier for localrelaxation scales as T (log c ) [31, 32], so one may alternatively write (6) as τ East ∼ exp (cid:18) (log c ) b JT (cid:19) , (7)where b is a constant of order unity. The relaxation time of the softened East modelscales as τ soft ∼ min ( τ East , /(cid:15) ) . (8)Note also that in Sections 3.3 and 4, we also consider persistence times, whichare typically of a similar order of magnitude to the relaxation time. We now introduce a fluctuating local softness. This is a qualitative departure fromprevious work on the soft East model, where the softness (cid:15) was spatially uniform andconstant in time. On each site i , we define an additional variable X i , with units ofenergy. We therefore replace the energy barrier U in (5) by a site-dependent barrierwhich we write as U i = (cid:26) B − X i , X i < B , otherwise (9)where B = B ( T ) is a parameter of the model, and X i is a (positive) energy associatedwith site i . Physically, the energy barrier for the soft constraint is at most B ; a largevalue of X i means that the local energy barrier is (relatively) small. The model obeysdetailed balance with respect to a Boltzmann distribution with ET = JT (cid:88) i n i + 1 v ( T ) (cid:88) i X i (10)where v ( T ) is a (temperature-dependent) parameter. This distribution is such thatthe spin variables and the local softness parameters are all independent with (cid:104) n i (cid:105) = c = 11 + e J/T , (cid:104) X i (cid:105) = v ( T ) . (11)The probability density for X i is (for X i > p ( X i ) = (1 /v )e − X i /v . (12) ccelerated relaxation in a kinetically constrained model with swaps The dynamics of the model obeys detailed balance withrespect to a Boltzmann distribution with energy (10). The spin n i has the dynamicsof the soft-East model, with constraint function C i = n i − + min(1 , e − ( B − X i ) /T ) (13)consistent with (2,9). Note that flips of spin i leave the associated X i unchanged.The dynamics of the X -variables is as follows: if n i = 1 then X i is updated(with rate r X ) to a new value drawn from the (equilibrium) exponential distribution p ( X ) = (1 /v )e − X/v . If n i = 0 then X i does not update. The rate r X is a parameterof the model. The physical interpretation is that if there is an excitation on site i ( n i = 1) then the liquid structure at that site is exploring configuration space rapidly,so the local barrier X i is free to change. On sites with n i = 0 then the local dynamicsare very slow, and the X i cannot change, until such time as the n i variable flips. For atomistic systems, the swap algorithm of [13] meansthat the local softness in a given region of the system can change without requiringstructural relaxation. For example, the size of the particles in that region mightbe reduced, which effectively reduces the softness barrier, and facilitates structuralrearrangement. To mimic this, we introduce an extra process by which the softnessvariables X i can be updated.We consider three possible choices for this extra process. The first is called “s-swaps” (short for softness-swaps): with rate N r s , we choose two sites at random,and exchange their values of X i . The second is called “s-updates”: with rate N r u we choose a single site at random and update the local softness X i to a new valuedrawn from the (equilibrium) exponential distribution p ( X ) = (1 /v )e − X/v , similarto what was proposed in [18]. Note that this can happen on any site, independentof the value of n i . The factors of N in these rates are chosen so that the rates forindividual spin flips are independent of system size. The third type of swap process isconsidered in Section 4, namely, local swaps, which are s-swaps that only take placebetween neighbouring sites. In this case, a random site is chosen with rate N r l , andits X -value is swapped with one of its neighbours (chosen at random).In a large system, s-swaps and s-updates lead to identical behaviour in theequilibrium state: this will be demonstrated explicitly in Fig. 5, below. To understandthis, note first that they both obey detailed balance with respect to the Boltzmanndistribution of (10). Second, note that if one considers separately the two spinsparticipating in a swap, each one is updated to a value that is copied from the otherspin, which will typically be far away (hence uncorrelated) and for which the softnessdistribution is the just same exponential used in an s -update. In other words, fromthe point of view of a single spin i , an s-update is equivalent to swapping X i with avalue chosen at random from a (fictitious) reservoir of softness values; alternativelyone may swap the X i with a value from a (real) reservoir that is constituted by theother particles in the system (this is an s-swap). If the reservoirs are large enough thencorrelation effects can be neglected and these two processes are equivalent. Based onthis equivalence, we refer to both s-swaps and s-updates as types of (non-local) swapmoves.Finally, note that an s-swap conserves the total softness (cid:80) i X i while an s-updatedoes not. However, the local (no-swap) dynamics of the X -variables already allows thetotal softness to fluctuate, so s-updates do not cause the breakdown of any (global)conservation law. For this reason, the fact that s-swaps conserve the total softness is ccelerated relaxation in a kinetically constrained model with swaps In the following, we will show that the behaviour of the East model with localsoftness can be captured by an even simpler model, which we now describe. Forlow temperatures, the effect of the local softness is dominated by sites with X i > B .The fraction of sites that have this property is easily verified to be e − B/v .To exploit this fact, we replace each X i by a binary softness variable s i ∈ { , } ,such that sites with s i = 1 are soft. For consistency with the previous model, theconstraint function is C i = n i − + s i (14)and the Boltzmann distribution for the ( n i , s i ) has ET = JT (cid:88) i n i + B ( T ) v ( T ) (cid:88) i s i (15)The dynamics of the s -variables are the same as those of the X -variables, except thatwhere X i is updated with an exponentially-distributed random number, s i is updatedto zero or 1 with probability (1+e − B/v ) − and (1+e B/v ) − respectively. The no-swapdynamics and the three varieties of swap dynamics are all generalised in this way.Compared to the model with real-valued softness, this model introduces twosimplifications: First, the binary variables s i enable efficient computer simulationof the model, see the first paragraph of Sec. 4 below. Second, the two parameters B, v enter the binary model only through their ratio
B/v . Before continuing, we summarise the parameters introduced so far. We have chosen tomaintain all rates as free parameters, so that our analysis is general. For this reason,our summary here also indicates which parameters are most important for controllingthe qualitative behaviour of the model. • The parameter J is the (free)-energy cost required to create an East-likeexcitation, n i = 1. This sets the fundamental energy scale in the model. Inthe simulations below we set J = 1. • The parameter B = B ( T ) is the maximal energy barrier associated with the softconstraint. The temperature-dependence of B must be determined from physicalarguments. In numerical simulations we typically take B ( T ) ∝ /T – we expectit to grow at low temperatures since the soft process is expected to be very slowin the glassy regime. • The parameter v = v ( T ) determines the mean and the standard deviation ofthe softness parameter X i . In general v may depend on T but in numericalsimulations we take v to be independent of temperature. ccelerated relaxation in a kinetically constrained model with swaps • The parameter r X controls the rate with which the softness is updated on siteswith n i = 1. The time unit in the model is fixed by the facilitated process: therates for facilitated spin flips are 1 and c (for flips into state with n i = 0 and n i = 1 respectively). Since this is an excited site, one expects r X to be relativelyfast. In numerical simulations we take r X = e − J/T which is much faster thanthe structural relaxation but small enough that adding the soft process to thesimulation does not result in too much of an increase in the simulation efficiency. • The parameters r s , r u control the rates of (non-local) swap processes (Sec. 2.2.2).As for r X , it is convenient (both numerically and analytically) to assume thatthese rates are small compared to 1 but (very) large compared to the rate ofstructural relaxation. In numerical simulations we mostly take r s , r u ∝ e − J/T , inwhich regime the results depend weakly on the specific choice of r . • The parameter r l controls the rate of local swap processes. We take r l of the sameorder as r s , that is r l ∝ e − J/T . The results do depend significantly on this choice,as it determines the time scale of the diffusion of excitations, see Section 4.3.Note that the parameters r s , r u , r l appear in different variants of the model. Forany given variant, only one of these parameters needs to be specified.Anticipating the results that we derive below, we comment that the behaviourof the model depends strongly on B : if this parameter is too small then the softrelaxation process dominates the system and the system does not behave in a glassyway, regardless of whether swap moves are included. The dependence on otherparameters is much weaker. In particular, our main result – that s-swaps and s-updates dramatically accelerate relaxation – does not require tuning of parameters.This effect is robust as long as the relevant rates r are not too small and v is nottoo large. (These are the minimal constraints that are consistent with the underlyingphysics: If the r -rates are extremely small then the swap processes hardly ever happenso no acceleration is possible; also if v is too large then soft process completely destroysthe glassy behaviour, independent of whether swaps are present.) These matters arediscussed further in Sec. 3.
3. Theory
We first consider the behaviour of the East model with local softness, in the absenceof swaps. The typical barrier for the soft constraint is B − v . We assume that thisbarrier is large enough that the no-swap dynamics are controlled by the facilitated(East) relaxation process. This requires (as a necessary condition) that the typical barrier for the soft process is large, specifically B ( T ) − v ( T ) (cid:38) T log τ East . (16)For this condition to hold at low temperatures, it is necessary that B must increase as T is reduced. In this regime, the barrier for the soft process is large, and one expectsthe behaviour to be dominated by the facilitated process. However, the soft processmay still have important effects, which come from (non-typical) sites where X (cid:38) B ,as we now discuss.Any site with X i > B that has n i = 0 will flip with a rate of order unity intoa state with n i = 1. At that point, the value of X i will be updated, so it will(most likely) revert to a typical value X i (cid:39) v . This means that sites with X i > B ccelerated relaxation in a kinetically constrained model with swaps n i = 1 , X i ' v
Prob ⇠ e J/T B/v
Prob ⇠ e B/v
Prob ⇠
Effects of the soft process and the swap process on East modelconfigurations ( x indicates a spatial co-ordinate). Vertical bars indicate spins with n i = 1 (as indicated) and circles indicate n i = 0. Black colour indicates typicalvalues of X i , of order v . Grey colour indicates anomalously soft sites, X i > B . Forno-swap dynamics, the three configurations in the top row can interconvert by the r X process (which refreshes X i with rate r X ) and the soft process (which leadsto “fast” flips if X i > B ). The probabilities indicate the equilibrium probabilitiesof these states, assuming that e − J/T , e − B/v (cid:28)
1. If this interconversion is fastcompared to structural relaxation, one may interpret the three states as differentmanifestations of a single “effective excitation”, which spends a fraction f ofits time in the top right configuration. If swapping is enabled, the “vertical”process shown at right is also possible: this allows removal of the “effectiveexcitation” by a purely local process and dramatically accelerates the relaxation.[For no-swap dynamics, the effective excitation can only be destroyed by a co-operative (facilitated) process, or by a rare event where a spin with X i (cid:39) v flipsspontaneously by the soft process. If (25) is satisfied, then such events are muchrarer than the ones shown in this figure, and the swap move strongly acceleratesthe dynamics.] quickly convert into excited sites (with n i = 1). The reverse process is also possible:an excitation ( n i = 1) can convert to an unexcited but softened state ( n i = 0 but X i > B ). These processes are illustrated in the top row of Fig. 1. The overall effectof this interconversion is that long-lived “effective excitations” (or “superspins” [31])spend some of their time in the unexcited but softened state. (This argument requiresthat 1 /r X is much smaller than the lifetime of a typical effective excitation; that is,interconversion between the two states is faster than structural relaxation. We take r X ∼ e − J/T but structural relaxation at low temperatures is much slower, so this issatisfied in practice.)Now define θ i to be equal to unity if site i is soft ( X i > B ) and zero otherwise,so that (cid:104) θ i (cid:105) = e − B/v . Then the density of the effective excitations is the fraction ofsites with either n i = 1 or θ i = 1 (or both), which is c eff = (cid:104) n i + θ i − n i θ i (cid:105) . (17)Since both e − ( B/v ) and e − ( J/T ) are small then the (cid:104) n i θ i (cid:105) is negligible and we obtain c eff ≈ e − B/v + e − J/T . Hence one sees that the fraction of time spent by a effective ccelerated relaxation in a kinetically constrained model with swaps θ i = 1) is f = (cid:104) θ i (cid:105) c eff ≈ e − B/v e − B/v + e − J/T . Using (17) to write (cid:104) θ i (cid:105) ≈ c eff − c , one obtains c eff ≈ c − f ≈ c (cid:16) ( J/T ) − ( B/v ) (cid:17) where the approximate equalities are all valid up to terms of order e − ( B/v ) − ( J/T ) . Forthe model with binary softness, the behaviour is the same on replacing θ i with s i .We see that for the system to behave similarly to an East model (with c eff ≈ c ),we require at low temperatures thate − B/v (cid:28) e − J/T , (18)which also implies f ≈ e ( J/T ) − ( B/v ) (cid:28)
1. Hence, while (16) is necessary for the systemto behave similarly to the original (hard) East model, it is not sufficient, since (18) isalso required.Given (16,18) we expect [by analogy with (7)] that the relaxation time of themodels with local softness is τ noswap ∼ exp (cid:18) (log c eff ) b JT (cid:19) (19)This corresponds to a speedup of the East dynamics, due to the softening of theconstraint. For the parameters considered here we will have c eff /c ≈
1, so this effectis weak. That is, we focus in the following on a dynamical speedup effect that is dueto swaps (or updates) of the softness; it is not a simple consequence of the softenedconstraint. We note however that the very strong dependence of τ noswap on c eff meansthat this weak speedup is still observable even when the differences between c eff and c are small, see Sec. 4 and in particular Fig. 3. When swap dynamics are included, a new process becomesimportant. In this case, a spin with n i = 0 and X i > B can still interconvert with anexcitation by the mechanism described above. However, it is also possible that thisspin can swap its X -value with another spin for which X has a typical value (of order v ). This provides a mechanism by which an effective excitation can be converted toan unexcited state with ( n i , X i ) = (0 , v ). This is shown as the rightmost (vertical)process in Fig. 1. The net effect of the chain of processes in Fig. 1 is the same as that ofthe original soft process of Sec. 2.1: a site with n i = 1 can convert to n i = 0, withoutever becoming facilitated, and with typical values of the local softness in both theinitial and final states. For the model with s -update moves, the rate for spontaneousdestruction of an excitation can be estimated as (approximately) (cid:15) eff = r X e − B/v ·
11 + r X · r u r u + e − J/T . (20)To see this, we consider the sequence of processes obtained by reading from left toright in Fig. 1): the first factor in (20) is the rate for the initial transition in Fig. 1, thesecond factor is the probability that the excitation is destroyed (second step) before ccelerated relaxation in a kinetically constrained model with swaps s -swaps, a similar formula holds, with r u replaced by r s .By analogy with (8), one infers that the relaxation time in a system with (nonlocal)swaps is τ swap ∼ min (1 /(cid:15) eff , τ noswap ) . (21)The estimate in (20) is not expected to give quantitative predictions but it shouldcapture the scaling of (cid:15) eff . From Sec. 2.4 we have r X (cid:28) r u and e − J/T are of a similar order. The result is that the “effective softness parameter”is (cid:15) eff ≈ r X e − B/v . (22)This is valid for r u (cid:38) e − J/T and r X (cid:28)
1. If on the other hand r u (cid:28) e − J/T then (cid:15) eff ≈ r X r u e − B/v + J/T . (23)In the former case (which includes r u ≈ e − J/T ), we predict that τ swap ∼ min (cid:16) r − X e B/v , τ noswap (cid:17) . (24) Suppose that (16,18) hold, so that the dynamics without swapsdepends very weakly on the fact that the model is soft; and assume also that( τ noswap ) − (cid:28) (cid:15) eff (cid:28) . (25)Then (21) shows that the swap dynamics will be accelerated dramatically, comparedto the no-swap dynamics. That is1 (cid:28) τ swap (cid:28) τ noswap (26)This is the key theoretical result of this paper, which we verify in Sec. 4 bynumerical simulations. It compares two models which have the same (softened) kineticconstraint, the only difference is whether there are swap moves that update the localsoftness.As anticipated in Sec. 2.4, Equ. (26) requires some assumptions on the rates inthe model, particularly (16,18) which require that the constraint is not so soft thatall glassy behaviour is destroyed, independent of swaps. We also require (25) whichis an assumption on the r -rates (that is, r s , r u , r X ). From (18) we have e − B/v (cid:28) r -rates is extremelysmall, of the order of ( τ noswap ) − . Recall that ( τ noswap ) − is the rate for a highly-collective relaxation process and has a super-Arrhenius dependence on temperature.On the other hand, there is no physical justification for choosing very small values forthe r -rates, which are microscopic parameters, and should therefore be much largerthan ( τ noswap ) − at low temperatures. Hence (25) holds very generally, and (26) is arobust result, independent of specific choices of the rates r .As a specific case, we take (consistent with Sec. 2.4) that B ( T ) ∼ aJ /T with a > v is independent of temperature, then (16,18,25) are all satisfied, and wefind B/v ∝ /T . In this case the swap dynamics will have an Arrhenius temperaturedependence, while the no-swap dynamics would be super-Arrhenius. This case isdiscussed in more detail in Sec. 4.1, below.More generally, it is useful to consider the mechanism of acceleration by swaps.We assume that with some small probability, the system has a reduced energy barrier ccelerated relaxation in a kinetically constrained model with swaps n i directly. Rather, the (non-local) swap moves change the softness, which then triggers(local) relaxation. Note also that this mechanism is relatively insensitive to theprecise route by which softness is imported – this might be a non-local swap [13]or a deterministic local change in particle diameter [33], or any other mechanism foraccessing local configurations where the barrier for structural relaxation is reduced.It is also noteworthy that for the models considered here, the swap mechanismdoes not have any collective character. Just like the soft process in the model ofSec. 2.1, the swap-mediated relaxation can destroy or create excitations, independentlyof their environment. This means that (for example) the relaxation will be lessdynamically heterogeneous in the presence of the swaps: see also Fig. 4 below.However, the microscopic mechanism for the soft process in liquids should have somecollective character. The nature of this process is not explicit within the model: itpresumably determines parameters such as the size of the large energy barrier B andthe temperature-dependence of B/v . It is not possible to explore these effects withinthis model – one would (presumably) have to consider the liquid structure and theway that the particles are packed in space.
So far we have concentrated on the relaxation time, which we defined as the time scaleassociated with the decay of the the spin correlation function (cid:104) n i ( t ) n i (0) (cid:105) − (cid:104) n i (cid:105) . Fornumerical work it is also useful to consider the persistence function. This is defined interms of the local persistence field : the local variable p i ( t ) takes value p i ( t ) = 1 if n i has not flipped from its initial state at time 0 up to time t , otherwise p i ( t ) = 0. Thepersistence function is P ( t ) = (cid:104) p i ( t ) (cid:105) (27)where the right hand side is independent of i , by translational invariance. Thebehaviour of this function has been studied extensively in kinetically constrainedmodels [23, 34, 35, 36, 37]. The persistence time τ p corresponds to the typical timescalefor the decay of P ( t ). We define it through a threshold P ( τ p ) = 10 − . (28)(Other definitions, such as through the time integral of P , give results that scalesimilarly with the parameters of the model.) Our numerical work mostly focusses onthe persistence function instead of the autocorrelation, as it is numerically easier toestimate and contains similar information. For (hard) kinetically constrained models,the persistence and correlation times are similar. However, it is important to notethat when relaxation occurs by the soft process, the persistence time is larger thanthe relaxation times that we have considered so far. ccelerated relaxation in a kinetically constrained model with swaps J/T . The physicalreason is that the autocorrelation time is equal to the typical time taken for a typicalspin with n i = 1 to decay to n i = 0, which is the inverse of the rate for the slow process, (cid:15) − . After a few multiples of this time, the state of the system has decorrelated.However, spins with n i = 0 are much more numerous at low temperatures so theydominate the persistence time, which is comparable to the time taken for a typicalspin with n i = 0 to flip to n i = 1. This time is of order (cid:15) − e J/T .In the softened East models considered here, the decoupling between persistenceand autocorrelation is less strong than this extreme case, where spins are completelyindependent. We will see in the following that the persistence time is significantlylarger than the correlation time. It is also notable that if relaxation takes place bythe soft mechanism, one expects exponential relaxation of both the correlation andthe persistence function P ( t ). We briefly discuss effects of local swap dynamics. In this case Fig. 1 is modified onlyin the final (vertical) process, in which the anomalously soft (grey) site would notdisappear. Instead, it would hop to an adjacent site. For r l (cid:28) e − J/T , the excitationtypically converts back to n i = 1 after hopping, by reversing the processes in the toprow of Fig. 1. This leads to excitation diffusion with a hop rate of order r X r l e − B/v + J/T [similar to (23)] and a hop size of one lattice spacing, so the resulting diffusion constantis D eff ∼ r X r l e − B/v + J/T . (29)On the other hand, for r l (cid:38) e − J/T , the excitation may hop several times beforeconverting back to n i = 1 (this conversion happens with rate e − J/T ). In this casewe expect that excitations hop with a rate of order r X e − B/v [by a similar argumentto (22)]; the typical size of a single hop is of order (cid:112) r l e J/T . This leads to the samediffusion constant as (29). Note that if the hop size is larger than the typical excitationspacing e
J/T , we expect the behaviour of the model to resemble that for non-locals-swaps (with r s = r l ). This requires a very large rate r l (cid:38) e J/T . However, inhigher dimensions ( d ≥ J/ ( T d ) . We expect the other arguments of this section to depend weakly on dimension:in this case local and non-local swaps should lead to similar behaviour as long as r l (cid:38) e J ((2 /d ) − /T . This constraint is much weaker in higher dimensions.We focus on r l ∝ e − J/T consistent with Sec 2.4, which leads to short-rangedhopping of the excitations. The resulting dynamics resembles a Fredrickson-Andersen(FA) model [23], with excitations that diffuse. A notable signature of this behaviouris the persistence function – one expects that for P ( t ) = O (1) then P ( t ) ∼ exp( − (cid:112) D eff c t ) (30)where D eff is the excitation diffusion constant and c their separation [34]. For verylarge times, P ( t ) crosses over to an exponential decay [23], but P itself is very small inthis limit. The prediction of this analysis is that persistence time decays significantlyslower than the relaxation with local swaps: one may estimate τ loc − swapp ∼ min (cid:18) r l r X e B/v + J/T , τ noswap (cid:19) (31) ccelerated relaxation in a kinetically constrained model with swaps Figure 2.
Persistence function for the East model with real-valued local softness(squares, dashed lines) and binary softness (circles, continuous lines). We take
B/v = 2
J/T (in the case of the real-valued model, B = 10 J/T and v = 5), forvarious values of J/T (see legend), in a system of N = 512 spins. (a) Dynamicswithout swaps (b) Dynamics including swaps (s-updates) (c) Dependence of thepersistence time τ p on temperature for both variants of the model, with fits τ p = τ exp (cid:2) ( bJ/T ) + J / (2 T ln 2) (cid:3) for dynamics without swaps [see (19)] and τ p = τ exp ( bJ/T ) with swaps [see (24)] (the fitting parameters are b, τ ). Thefits are excellent for J/T > This prediction is rather simplistic because it assumes that the diffusive (FA) processdominates the relaxation. In practice, the original facilitated process of the East modelis also expected to play a role, so the relaxation is a mixture of FA and East dynamics.Compared to the case where the FA dynamics dominates, the East dynamics acts asan additional relaxation channel, so one expects relaxation to be somewhat faster thanthat predicted by (31).
4. Numerical results
We have performed numerical simulations of the East models with local softness,using variants of the Bortz-Kalos-Lebowitz (BKL) algorithm, which is also known as ccelerated relaxation in a kinetically constrained model with swaps s i ,where the set of rates of local moves is finite, the standard rejection-free BKL methodis used. For the version of the model with real-valued softness X i , and with energybarrier as in Eq. (9), where the set of possible transitions is not finite, we adapt theBKL algorithm: the different types of moves ( n i = 0 → n i = 1, n i = 1 → n i = 0, andupdates of the local softness X i ) are proposed with rates that depend on n i and n i − but not on X i . In order that MC moves occur with the correct rates, the moves thatare proposed in this way are then accepted with a probability that accounts for thedependence of the rate on X i .The remainder of this section presents numerical results, with a special emphasison the impact of swaps on the relaxation dynamics at low temperatures. First, wefocus on the temperature dependence of relaxation and persistence times for dynamicswith and without swaps. We then study the implications of including swaps for thedynamic heterogeneity of the relaxation process. To conclude, we compare differenttypes of swap processes. Throughout the section, numerical results are related to thetheoretical arguments provided in the previous sections. A more general discussion ofthese results (including comparisons with atomistic systems) and their implications isgiven in Section 5. To illustrate the main behaviour, we consider the time dependence of the persistencefunction for a range of temperatures in the East model with a real-valued local softness(Sec 2.2) and with binary local softness (Sec 2.3). We consider dynamics withoutswaps, and also a model with s-updates as swap processes; results for s-swaps andlocal swaps are given in Sec. 4.3, below. We take
B/v = 2
J/T : this ratio is itself aparameter of the model with binary softness; in the model with real-valued softness, wetake B = 10 J/T and v = 5. The swap rate is r u = e − J/T / J ) in theabsence of swaps. We show data for the model with real-valued softness, and for theversion with binary softness. The two variants of the model behave almost identicallyat low temperatures (but the binary-softness can be simulated to much longer timesdue to its simplicity). Figure 2(b) shows the persistence for the same conditions butnow including swap dynamics (corresponding to s-updates). As predicted, one sees adramatic acceleration of relaxation by the swap dynamics, consistent with Eqs. (19,24).Figure 2(c) shows the corresponding persistence times τ p as functions of J/T for bothvariants of the model. The data are compatible with a super-Arrhenius scaling for thedynamics without swaps [see (19)] and an Arrhenius dependence in the presence ofswaps [see (24)]. As in the case of atomistic models [13] the inclusion of swaps allowsto significantly increase the range of low temperatures accessible to the numerics. Weemphasise that the results of Fig. 2 are all for models with softened constraints –the speedup is not a simple consequence of the softening, it comes instead from thes-updates, which trigger local relaxation of the excitations.Since the models with real-valued and binary softness behave the same, we focuson the model with binary softness in the following. Fig. 3(a) shows how the persistence ccelerated relaxation in a kinetically constrained model with swaps Figure 3.
Persistence time (a) and autocorrelation time (b) as functions of
J/T ,for the East model with binary local softness. The softness parameter
B/v isvaried, as shown in the legend. The system size is N = 512 spins. The rate forthe soft process is smallest when B/v = 10
J/T – in this case, the dynamics isalmost unaffected by the soft process, both with and without swaps. By contrast,the soft process is relatively fast when
B/v = J/T – this has a strong effecton the dynamics with swaps, and a weak (but significant) effect on the no-swapdynamics. The solid lines are fits either to τ = τ exp (cid:2) ( bJ/T ) + J / (2 T ln 2) (cid:3) , inthe case of the dynamics without swaps or dynamics with swaps for B/v = 10
J/T ,or τ = τ exp ( bJ/T ), in the case of the dynamics with swaps for B/v = J/T and2
J/T . See the discussion in the main text. time depends on the choice of
B/v : we take
B/v = yJ/T with y = 1 , ,
10. Aspredicted by (24), for large y (in this case y = 10) the swap mechanism is negligibleand the systems with and without swaps behave the same. For y = 2 one observesa dramatic acceleration by the swaps, as in Fig. 2. For y = 1, the condition (18) isbreaking down, even though (16) is still satisfied at low temperatures. As predictedby (19), this means that the soft process leads to a reduction in the relaxation time,even in the absence of swaps (because c eff > c ). Nevertheless, the swaps still lead toa significant acceleration.Figure 3(b) shows that analogous results are obtained by considering thecorrelation time τ c extracted from the autocorrelation ( (cid:104) n i ( t ) n i (0) (cid:105) − (cid:104) n i (cid:105) ) with thesame threshold as in (28). From the fits in this Figure, it is useful to considerswap dynamics with B/v = J/T and
B/v = 2
J/T . In this case the fits are τ c = τ exp ( bJ/T ) with b = (1 . , .
0) for
B/v = (
J/T, J/T ) respectively. Recallingthat r X ∼ e − J/T , Equ. (24) predicts b = (2 , τ p yield b = (2 . , . b = (3 ,
4) – the observed values of b are intermediate betweenthese values and the exponents of the correlation time, consistent with Sec. 3.3.For the cases where the facilitated process is dominating, we observe good fits to τ = τ exp (cid:2) ( bJ/T ) + J / (2 T ln 2) (cid:3) , consistent with (19). ccelerated relaxation in a kinetically constrained model with swaps Figure 4.
Data for the dynamic susceptibility χ (see text for definition)without swaps (a) and with swaps (b) for different values of J/T (see legend) with
B/v = 2
J/T in a system of N = 512 spins. In the regime where the swaps leadto dramatic acceleration of the dynamics, the dynamics are not heterogeneous. We consider the effect of swaps on the extent of heterogeneity in the dynamics. Wequantify dynamical heterogeneity via the dynamic susceptibility χ ( t ) = 1 N (cid:42)(cid:88) ij [ p i ( t ) − P ( t )][ p j ( t ) − P ( t )] (cid:43) . (32)This function shows a maximum for some time t ∗ (cid:39) τ p . If relaxation is dominatedby the facilitated (East) mechanism then the relaxation is heterogeneous with acharacteristic length scale ξ ∼ e J/T and one expects (in one dimension) that themaximal value of χ is proportional to this length scale, that is χ ( t ∗ ) ∼ e J/T [39].If relaxation is dominated by a soft (non-collective) process then relaxation much lessheterogeneous, the length scale ξ = O (1) and we expect χ ( t ∗ ) = O (1). Hence, theEast relaxation process (which has a relaxation time growing faster than Arrhenius)also has a growing length scale and a growing χ , as expected on general grounds. Onthe other hand, the soft relaxation process has a small length scale and an Arrheniusgrowth of the relaxation time.Fig. 4 shows χ ( t ) for different values of J/T with
B/v = 2
J/T (cf. Figs. 2 and 3).Results for no-swap dynamics are in Fig. 4(a), while in Fig. 4(b) shows the dynamicsincluding swaps (s-updates). We observe the following: (i) the peak times t ∗ arereduced in the presence of swaps, compatible with the results from the persistenceand autocorrelation functions; (ii) dynamic heterogeneity is strongly suppressed inthe presence of swaps (note that the vertical scale of both panels differs by an orderof magnitude), because the soft process does not lead to heterogeneous relaxation;(iii) for the system with swaps, the peak height χ ( t ∗ ) depends non-monotonicallyon temperature, because the soft process becomes increasingly dominant at lowtemperature. ccelerated relaxation in a kinetically constrained model with swaps Figure 5. (a) Persistence curves for different swap processes with
B/v = 2
J/T in a system of N = 512 spins. Different curves [corresponding to (inverse)temperatures J/T = 1, 2, 3, 4 and 5] are shown for (non-local) s-swaps,s-updates and local swaps. While s-swaps and s-updates give results thatare essentially identical, as expected for sufficiently large system sizes, localswaps lead to a reduced accelerating effect on the relaxation dynamics. (b)Persistence curves for a dynamics with local swaps with
B/v = 2
J/T and
J/T = 0 . , , . , , . , . , , . , . , . , P ( t ) = e − √ t/τ s . (c) Characteristictime τ s [free parameter of the fit in panel (b)] as a function of J/T . A fit τ s ∼ e bJ/T shows excellent agreement with the numerical data. Up to now we have considered swaps corresponding to s-updates, for both the real-valued and binary-softness models. Fig. 5(a) shows persistence curves for (non-local)s-swaps and for local swaps as well. We take r s = r l = r u / − J/T /
8, The valueof r u is consistent with previous sections. The factor of two between r s and r u leadsto similar behaviour for s-updates and s-swaps, because each s-swap is equivalent toan update of two spins. The figure shows that s-swaps and s-updates behave almostidentically, but local swaps lead to slower relaxation. As argued in Subsection 2.2.2, ccelerated relaxation in a kinetically constrained model with swaps P ( t ) = exp( − (cid:112) t/τ s ). Using the scaling of r X , r l withtemperature, (31) predicts τ s ∼ e bJ/T with b = 5: fits give b = 4 .
4, see Fig. 5(c) where τ s as a function of J/T is shown. This constitutes reasonable agreement, given that(31) comes from a rather simplistic analogy with the FA model (recall the discussion inSec. 3.4). We note that this (significant) difference in relaxation time between systemswith local and non-local swaps is not observed in atomistic models [40, 17], as furtherdiscussed in Section 5.
5. Discussion
It is an essential characteristic of KCMs that the relaxation time depends on thedynamical rules of the model, and cannot be determined by thermodynamic propertiesalone. In this sense, the fact that the swap algorithm accelerates the structuralrelaxation of glass-formers [8, 9, 10, 11, 12, 13] is entirely consistent with the dynamicalfacilitation theory of glasses [6]. Our contribution here is to make this observationexplicit by considering one specific mechanism by which swapping of local degrees offreedom can accelerate relaxation. In the KCM that we study, the relevant degree offreedom for the swap moves is the local softness, which can be interpreted as a localenergy barrier associated with unfacilitated relaxation.The mechanism of acceleration operates as follows: while the barriers at lowtemperature may be very high far away from excitations, in a particular region theycan be temporarily reduced by swaps that increase the local softness (that is, byreducing the local barrier for the soft process which overcomes the constraint). In themodels that we have discussed, the swaps have a simple character, in that all sites haveequal probability of participating in a swap move, and all such moves are accepted,independent of the local state. The situation in atomistic models is more complicated,in particular, the ability of a region of the system to participate in a swap is likelyto depend on its local structure. Such an effect might help to explain some of thedifferences between the simple models presented here and atomistic liquids (see alsobelow).This possibility might be incorporated in the approach shown here by modifyingthe acceptance probability of swap moves, so that it depends on the local X-values(see also [41]). As long as the dynamical rule always respects detailed balance withrespect to the underlying equilibrium distribution, the theoretical arguments that wehave given are easily generalised to this case, and we argue that the general picturethat we present should be robust. We also observe that the acceleration by swaps inour results is controlled by a small fraction of sites, which have X i > B (or s i = 1).This aspect of the mechanism is natural within the framework considered here, butthere might be other ways to combine KCMs with swap dynamics which would nothave this property.Within the specific framework we considered here, the speedup by swaps is robust.Nevertheless, given the simplicity of our setting, there are some differences between ccelerated relaxation in a kinetically constrained model with swaps B/v , which we chose as proportional to 1 /T in Fig. 2. It is not clearhow the scaling of this quantity can be derived within dynamical facilitation theory,and a swap dynamics with super-Arrhenius temperature dependence could also bereproduced with soft KCMs by taking a different scaling of B/v with T . However,this would require an explanation of why the energy barrier B ( T ) should increase [aquestion that shares some similarities with the glass transition problem in general:how does A ( T ) in (1) change on cooling?].A related point is that the non-monotonic dependence of χ ∗ = χ ( t ∗ ) ontemperature in Fig. 4 is not seen in Fig. 14 of [13], which shows analogous resultsfor atomistic systems. The residual growth of χ ∗ could naturally be associated witha residual collectiveness of the dynamics not completely removed by the swaps. Ittherefore seems plausible that a theory that could explain the temperature dependenceof B ( T ) would also explain the growth of χ ∗ that occurs for dynamics with swaps.A final remark concerns the behavior of the model with local swap dynamics. Inatomistic systems, local and non-local swaps lead to almost identical behaviour [40, 17].In Fig. 5, there is a significant difference in relaxation time between local and non-local swaps. As discussed in Sec. 3.4, if both r l and r s are very large, one expectslocal and non-local swaps to lead to the same behaviour. This limit is howeverdifficult to study numerically as simulations become very inefficient. In contrast, inatomistic simulations it is natural to choose larger swap rates than those consideredhere because the simulation of the atomistic dynamics is much more expensive thanthe swaps. A second consideration is that of dimensionality: Relaxation in thesesoft KCMs is mediated by rare anomalously soft sites, which move diffusively in themodel with local swaps. As discussed in Sec. 3.4, mixing by diffusion is slow inone dimension, compared with higher dimensions. It therefore seems likely that thedifferences between local and non-local swaps in Fig. 5 would be much smaller in acorresponding three-dimensional version of the model. (The change of dimensionalitywould not affect the hard constraint, as the features of the East model are mostlyindependent of spatial dimension [39, 27].)To conclude, acceleration by swaps can be reproduced quite naturally inkinetically constrained models. Our work here provides a concrete and specific exampleof a model that reproduces this important phenomenon [13]. The acceleration dueto swaps should be robust within KCMs, which makes the observation of enhancedrelaxation in more realistic glass formers compatible with dynamic facilitation. Inorder to account more accurately for features such as the residual temperaturedependence of timescales or the remnant growth of dynamical correlations (which inthe context of KCMs are related to less coarse-grained features of supercooled liquids)further work is needed. This would be the natural direction in which to extend themodels considered here. ccelerated relaxation in a kinetically constrained model with swaps Acknowledgments
RLJ thanks Ludovic Berthier for helpful discussions about swap dynamics. This workwas funded by EPSRC Grants no. EP/M014266/1 and EP/R04421X/1. We are alsograteful for the computational resources and assistance provided by CRESCO, thesuper-computing center of ENEA in Portici, Italy.
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