Effective Trap-like Activated Dynamics in a Continuous Landscape
EEffective Trap-like Activated Dynamics in a Continuous Landscape
Matthew R. Carbone, ∗ Valerio Astuti, and Marco Baity-Jesi
1, 3 Department of Chemistry, Columbia University, New York, NY 10027, USA Dipartimento di Fisica, Sapienza University of Rome, P. le A. Moro 5, Rome 00185, Italy Eawag, ¨Uberlandstrasse 133, CH-8600 D¨ubendorf, Switzerland (Dated: May 5, 2020)We use a simple model to extend network models for activated dynamics to a continuous land-scape with a well-defined notion of distance and a direct connection to many-body systems. Themodel consists of a tracer in a high-dimensional funnel landscape with no disorder. We find anon-equilibrium low-temperature phase with aging dynamics that is effectively equivalent to thatof models with built-in disorder, such as Trap Model, Step Model and REM. Finally, we compareentropy- with energy-driven activation, and we remark that the former is more robust to the choiceof the dynamics, since it does not depend on whether one uses local or global updates.
I. INTRODUCTION
Glasses display a extraordinarily slow dynamics astemperature is decreased [1]. The mean-field picture ofthis slowing down is elegantly explained as a topologicaltransition in the energy landscape: while at high temper-ature T the typical configurations are close to the saddlepoints of the energy landscape, under the dynamical tem-perature T d the system is confined near the minima of thelandscape. Since in mean-field models the energy barriers∆ E diverge in the thermodynamic limit, for T < T d thesystem remains confined near the local energy minimaand ergodicity is broken [2].In non-mean-field systems, the barrier heights remainfinite, and can be overcome in time scales τ that fol-low the Arrhenius law, τ ∝ exp( ∆ Ek B T ) [3]. This barrier-hopping dynamics, which correspond to collective rear-rangements of particles, are also called thermally acti-vated . The ergodicity breaking induced by the topolog-ical transition in the mean field model is hence avoidedin many systems of interest, such as 3D glass formers.Activated dynamics must thus be understood in order tocharacterize the slowing down of glasses.Given the overwhelming difficulties in the theoreticaldescription of low-dimensional glass formers, a first steptowards the understanding of activated dynamics shouldbe done in the mean field approximation. Barrier cross-ing is in fact also possible in the mean field approxima-tion, provided that the system size N is kept finite [4, 5].Keeping N finite makes calculations especially hard, sosave for some exceptions [6–8], most work on activateddynamics consists of numerical simulations [9–11].The most popular theoretical framework for the in-terpretation of activated dynamics is the Trap Model(TM) [12, 13], which consists of a simplified, solvable,version of the energy landscape of glasses, in which the ∗ [email protected] From here on we set the Boltzmann constant k B = 1 and note thatunless written with a specified base, all logarithms are the naturallogarithm. only way to explore the phase space is a purely activatedmotion between minima in the landscape (called traps),with no notion of distance and with a fixed threshold en-ergy that needs to be reached in order to escape a trap.The TM yields a wide set non-trivial quantitative andqualitative predictions that have been used to rational-ize numerical simulations of low-dimensional glass form-ers [10, 14, 15], and has been recently shown to serve asan accurate representation of the dynamics of some sim-ple models of glasses in which a threshold energy can beeasily identified [7, 8, 16]. A TM description of the dy-namics was also shown to be accurate in the Step Model(SM), a model with a single energy minimum, providedthat one identifies traps in a dynamical way [18].Despite these successes, the TM suffers from limita-tions. On one side, it pictures a phase space motion thatis completely unrelated to real-space degrees of freedom,and it is defined on a discrete space of configurations.The first of these two issues was successfully addressedby showing that some problems from number theory canbe reformulated as physics problems on a lattice, whichbehave like the TM [19, 20]. On the other side, the TMparadigm of activated dynamics is probably not suitablefor the description of most systems with strong enoughcorrelations [11, 21, 22]. One must therefore try to under-stand the limits of the applicability of the TM paradigm,and whether it is possible to create a connection betweenthe TM and other systems such as sphere packings.Much progress along these lines was made in a series ofworks culminating with the proof that the Random En-ergy Model (REM), a simple model with glassy behavior,exhibits trap-like dynamics [6–8, 23, 24]. Another con-sists of studying the influence of phase space connectivityon the dynamics [25, 26]. In our approach, we show thatthe TM paradigm also applies to a very simple model ofa continuous N -dimensional landscape, where each di-mension represents an independent coordinate in a fic-titious space with well-defined metrics. This is done by This connection between trap models had been anticipated decadesearlier (see e.g. Refs. 6 and 17). a r X i v : . [ c ond - m a t . d i s - nn ] M a y noting that a TM-like activated behavior can arise dueto entropic effects also in the absence of multiple localminima, as was shown for the SM [18].Our work is thus organized as follows: In Sec. II wemake simple preliminary observations on how dimension-ality induces entropic effects, and in Sec. III we introducethe physical model. Furthermore, we study its out-of-equilibrium behavior in Sec. IV and in Sec. V we showthat TM-like dynamics arise. Finally, in Sec. VI we sum-marize and discuss our results. We also provide a dis-cussion of the details of our numerical simulations, andmathematical derivations, in the appendices. II. DIMENSIONALITY AND ENTROPICEFFECTS
We study the dynamics of a tracer in an N -dimensionalspace. This is meant to represent the phase space dynam-ics of a many-body system in a central potential. Thepotential energy depends only on the distance r of thetracer from the origin,˜ E ( r ) = log r, (1)and we restrict the phase space to r ∈ (0 , N is large, the system would barely feelthe presence of the energy funnel, despite its negativedivergence. As an example, let us take a Monte Carlodirect sampling dynamics on the N -dimensional unit hy-persphere. At every time step a new configuration isproposed with a uniform probability, and a transition to-wards it is accepted with probability p MC = min(1 , e − β ∆ E ) , (2)where β = T − is the inverse temperature and the energydifference ∆ E is negative if the transition decreases r .From any position x = ( x , , . . . , x ,N ) in the landscape,the probability P ↓ of moving towards lower energy is setby the relative volume of the sphere of radius r = | x | , P ↓ ( r ) = V N ( r ) V N (1) = r N , (3)where V N ( r ) is the volume of the N -dimensional sphereof radius r . As also depicted in Fig. 1, the probability ofproposing a move that decreases the energy goes downexponentially with the dimension of the system N , and Note that these are N one-dimensional interacting particles, or asingle particle in an N -dimensional space. this decrease is more severe the closer x is to the origin.In Fig. 1 we can remark that, already for dimensions assmall as N = 10, P ↓ is smaller than single floating pointaccuracy. N − − − − − P ↓ ( N ) / [ − P ↓ ( N ) ] r = 0 . r = 0 . r = 0 . FIG. 1. Ratio of the probability of the tracer moving towardsthe well singularity, P ↓ , and the tracer moving either awayfrom it or staying in the same location, 1 − P ↓ . Each curverepresents a different starting radius, r . From Eq. (3) we can extract a characteristic time scalefor decreasing the energy τ ∼ /P ↓ = e − N log r . (4)This kind of slowing down generated by the rarefactionof directions that decrease the energy is called entropicaging [27, 28].At non-zero temperature, for large-enough N the dy-namics will always be pushed outwards, because theprobability of accepting moves will stay finite (i.e. in-dependent from N ), while the probability of propos-ing moves that decrease the energy is dramatically sup-pressed. In other words, unless the potential energy alsoscales with N , at any fixed r there will always existan N over which the attraction that ˜ E ( r ) exerts on thetracer becomes irrelevant. One therefore needs to coun-terbalance by either taking temperatures of order 1 /N ,or giving the potential the right scaling N , in order toaccount for the energetic push towards the center of the N -dimensional sphere with the entropic effects inducedby large dimensionality that push the system towardsthe boundary. This is standard practice in statisticalmechanics problems. III. PHYSICAL MODEL
If we recall that our model describes the phase spaceof an N -body system, we should not be surprised thatplausible dynamics require the potential to be rescaledwith N , as the energy should be an extensive quantity.Thus, in the rest of this work we will use the properlyre-scaled potential E ( r ) = Nβ c log r , (5)where β c > r ∈ (0 , d ˜ V = dVV N (1) = Ω N Nr N − drV N (1) = N r N − dr = β c e β c E dE , where Ω N is the N -dimensional solid angle. As a consequence, the den-sity of states g ( E ) ≡ | d ˜ VdE | is equal to g ( E ) = β c e β c E Θ( − E ) , (6)where Θ( x ) is the Heaviside step function.From g ( E ) we can calculate the partition function ofthe Canonical Ensemble, Z ( β ; β c ) = (cid:90) −∞ β c e ( β c − β ) E dE = β c β − β c , (7)which is well-defined only for β < β c . Thus, the equi-librium phase, with average energy (cid:104) E ( β ) (cid:105) = β c − β andradius (cid:104) r ( β ) (cid:105) = exp (cid:16) β c N ( β − β c ) (cid:17) , only exists for β ≤ β c .For β > β c , the system is out of equilibrium and theenergy will eventually diverge to −∞ as time goes toinfinity. IV. OFF-EQUILIBRIUM DYNAMICS
Although the equilibrium phase is trivial, the out-of-equilibrium phase of this model displays rich behav-ior. Since we are out of equilibrium, we need to definethe kind of dynamics used: we analyze both non-local(global) and local dynamics, which are generally equiv-alent for equilibrium simulations. For global dynamicswe use Direct Sampling Monte Carlo (DSMC), and forlocal dynamics, Markov Chain Monte Carlo (MCMC).Details on simulations and measurements are given inAppendix A.
A. Direct Sampling Monte Carlo
With DSMC, at every time step a point in the hy-persphere is proposed as a move for the algorithm. Allpoints of the phase space are proposed with equal proba-bility, and the moves are accepted with probability p MC [Eq. (2)].In the following, we show that according to the value of β there are several regimes in the dynamics, which werealready found in the SM. For β > β c the energy de-creases slowly and steadily, at a rate that follows Eq. (4); we call this the entropic aging (EA) regime [27, 28]. Forintermediate β , even though the energy as a functionof time is decreasing on average, the trajectory inter-mittently returns to high energies. Following previousliterature on the SM, we call this regime thermally acti-vated [18, 29, 30]. In this regime, one can identify a finite threshold energy (or, equivalently, radius) towards whichthe dynamics is spontaneously driven. Even though thedynamics intermittently returns to the threshold, E ( t ) isdecreasing because the system spends short times at highenergy, and increasingly longer times at lower energy.Following Refs. [18, 29], we define the threshold radius r th as the radius from which the probability P ↑ of in-creasing the energy equals the probability of decreasingit, P ↑ ( r th ; β ) ≡ P ↓ ( r th ) . (8)With Monte Carlo dynamics, in general P ↑ + P ↓ <
1, sincethere is also a non-zero probability P that the tracer doesnot move due to the rejection of movement proposals.However, our calculations of r th are static, so P does notinfluence them. Neglecting P in a dynamic calculationis equivalent to saying that time does not advance whena move is rejected: this does not change the probabilityof increasing or decreasing the energy once the move getsaccepted.The probability of increasing the energy from a radius r with DSMC is P ↑ ( r ; β ) = Ω N V N (1) (cid:90) r r N − e − N ββ c (log r − log r ) dr = (9)= r N − r Nβ/β c ββ c − , (10)while P ↓ ( r ) is given in Eq. (3). Equating the two, oneobtains no real solution for β > β c . For β c ≤ β ≤ β c there is a solution growing continuously from 0 at β = 2 β c to 1 at β = β c , r th ( β ) = (cid:18) β c − ββ c (cid:19) β c N ( β − β c) . (11)For β < β c we are in the equilibrium phase: r th is atdistance ∼ /N from the system boundary and from (cid:104) r (cid:105) .Summarizing, in our simple funnel model we have threeregimes (Fig. 2): • β < β c : Equilibrium Phase (EP) We mean an average over the trajectories, not an ensemble average,which is not well defined for β > β c . We use an overbar, ( . . . ), todenote average over trajectories. This concept of threshold energy is related to its definition in theTM, as the energy to reach in order to jump barriers and haverenewal dynamics [13]. Operative methods for the detection of thethreshold energy based on the behavior of p -spin-like models (e.g.those used in Refs. 11 and 31) are not good methods in this context. • β c ≤ β < β c : Thermal Activation (TA) • β ≥ β c : Entropic Aging (EA) . . . . . . β/β c − − − − − h E i . . . . r t h EP TA EA
FIG. 2. Threshold radius (green) and equilibrium value ofthe energy (black) plotted as a function of β/β c for N = 10.The equilibrium phase (EP) regime, where the system en-joys both a well-defined threshold radius and equilibrium en-ergy is shown in blue (0 < β < β c ). The thermal activation(TA) regime is shown in red. Finally, the entropic aging (EA)regime is shown in grey, where the tracer is relentlessly at-tracted towards the center of the well ( β > β c ). Note that the threshold is an attractor of the dynamics,in the sense that the tracer’s distance from the centertends to shrink when r > r th , and to expand when r < r th (Fig. 3). This can be seen clearly from Fig. 3, where weshow that P ↑ > P ↓ ∀ r < r th , and P ↑ < P ↓ ∀ r > r th . Wealso show P ( r ; β ), that goes to 1 as r decreases and canbe used as an indicator of the slowness of the dynamics. . . . . . . r . . . . . . P r o b a b ili t y (a) r th P ↑ P ↓ P . . . − − l og ( P ↑ − P ↓ ) (b) FIG. 3. Main set: (a) Probability of the tracer moving up( P ↑ ( r ; β )), down ( P ↓ ( r )) or not moving ( P ( r ; β )) for N = 10and β = 1 . β c (TA regime). The threshold radius is identifiedby P ↑ = P ↓ . For r > r th , we have P ↓ > P ↑ . For r < r th , wehave P ↓ < P ↑ (see the inset (b) for a closeup of the differencebetween the two). The slowdown of the dynamics with small r is encoded in P going to 1. − − − − − − E ( t ) h E i = ( β − β c ) − (a) DSMC ( N = 100) β = 0 . β = 1 . β = 2 . h E i = ( β − β c ) − (b) MCMC ( N = 10) β = 0 . β = 1 . β = 1 . β = 1 . β = 2 . time − − − − ψ ( τ ) ∼ τ − (2 − β/β c ) (c) ψ B ( τ ) , β = 1 . ψ C ( τ ) , β = 1 . time ∼ τ − (2 − β/β c ) (d) β = 1 . β = 1 . β = 1 . FIG. 4. Top: We show the energy of three trajectories asa function of the Monte Carlo timestep t , for N = 100 and δ = 0 .
1, each in a different dynamical regime. On the left weshow DSMC dynamics (a), and on the right we show MCMCdynamics (b). The EP curve ( β = 0 .
2) converges quickly toits equilibrium value (cid:104) E ( β ) (cid:105) = − .
25 (horizontal blue dashedline), while TA and EA curves ( β = 1 . −∞ . Bottom: Configura-tion and basin trapping time distributions, ψ C ( τ C ) (dashed)and ψ B ( τ B ) (solid), for DSMC (c) and MCMC (d) dynamics.Results are averaged over 200 identical simulations, each with500 tracers. In the entropic aging regime the tracer is attracted tothe center of the sphere, which it approaches over an in-finitely long amount of time. In the thermally activatedregime the system fluctuates around 0 < r th <
1, but astime passes it becomes increasingly probable that low-energy configurations are reached, where the system willspend very long times before rising to the threshold again.Finally, in the equilibrium phase the system is squeezedon the surface of the hypersphere [ r = 1 − O (1 /N )], andwhen low-energy configurations are reached the thermalagitation is strong enough to allow for the system toquickly go back to the surface of the hypersphere.The described dynamical scenario is also encounteredin the SM [27, 29, 30]. This is due to the combination oftwo ingredients: on one side the density of states g ( E )in Eq. (6) is the same of the SM, and on the other theDSMC algorithm samples directly from g ( E ). As a con-sequence, the dynamical succession of energies in our fun-nel is statistically the same as that of the SM, so we notethat several results from the SM are realized in DSMCdynamics. For example, in the SM, the distribution ofpersistence times in a configuration, ψ C ( τ C ) (distributionof times spent in a configuration), and in a basin, ψ B ( τ B )(distribution of times spent under the threshold) bothdecay as [18, 30] ψ ( τ ) ∼ /τ µ , (12)with µ = 2 − β/β c , which is what we find in our high-dimensional funnel (Fig. 4, left bottom), and the energydecays logarithmically (Fig. 4, left top).Note, however, that finite-size effects are now different,because in network models such as TM, SM and REM,the lowest available energy depends on N . Therefore, forany finite N , E ( t ) eventually saturates, whereas in ourmodel the energy decreases to −∞ at any system size, sothe dynamical phase diagram [24] can differ. B. Markov Chain Monte Carlo
Especially given that we are introducing spatial effectsin the dynamics of network models, it is perhaps moreinteresting to study also local dynamics. We analyzea Markov Chain sampling of our N -dimensional funnel.From every point x t in the hypersphere, a new point x t +1 is proposed by making a Gaussian shift x t +1 = x t + ∆ , (13)where ∆ ∼ N N (0 , δ ) is an N -dimensional Gaussian ran-dom variable with variance δ (meaning a diagonal co-variance matrix with δ in each entry) centered at theorigin and randomly sampled at every time step. Themove is accepted with the Monte Carlo rate in Eq. (2).The initial configuration is uniformly drawn from the ra-dius 1 hypersphere.Also in this case, we find the same three dynamicalregimes that we found for the DSMC. Since the equilib-rium properties are independent of the type of dynamics(provided that it obeys detailed balance), we still havean EP for β < β c . For higher β , there is a TA regimedefined by the presence of a positive threshold radius.In Appendix B we show that the TA regime terminatesat β = 2, where the dynamics is not intermittent any-more, and one reaches an EA regime, where the energydecreases steadily.We can derive the threshold radius that is valid forlarge N by imposing that the probability of increasingand decreasing the energy is equal, P ↓ ( r th ; δ ) ≡ P ↑ ( r th ; β, δ ) , (14)where now Eq. (14) also accounts for the size of theMCMC step, δ , in the upwards and downwards proba-bilities, which can be formally written as P ↓ ( x ; δ ) ∝ (cid:90) < | x | 1 1.5 2 N = 200(c) β FIG. 5. Main set: (a) Threshold radius as a function of β ina system of size N = 50, for DSMC [Eq. (11)] and MCMC[Eq. (17)] dynamics. The y axis is truncated at 0.5 to improvethe figure’s clarity. Points are shown in the curves in order tomake overlapping curves visible. Insets: Ratio between theMCMC and the DSMC threshold radii for (b) N = 50 and(c) N = 200. V. TRAP-LIKE BEHAVIOR In Ref. 18, it was shown that the SM displays an ac-tivated aging dynamics that is effectively like that of theTM. In order to do so, we studied the time evolutionof the energy, and defined energy basins dynamically, asthe periods of time that the system remains at E < E th .The distributions of trapping times are shown in Fig. 4–bottom.We can use the exponent µ [Eq. (12)] to show that thefunnel model has TM dynamics, as was done in Ref. 18for the SM, by studying the aging function Π B ( t w , t w + t ),defined as the probability of not changing basin betweenthe times t w and t w + t . To define the basins’ thresholdwe used Eq. (11) for both DSMC and MCMC.In the TM, the aging function has a well-defined limit-ing value which depends only on the exponent µ [Eq. (12)]and on the ratio w = t/t w [13, 32], H µ ( w ) = sin( πµ ) π (cid:90) ∞ w du (1 + u ) u µ . (18)In Fig. 6–top we show that the aging function in the TAregime converges clearly to the TM prediction in DSMC Details on definition and computation of ψ ( τ ) and Π B ( t w , t w + t )are given in Appendix A. dynamics. The same is valid for MCMC dynamics which,being local, is much slower than DSMC, so our simula-tions are restricted to lower N and β . . . . . . . Π D S M C B ( t w , . t w ) (a) β = 1 . β = 1 . β = 1 . β = 1 . t w . . . . . . Π M C M C B ( t w , . t w ) (b) β = 1 . β = 1 . β = 1 . FIG. 6. Top: (a) Aging functions Π B ( t w , . t w ) for DSMCdynamics, with N = 100, β = 1 . , . , . , . 75. Bottom:(b) Aging functions Π B ( t w , . t w ) for MCMC dynamics for N = 10 and β = 1 . , . , . 25. The dashed horizontal linescorrespond to the trap value H − β (0 . . Results are averagedover 200 identical simulations, each with 500 tracers. The strong slowing down in the MCMC dynamics canbe appreciated from Fig. 7. As one can expect, the globalupdate dynamics has no finite-size effects, whereas thelocal dynamics is increasingly slower as the system sizeincreases. This slow down is exponentially large withthe system size (Fig. 7, inset), which suggests that ourfunnel model is correctly capturing the nature of acti-vated processes. VI. DISCUSSION & CONCLUSIONS We investigated the out-of-equilibrium dynamics of atracer in an N -dimensional funnel landscape. The dy-namics is dominated by the competition between an en-ergetic pull towards the center, and an entropic push out-wards due to the dimension of the space, which turnsout to be equivalent to increasing the thermal noise by afactor N . As a consequence, the energetic contributionneeds to scale with N (or the temperature needs to berescaled by 1 /N ) for it to be relevant. This is true if δ is independent from the system size. If δ in-creases with the system size (which is not the case in typical localalgorithms of many-body systems), the dependence on N can besuppressed. To obtain that the ratio of the volume of configura-tions accessible in one step, divided by the total volume, shouldstay constant. We can use Eq. (3) to obtain δ ∼ e − const .N . . . . . . . Π D S M C B ( t w , . t w ) (a) N = 3 N = 6 N = 10 N = 30 N = 60 N = 100 t w . . . . . . Π M C M C B ( t w , . t w ) (b) 20 40 60 N t ∗ ( N ) (c) FIG. 7. Top: (a) Aging functions Π B ( t w , . t w ) for DSMCdynamics with β = 1 . N = 3 , , , , , . Bottom:(b) Aging functions Π B ( t w , . t w ) for MCMC dynamics with β = 1 . 05 and the same values for N. The dashed horizontallines correspond to the trap values H − β (0 . . Results areaveraged over 100 identical simulations, each with 200 tracers.Inset: (c) The time at which the average Π MCMCB ( t w , . t w )function crosses 0.4 as a function of N (obtained throughlinear interpolation). Note the semi-log scale on the y -axisimplies exponential scaling in N. The properly rescaled model has a high-temperatureequilibrium phase, and two low-temperature out-of-equilibrium regimes: a thermally activated regime inwhich the tracer intermittently comes up to the surfaceeven though, on average, its energy decreases indefinitely,and an entropic aging regime in which this intermittencydisappears. This same phenomenology is found in theStep Model (SM), a network model with random ener-gies and no notion of distance [27, 28, 30], which in ourmodel is recovered in the limit of maximally delocalizedand uncorrelated updates.We find that, besides the system size dependence, thisnon-equilibrium behavior is independent of the chosendynamics. We examined a global update method, Di-rect Sampling Monte Carlo (DSMC), and a local update,Markov Chain Monte Carlo (MCMC), which turned outto be equivalent, providing an example of the equivalenceof equilibrium algorithms in out-of-equilibrium contexts.Extensions to other kinds of physical dynamics wouldarguably give the same results.Unlike the SM, quenched disorder is not needed in or-der to have glassy TM-like activated dynamics. This isunderstood by comparing the SM with our funnel modelwith DSMC dynamics: the randomness due to disorderin the SM can be incorporated into that due to thermalfluctuations, giving the same kind of long-time activateddynamics.The funnel model can be seen as an extension of the SMto a continuous landscape, where a notion of space, dis-tance and dimension are now well-defined. This makes itviable to extend the TM paradigm (or the suitable modi-fications of it) to more realistic models, such as structuralglasses. A critique suffered by models such as TM and SM isthat the excessive simplicity of their phase space makesit impossible to use them to describe any Hamiltoniansystem in realistic terms, which is solved by our funnelmodel. An alternative approach to associate the TMto models with microscopic degrees of freedom was pro-posed in Ref. 19, by reformulating the number partition-ing problem (NPP) as a Mattis spin glass, and focusingon single spin flip dynamics. This TM-like behavior isdue to the presence of few low-energy configurations thatare one spin flip away from typical energies. By changingthe dynamics to multiple spin flips, the TM behavior dis-appears, and the dynamics mimics the SM when all thespins are updated simultaneously [20]. This is conceptu-ally different from what we find for two main reasons: • In Ref. 19, the TM dynamics is due to an equiv-alence at a level of the landscape , which is com-posed by rare, point-like, regions with a very lowenergy in an environment where the system canotherwise move freely. Instead, in the funnel model(and in the SM), the origin of the TM-like behavioris purely due to entropy, and it is effective in thesense that the trapping time distributions are dif-ferent from those of the TM, but their relationshipwith the aging functions is the same as that of theTM. We thus have two very different kinds of TM-like dynamics, one which is energy-driven, and an-other which is entropy-driven, and they should betreated differently. Defining basin hopping throughthe dynamics has also been done in experiments ofglass-formers by looking at particle movement [37].However, the possible entropic origin of the ob-served activated dynamics is either dismissed, infavor of energy-based arguments, or it is incorpo-rated into kinetic constraint arguments [38]. Bothkinds of barriers (energetic and entropic) inducelogarithmically slow dynamics [10], and in realisticsystems there is likely competition (or synergy) be-tween the two kinds of effects, due to the presenceof a collection of deep wide minima [21]. One may argue that most models of structural glasses enjoy trans-lation invariance, which the funnel model lacks. However, eventhough we do not claim that this funnel model as is can faithfullyrepresent a glass, translation invariance, or lack thereof, does notpresent with cause for concern. Indeed, translation invariance isnot necessarily a feature of glasses, and breaking it often leads toan increased glassiness. For example, randomly pinning particlesin a supercooled liquid breaks translation invariance, but exacer-bates the glassy behavior [33, 34]. Both in structural and in spinglasses, symmetries such as translation and rotation give rise toDebye modes in the density of states, which are not a signature ofglassiness. To the contrary, the glassy low-frequency modes emergewhen those symmetries are explicitly broken [35, 36]. • We find a TM-like behavior for both local andglobal dynamics. The type of activation ana-lyzed in the NPP is energy-driven [19], and in thelimit of global updates the model resembles theSM [20], which exhibits entropic TM-like activa-tion [18]. Our analysis suggests, therefore, thatentropy-driven activation is more robust to changesin the dynamics, and that the NPP is likely to ex-hibit both energetic and entropic trap-like behav-iors at the same time. We highlight these simplemodels and their activated behaviors in relation totheir dynamics in Table I. TABLE I. For each model and dynamics, we specify whetherwe have energy- or entropy-driven activation. Trap model(TM) and Step model (SM) live in a fully connected phasespace, so they cannot have local dynamics. The funnel modelstudied in this paper consists of a single well, so it cannothave energy-driven trap-like behavior. It does have entropy-driven activation for both local and global dynamics. Thenumber partitioning problem (NPP) and the random energymodel (REM) have energy-driven trap-like activation whenusing single spin flip dynamics. With global dynamics, theNPP has (as well as the exponential version of the REM)entropy-driven activation. This makes the NPP and REMmodels good candidates for having both kinds of trap-likebehaviors simultaneously (i.e. with the same dynamics), sincewe can expect the same phenomenology of the funnel model.Global LocalModel Energy Entropy Energy EntropyTM Yes No No a No a SM No Yes No a No a Funnel No Yes b No Yes b NPP No Yes Yes ? c REM No Yes d Yes ? ca The trap and step models cannot have local dynamics. b The subject of this work. c Models like NPP and REM could have entropically-activatedtrap-like behavior also with local dynamics. d An exponential REM with global dynamics is a SM. The usual,Gaussian, REM has not been examined. Our funnel model introduces Euclidean space in theSM, and shows an alternative way of introducing local-ity, providing the possibility of local moves but with-out the multiplicity of local minima that characterizeenergy-driven trap landscapes. As it also happens forthe NPP [20], the SM is only the limit for maximallyglobal dynamics of our model, but now locality is differ-ent than in the NPP, since it involves a Euclidean metric,and therefore, finite-size effects are also different. In fact,in the SM and the other aforementioned lattice models, N determines the lowest reachable energy, whereas here N is related to the amplitude of the noise. This impliesthat SM and funnel model are the same model only in the N → ∞ limit using DSMC dynamics. Another way tosee this is that, even for finite N , in the funnel model, theground state is at −∞ , and it is almost impossible for anyalgorithm with any amount of noise to reach the center ofthe hypersphere. A direct consequence of this is that, un-like TM, SM, REM and NPP, a finite-size funnel modelat β > β c will never reach equilibrium. Furthermore,the introduction of local dynamics and the connectionto particle systems uncover that entropic TM-like agingalso displays an exponential slowing down with the sys-tem size, which is a fundamental trait of activation whichneeded to be observed.The interaction potential to which these particles aresubject, although exotic, can be found in several situa-tions, such as in bosonic systems [39], when convertinginto extensive problems with an exponential scaling inthe system size [19], or by reformulating number theoryproblems in terms of a cost function [19]. An interest-ing development of our work would be the explorationof entropy-driven activation in r α potentials (for exam-ple the case α = − N particles in aCoulomb potential). Such further directions will be thesubject of future work. ACKNOWLEDGMENTS MBJ and MRC are very grateful to D. R. Reichmanfor supporting them in doing independent research in hislab. MBJ thanks S. Franchini for pointing out the con-nection to bosonic systems. This work was funded bythe Simons Foundation for the collaboration Crackingthe Glass Problem (No. 454951 to D. R. Reichman).MRC acknowledges support from the US Department ofEnergy through the Computational Sciences GraduateFellowship (DOE CSGF) under Grant No. DE-FG02-97ER25308. MBJ thanks MINECO for partial supportthrough research contract No PGC2018-094684-B-C21(contract partially funded by FEDER). Appendix A: Simulation Details In this Appendix we explain our simulation procedures,and explain our parallel code, that we provide for openaccess at https://github.com/x94carbone/hdwell .For each choice of the parameters and dynamics wesimulate M batches (usually 100 to 200) of trajectoriesof τ max = 10 to 10 time steps. Each of the m tracersper batch (usually 200 to 500) is computed in parallel,and are used to compute distribution averages effectively.From each batch we obtain the whole curves ψ ( τ ) andΠ( t w , t w + t ) which can then be averaged among batches.The specific details of how these values are calculated aresummarized in this Appendix. 1. Monte-Carlo Procedure The details of the simulation algorithms are henceforthsummarized:1. Initialize on the N -dimensional sphere with a uni-form distribution (so for large N the tracer is ini-tially at r (cid:39) t , make a proposal movetowards x ∗ t +1 as follows:(a) If DSMC: x ∗ t +1 uniform in the hypersphere(using Ref. 40).(b) If MCMC: x ∗ t +1 according to Eq. (13).3. Accept or reject the move with the Metropolis ruleEq. (2).4. If the timestep is designated for recording quanti-ties of interest, save the value of the energy, ψ andΠ-values for each tracer.5. Update the timestep: t ← t + 1 . 6. Repeat 2-5 until t = t max . 2. Calculation of the Trapping Time Distributions To compute ψ C , the following procedure is used: Acounter is initialized for each tracer in a simulation whichkeeps track of the number of time steps that tracer re-mains in a single configuration. Since this is a continuouslandscape, the distance from the center of the well is asufficient proxy for the exact configuration, since we canneglect the probability of changing configuration main-taining exactly the same radius. Therefore, the trappingtime is measured as the number of steps during whichthe system is at the same r . At every time step, theconfiguration of each tracer is queried. If r t = r t +1 , thattracer’s counter increases by 1. If r t (cid:54) = r t +1 , we imme-diately update a histogram (with log -spaced bins) withthe the value of the trapping time. This procedure al-lows for a sizable reduction of the memory devoted tothe measurements [16].A similar procedure is used to calculate the values for ψ B . A separate counter keeps track of the number oftime steps that a tracer is below E th in a basin. As soonas the tracer rises above E th this counter is logged andreset in the same way we described for ψ C . 3. Calculation of the Aging Functions Finally, we make note of how we calculate values forΠ B during the simulation. Note that the calculation ofΠ C is analogous, where instead of the basin index de-scribed further on, the configuration proxy r t is used, inthe same way that we described for ψ C . The quantityΠ B is the probability of not changing basin between twotimes. Stated another way, Π B ( t w , t w + t ) is the proba-bility, being the tracer in some basin at t w , that it is inthe same basin at ( t + t w ) = t w (1 + w ) that the traceris in that same basin (having not left). In this work wetake w = 0 . B j is kept for every tracer j and has thefollowing properties.1. If at time step t w , tracer j is in its n th basin (mean-ing it has entered and left n − t w ),then B j = n. 2. If the tracer has just left its n th basin at t w , then B j = n + i, where i is the imaginary number. Thechoice of using complex numbers to index whethera tracer is in or out of a basin is arbitrary, butallows for simpler notation in the code. 3. When the tracer reenters a basin, the imaginarycomponent of B j is set back to 0, and the real partincrements: n ← n + 1 . Thus in summary, the real part of B j references the in-dex of the last basin that tracer was in, and the pres-ence of an imaginary component is used to index whetheror not that tracer is currently in or out of a basin. If (cid:61){B ( t w ) } (cid:54) = 0, the measurement is discarded in comput-ing the normalization of Π B , since the tracer is initiallynot in a basin. If instead (cid:61){B j ( t w ) } = 0, then for aparticular tracer, • if B j ( t w ) = B j ( t w + t ) ⇒ Π B ( t, t (cid:48) ) = 1 • if B j ( t w ) (cid:54) = B j ( t w + t ) ⇒ Π B ( t, t (cid:48) ) = 0As we described for the trapping time distributions, weextract a curve Π( t w , t w (1 + w )) from each batch of runs,and compute statistical error bars by comparing batches. Appendix B: MCMC calculation of the threshold To evaluate the threshold radius in the MCMC approach we need to solve the equality P ↓ ( r th ; δ ) = P ↑ ( r th ; β, δ ) , (B1)with P ↓ ( x ; δ ) = N δ (cid:90) < | x | 1. Threshold Radius for MCMC steps We are interested in the ratio between the two probabilities, which can be expressed as P ↑ ( x ; β, δ ) P ↓ ( x ; δ ) = R ( x ; β, α ) = (cid:82) < | x | < x d x | x | − βN e − ( x − n )22 α (cid:82) < | x | < d x e − ( x − n )22 α , (B13)or, keeping only the radial coordinates, R ( x ; β, α ) = (cid:82) /x dy y N − − βN (cid:18) e − ( y − α − e − ( y +1)22 α (cid:19)(cid:82) dy y N − (cid:18) e − ( y − α − e − ( y +1)22 α (cid:19) . (B14)In order to find the threshold radius, we need to solve R ( r th ; β, α ) = 1 . (B15) We assume this scaling for δ and N to be valid through the remaining of the appendix. x → N − corrections): lim x → R ( x ; β, α ) = 1 β − , (B16)which gives a threshold radius r th = 0 for β = 2. By imposing a null variation of R ( x ; β, α ) with respect to x and β it is easy to show that the threshold radius is a decreasing function of β . Since r th = 0 at β = 2, there cannot be anyentropically activated dynamics for β > 2. Using a saddle-point-like approximation the ratio R ( x ; β, α ) reduces to R ( x ; β, α ) = N ( x ; β, α ) D ( α ) , (B17) N ( x ; β, α ) ≈ (cid:16) − e − α (cid:17) N ∆ β (cid:104) − x N ∆ β (cid:105) + 2 α e − α − x N ∆ β [1 + (1 − x ) N ∆ β ] N ∆ β [1 + N ∆ β ] ++ 12 α (cid:18)(cid:18) − α (cid:19) e − α − (cid:19) − x N ∆ β − (1 − x ) N ∆ β N ∆ β [ N ∆ β + 1] [ N ∆ β − , (B18) D ( α ) ≈ (cid:16) − e − α (cid:17) N − − α e − α N ( N − 1) + 12 α (cid:18)(cid:18) − α (cid:19) e − α − (cid:19) N ( N + 1) ( N − 1) (B19)where we defined ∆ β = β − 1. As long as x (cid:29) δ the last expressions simplify to R ( x ; β, α ) ≈ (cid:16) − x N ∆ β (cid:17) ∆ β − βN ∆ β − x N δ − x N ∆ β (1 − x ) N ∆ β β (cid:16) − x N ∆ β (cid:17) − . (B20)Note that the condition x (cid:29) δ , given the scaling of δ with N that we assumed, is true in the whole sphere barring anegligible volume around the origin which becomes important only when β → 2. 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