Anomalous diffusion of driven particles in supercooled liquids
AAnomalous diffusion of driven particles in supercooled liquids
Carsten F. E. Schroer ∗ and Andreas Heuer † Westf¨alische Wilhelms-Universit¨at M¨unster, Institut f¨ur physiklaische Chemie,Corrensstraße 28/30, 48149 M¨unster, Germany andNRW Graduate School of Chemistry, Wilhelm-Klemm-Straße 10, 48149 M¨unster, Germany (Dated: November 1, 2018)We have performed non-equilibrium dynamics simulations of a binary Lennard-Jones mixture inwhich an external force is applied on a single tagged particle. For the diffusive properties of thisparticle parallel to the force superdiffusive behavior at intermediate times as well as giant long-timediffusivity is observed. A quantitative description of this non-trivial behavior is given by a continuoustime random walk analysis of the system in configuration space. We further demonstrate, that thesame physical properties which are responsible for the superdiffusivity in non-equilibrium systemsalso determine the non-Gaussian parameter in equilibrium systems.
Introduction.
Due to the distinct multi-particle dy-namics of glass-forming systems several interesting prop-erties can be observed like the occurrence of dynamicalheterogeneities [1] or the violation of the Stokes-Einsteinrelation [2, 3]. In the non-equilibrium situation, the ob-served phenomena can become even more complex. Re-cently, Winter et al. performed computer simulations ofa tracer particle which is driven by a constant externalfield trough a binary Yukawa fluid [4]. It was shown thatfor this microrheological simulation the diffusive proper-ties of the particle become highly anisotropic: While themean squared displacement (MSD) of the tracer parti-cle perpendicular to the force direction (cid:104) x ⊥ (cid:105) ( t ) increaseswith increasing force but still displays a diffusive behav-ior, the centered MSD parallel to the force direction σ ( t ) = (cid:104) x (cid:107) ( t ) (cid:105) − (cid:104) x (cid:107) ( t ) (cid:105) (1)displayed a superdiffusive behavior at the observedtime range. This result was rationalized in terms of aspecial type of biased trap model [5] in which a superdif-fusive behavior is predicted due to rising fluctuations.Therefore, it was stated that the diffusion constant forthe parallel direction of the tracer particle does not exist[4]. However, this model has to be regarded with care be-cause the rising fluctuations would lead to a permanentlyincreasing energy barriers. This scenario is difficult toreconcile with the observed stationary behavior. Mode-coupling theory has been very successful to predict, e.g.,the nonlinear mobility dependence on the applied forcein microrheological simulations [6–9]. Interestingly, thissuperdiffusive behavior could not be reproduced [9].A different approached is used by Jack et al. [10]. Mo-tivated by the analysis of an one-dimensional spin facil-itated models, they performed an analytical calculationfor a biased continuous time random walk (CTRW). Forthis model, a diffusive regime is predicted for long times.This diffusive regime is characterized by a strong depen-dence on the width of the used waiting time distribution.Broader waiting time distributions lead to a dramaticincrease of spatial fluctuations, denoted as ”giant diffu-sivity” [11]. The key goal of this paper is to elucidate the prop-erties of the superdiffusivity in the driven particle dy-namics. First, we present a formal expression which re-lates the superdiffusivity to dynamic heterogeneities inthe CTRW framework. Second, for the trajectories of aglass-forming model system we can extract the relevantobservables from an appropriate CTRW analysis and pre-dict the superdiffusive behavior in a quantitative way.For the long-time limit our expression reduces to the gi-ant diffusivity as calculated in Ref.[10]. Third, we areable to show that the superdiffusivity has a deep physi-cal connection to the non-Gaussian parameter (NGP) inequilibrium, thereby establishing a strong connection be-tween the non-equilibrium and the equilibrium dynamicsof glass-forming systems. Simulations.
We have performed computer simula-tions of a binary mixture of Lennard-Jones particles [12](BMLJ) which we have extended by applying a constantforce on one randomly selected particle. Constant tem-perature conditions are ensured by using a Nos´e-Hooverthermostat [13]. By applying a suitable minimizationprocedure it is possible to track minima of the PEL,called inherent structures (IS), which the system had ex-plored during its time evolution. Analogous to equilib-rium simulations [14–16] we have recently demonstrated[17], that the time evolution of a small stationary non-equilibrium system (consisting of 65 particles, denotedas BMLJ65) can be analogous to equilibrium systems[14, 15] described in terms of a continuous time randomwalk (CTRW) of the system between coarse grained min-ima, called metabasins (MB). This projection allows fora discretization of the system trajectory into dynami-cal events, which are characterized by the distributionof particle displacements during one transition, and abroad waiting time distribution. As shown in [17] thelinear and nonlinear response only shows very small fi-nite size effects. Here we will also show that the resultsof this work can be transferred to the properties of largesystems as well.
Results.
Focusing on the diffusive behavior of thetracer particle parallel to the force direction, our ap- a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t -3 -2 -1 σ / n n F = 7.50F = 5.00F = 3.00F = 1.00F = 0.00
FIG. 1. Centered MSD σ ( n ) of BMLJ65 as a function of thenumber of transitions n at a temperature T = 0 . a (cid:107) parallel to theforce direction. proach offers two different routes to define the centeredMSD: On the one hand, one can consider the centeredMSD after a certain number of MB transitions n on theother hand, it can be evaluated after a certain time t . Inthe following we will distinguish between these quanti-ties by writing σ ( n ) and σ ( t ), respectively. Similar tothe equilibrium dynamics [14], σ ( n ) grows linearly aftermore than ∼
20 transitions (see Fig. 1). In marked con-trast, σ ( t ) displays a superdiffusive behavior (see Fig. 2)as it was reported for the binary Yukawa fluid [4]. From σ ( n ) one can define for large n the diffusive length scale a (cid:107) via a (cid:107) = lim n →∞ σ ( n ) n ; (2)see also [17].To achieve a more quantitative understanding andto unravel the surprising qualitative differences between σ ( n ) and σ ( t ), we have performed an analytical calcu-lation of σ ( t ) within the CTRW framework. We startwith a one-dimensional CTRW with an elementary step x i, (cid:107) = a i, (cid:107) + ∆ x (cid:107) . ∆ x (cid:107) denotes the average translationthe particle performs along the force direction during oneMB transition. As shown in [17], ∆ x (cid:107) is basically propor-tional to F in the whole force interval considered in thiswork. a i, (cid:107) is considered to be the remaining translationallength with (cid:104) a (cid:107) (cid:105) = 0. Successive steps are regarded asuncorrelated so that (cid:104) a i, (cid:107) a j, (cid:107) (cid:105) = δ i,j (cid:104) a (cid:107) (cid:105) . For reasons ofconsistency with our previous work, we will further de-note (cid:104) a (cid:107) (cid:105) simply as a (cid:107) . Then, the MSD of the particle isgiven by the sum over all steps n which were performedup to a time t : (cid:104) x (cid:107) ( t ) (cid:105) = (cid:104) ( n ( t ) (cid:88) i =0 ( a i, (cid:107) + ∆ x (cid:107) )) (cid:105) . (3)This yields for the time evolution of the MSD (cid:104) x (cid:107) ( t ) (cid:105) = (cid:104) n ( t ) (cid:105) a (cid:107) + (cid:104) n ( t ) (cid:105) ∆ x (cid:107) . (4) -4 -3 -2 -1 σ / t t F = 7.50F = 5.00F = 3.00F = 1.00F = 0.00
FIG. 2. Centered MSD σ ( t ) of BMLJ1560 as a function oftime t at a temperature T = 0 . In Eq. 4 (cid:104) n (cid:105) denotes the average number of jumps theparticle performs in a certain time t and (cid:104) n (cid:105) the fluctu-ation of these, respectively. Considering σ ( t ) instead ofthe MSD one has to subtract the squared first momentof the particle displacement which is given by (cid:104) x (cid:107) ( t ) (cid:105) = (cid:104) n ( t ) (cid:105) ∆ x (cid:107) .After subtracting this expression from Eq. 4 one finallyobtains σ ( t ) = (cid:104) n ( t ) (cid:105) a (cid:107) + [ (cid:104) n ( t ) (cid:105) − (cid:104) n ( t ) (cid:105) ] ∆ x (cid:107) . (5)The first term of Eq. 5 can be identified with the one-dimensional equilibrium-like diffusive process2 D (cid:107) t = a (cid:107) t (cid:104) τ (cid:105) . (6)We use the term ”equilibrium-like” because, as we haveshown for our numerical data in [17], both a (cid:107) and (cid:104) τ (cid:105) dis-play a force dependence so that the actual value of D (cid:107) increases with increasing force. The superdiffusivity of σ ( t ) can be related to the latter term which is only vis-ible in driven systems. The expression [ (cid:104) n ( t ) (cid:105) − (cid:104) n ( t ) (cid:105) ]describes the heterogeneity of the number of occurringjumps in a certain time interval and can be directly ob-tained from our trajectories.For BMLJ65 one is able to perform a complete CTRWanalysis so that each observable in Eq. 5 is directly ac-cessible. As it can be seen in Fig. 2 this ansatz allows toquantitatively reproduce the behavior of σ ( t ).The time evolution can be divided into three regimes:At short times one observes a decay of σ ( t ) due to lo-cal caging until a short diffusive regime is reached at t ≈ (cid:104) τ (cid:105) which reflects the equilibrium-like diffusion pro-cess (Eq. 6). The initial decay of σ ( t ) /t can be qualita-tively understood by considering, that σ ( n ) /n displaysa similar decay due to slightly forward-backward corre-lations until the constant diffusive length a (cid:107) is reachedat n ≈
20 transitions (see [14] for further details). It isimportant to notice that only in case of small forces theminimum of σ ( t ) /t indicates the true value of D (cid:107) whileat higher forces it is already superimposed by superdiffu-sive contributions. At intermediate times, one observes asuperdiffusive behavior which is caused by the nonlinearevolution of [ (cid:104) n ( t ) (cid:105) − (cid:104) n ( t ) (cid:105) ]. At long times, when t islarger than the largest measured waiting time, one indeedobserves an indication, that the MSD becomes diffusiveagain but with a significantly larger diffusion constant.For this particular long time behavior one is able to yieldan analytical prediction by the present CTRW ansatz:The waiting time distribution ϕ ( τ ) of the CTRW canbe characterized by its average value (cid:104) τ (cid:105) and its variance V = (cid:104) τ (cid:105) − (cid:104) τ (cid:105) . Due to the central limit theorem thedistribution of the cumulated waiting time τ n of a largenumber of jumps n , P n ( τ n ), is given bylim n →∞ P n ( τ n ) ∝ exp (cid:18) ( τ n − n (cid:104) τ (cid:105) ) V n (cid:19) . (7)The corresponding probability to find n jumps in alarge time interval t , P t ( n ), is directly related to P n ( τ n ).With the substitution n = t/ (cid:104) τ (cid:105) and identifying τ n = t we obtain from Eq. 7 the expressionlim t →∞ P t ( n ) ∝ exp (cid:32) ( n − t (cid:104) τ (cid:105) ) V t (cid:104) τ (cid:105) (cid:33) . (8)Determination of the second moment of P t ( n ) yieldslim t →∞ [ (cid:104) n ( t ) (cid:105) − (cid:104) n ( t ) (cid:105) ] 1 t = V (cid:104) τ (cid:105) = (cid:34) (cid:104) τ (cid:105)(cid:104) τ (cid:105) − (cid:35) (cid:104) τ (cid:105) (9)and hence, by combining Eq. 9 and Eq. 5, for the long-time behavior of σ ( t )lim t →∞ σ t = D (cid:107) f = D (cid:107) (cid:34) x (cid:107) a (cid:107) (cid:32) (cid:104) τ (cid:105)(cid:104) τ (cid:105) − (cid:33)(cid:35) . (10)In this equation, f describes the factor which re-lates the equilibrium-like and the long-time diffusion con-stants. Independent from our derivation Jack et al. ana-lytically obtained a similar result for the giant diffusivityby considering the Montroll-Weiss equation of a biasedCTRW [10]. The long-time diffusion constant D (cid:107) f wasalready indicated in Fig. 2. For BMLJ65, it is possibleto explicitly compute the long-time diffusivity becauseof the direct access to the CTRW observables in Eq. 10.Importantly, it is also possible to estimate the long-timebehavior for larger system sizes. As shown in the sup-plementary material the numerically observed degree ofsuperdiffusivity is fully compatible with the theoreticalexpectation.The presented ansatz allows one to give an explicit cri-terion how long a particle requires to reach the diffusive FIG. 3. One-dimensional NGP α ( t ) multiplied by time t as afunction of t . The dashed lines corresponds to the theoreticalprediction in Eq. 12, the arrows indicate the structural relax-ation time τ α for the different temperatures. Inset: α ( t ) t at a temperature T = 0 .
475 together with its temporal andspatial contributions (see text). regime. It is related to the applicability of the centrallimit theorem and thus to the width of the waiting timedistribution: The narrower the waiting time distribution,the earlier the particle becomes diffusive. Since the ap-plication of a strong microrhological perturbation causesa narrowing of the waiting time distribution [17] one ex-pects an earlier advent of the long-time diffusivity at highforces. This behavior can be qualitatively observed inFig. 2 as well.Besides the MSD of a driven particle, the heterogeneityof MB transitions can also be observed in equilibrium sys-tems by analyzing the NGP α ( t ) of the one-dimensionalparticle displacement which is defined as α ( t ) = (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) (cid:104) x ( t ) (cid:105) . (11)Using the ansatz of an unbiased CTRW one obtainsfor the NGP α ( t ) = [ (cid:104) n ( t ) (cid:105) − (cid:104) n ( t ) (cid:105) ] (cid:104) n ( t ) (cid:105) + (cid:90) dn P t ( n ) A ( n ) . (12)The latter term describes the non-Gaussianity of thecumulated displacements a n after n transitions A ( n ) = [ (cid:104) a n (cid:105) − (cid:104) a n (cid:105) ]3 (cid:104) a n (cid:105) (13)weighted by the probability to find exactly n transitionsat a time t . In what follows we use the approximationthat (cid:82) dn P t ( n ) A ( n ) ≈ A ( (cid:104) n ( t ) (cid:105) ).Eq. 12 contains two contributions: The first term in-cludes the heterogeneity of the performed jumps n in acertain time interval. It is the same quantity which wasobserved to be responsible for the superdiffusive behav-ior in the non-equilibrium system. Because this term isindependent of any length scales, one can regard it asa measure for the temporal heterogeneities of the sys-tem dynamics. The latter term reflects spatial hetero-geneities of the elementary MB transition which becomeless important at a larger number of transitions becausethe distribution of the cumulated lengths approaches aGaussian shape.In Fig. 3 α ( t ) · t is shown at different temperatures to-gether with the theoretical prediction by Eq. 12. Pleasenote that, as it was also shown by Liao et al. [18], incase of transitions between adjacent MB, α ( t ) displaysa monotonic decay. This behavior can be understoodby considering that the initial growth of α ( t ) is causedby vibrational parts (short times) and the β -relaxationprocess (at intermediate times) [12, 19] while, by con-struction, MB trajectories only resolve the α -relaxationprocess [20]. As one can see, the theoretical predic-tion allows one to fully describe the NGP at each tem-perature. One further observes that for long times, α ( t ) · t approaches a constant which corresponds to adecay of α ( t ) ∝ /t which is exactly the expectation for[ (cid:104) n ( t ) (cid:105) − (cid:104) n ( t ) (cid:105) ] / (cid:104) n ( t ) (cid:105) when the central limit theorembecomes valid (see Eq. 9).In the inset of Fig. 3 the different temporal and spatialcontributions to α ( t ) · t are shown. At very short times,the behavior of α ( t ) · t is mainly determined by A ( (cid:104) n ( t ) (cid:105) )while above t ≈ , the temporal part of Eq. 12 is foundto be the major contribution. At t ≈ , one observes A ( (cid:104) n ( t ) (cid:105) ) ∝ /t which indicates that the central limittheorem starts to hold for the distribution of the spatialdisplacement. Indeed, for the waiting time distributionthe central limit theorem is only fulfilled on a larger timescale, so that there is still a growth of α ( t ) · t .It is know from a comparison between mode-couplingtheory and Brownian dynamics simulations of BMLJ [21],that mode-coupling theory tends to strongly underesti-mate the magnitude of α ( t ) in the diffusive regime. Thisdifferences between theory and simulation are also knownfor the hard-sphere system [22, 23]. Recently, it wasfurther reported, that simplified mode-coupling theorymodels cannot reproduce the superdiffusive behavior ofa driven particle along its force direction [9]. Therefore,it is quite remarkable that the CTRW approach enablesto relate both, the non-Gaussianity of the equilibriumsystem and the superdiffusive behavior of the stationarynon-equilibrium system to have the same origin, reflect-ing the presence of the dynamic heterogeneities. Onethus might argue whether mode-coupling theory, possi-bly due to its dependence on average quantities [22], isnot able to fully describe these fluctuations. As a conse-quence both effects cannot be qualitatively reproduced. Summary.
In the present paper we have demonstrated,that a model of a biased CTRW allows to fully predictthe anomalous diffusion of a driven particle in a super-cooled medium which is characterized by equilibrium-like diffusion, superdiffusivity and long-time diffusivity. It was further shown, that the origin of the superdiffu-sivity results from temporal fluctuations of the systemdynamics which become visible due the applied bias.Indeed, also in equilibrium the same fluctuations arepresent and determine the evolution of the NGP α ( t ).Therefore, the connection between superdiffusive behav-ior and non-Gaussianity is a remarkable example, hownon-equilibrium dynamics also enables a deeper physicalunderstanding of the equilibrium system, by uncoveringessential underlying physical properties.This work was supported by DFG Research Unit 1394”Nonlinear Response to Probe Vitrification”. Further-more, C. F. E. Schroer thanks the NRW Graduate Schoolof Chemistry for funding. We acknowledge helpful dis-cussions with J. Horbach, C. Rehwald and D. Winter. LONG-TIME DIFFUSIVITY OF LARGERSYSTEMS
As we had demonstrated, the continuous time randomwalk (CTRW) analysis allows one to predict the long-time diffusivity of the driven particle aslim t →∞ σ ( t ) t = D (cid:107) f = D (cid:107) (cid:34) x (cid:107) a (cid:107) (cid:32) (cid:104) τ (cid:105)(cid:104) τ (cid:105) − (cid:33)(cid:35) . (14)For a binary mixture of N = 65 Lennard-Jones parti-cles (BMLJ65) the required CTRW quantities can be ex-plicitly computed from the metabasin (MB) trajectories.For systems larger then BMLJ65 it is not possible be-cause there is no direct access to the required quantities.Unfortunately it is known, that especially the structuralrelaxation time τ α , which reflects the higher moments ofthe waiting time distribution, displays significant finite-size effects [16, 24]. Therefore, one must expect, thatthe degree of superdiffusivity will be different at largersystem sizes. It is nevertheless possible to estimate themagnitude of long-time diffusivity from the real spacetrajectories of larger systems.We start with a relation between the average waitingtime (cid:104) τ (cid:105) and the diffusion constants D (cid:107) / ⊥ of the tracerparticle parallel and perpendicular to the direction of theforce. It is given by D (cid:107) / ⊥ = a (cid:107) / ⊥ (cid:104) τ (cid:105) (15)where a (cid:107) / ⊥ denote the apparent diffusive lengths dur-ing one MB transition [14, 17]. Furthermore, the secondmoment of the waiting time distribution (cid:104) τ (cid:105) is connectedto τ α by [15, 24] lim q →∞ τ α ( q ) = τ α = (cid:104) τ (cid:105) (cid:104) τ (cid:105) . (16)Please note, that in a strict sense the relation betweenlim q →∞ τ α ( q ) and (cid:104) τ (cid:105) is only valid for BMLJ65. We willidentify the relevant value of τ α further below. InsertingEq. 15 and Eq. 16 in Eq. 14 yields D (cid:107) f = a (cid:107) a ⊥ D ⊥ (cid:34) ∆ x (cid:107) a (cid:107) (cid:18) τ α D ⊥ a ⊥ − (cid:19)(cid:35) . (17)In contrast to Eq. 14, Eq. 17 only depends on micro-scopic lengths and quantities which are measurable inreal space. Please note, that the diffusion constant D (cid:107) was substituted by a (cid:107) a ⊥ D ⊥ because D (cid:107) is not accessibledue to the superdiffusive behavior of σ ( t ). -1 -1 F ⊥ / v , BMLJ1560a ⊥ / < ∆ x || >, BMLJ65 FIG. 4. Ratios of the diffusion coefficient D ⊥ perpendicular tothe force direction and the drift velocity v of BMLJ1560 andthe corresponding microscopic length scales a ⊥ and ∆ x (cid:107) ofBMLJ65 versus the applied Force F . Both systems were sim-ulated at T = 0 . We now assume that the microscopic length scales inEq. 17 are the same for BMLJ65 and larger systems. Thisassumption is made because the microscopic lengths referto the local spatial displacement of single particles duringsingle relaxation processes which should be largely inde-pendent from the system size. To verify this, we havecomputed the ratio of the drift velocity v and the perpen-dicular diffusion constant D ⊥ of a system with N = 1560particles (BMLJ1560). Because v = ∆ x (cid:107) (cid:104) τ (cid:105) and Eq. 15 (forBMLJ65), this ratio can be identified with a ⊥ / x (cid:107) . Ifthe assumption holds, a ⊥ / ∆ x (cid:107) of BMLJ65 should be es-sentially the same as 2 D ⊥ /v of BMLJ1560. The corre-sponding plot is shown in Fig. 4. One can observe, thatthe mismatch between these curve is less than 20% sothat the independence of the microscopic length scaleson system size indeed seems to be reasonable.The missing observable τ α in Eq. 17 is determined bycomputing the incoherent scattering function S ⊥ ( q, t ) = (cid:104) cos ( q ( x ⊥ ( t ) − x ⊥ ( t )) ) (cid:105) (18)of the tracer particle perpendicular to the force direction.The reason for the choice of the perpendicular directionis, that one would expect, analogous to the superdiffu-sive behavior of σ ( t ), an additional relaxation along theparallel direction which does not reflect the underlyingwaiting time distribution. Following the procedure ofRehwald et al. [24] we have computed the apparent wave-vector relaxation τ ( q ) = (cid:90) ∞ dt S ⊥ ( q, t ) . (19)and fitted the resulting curves in the range q ≤ τ ( q ) = τ α + 1 q · D ⊥ , (20) -4 -3 -2 σ / t t F = 7.50F = 5.00F = 1.75F = 1.00
FIG. 5. Centered MSD σ ( t ) versus time t for BMLJ1560 at T = 0 . finally allowing us to determine the local relaxation time τ α .In Fig. 5 σ ( t ) of BMLJ1560 is shown together with thepredicted equilibrium-like and long-time diffusion con-stants. One observe a quantitative agreement betweenthe numerical data and the theoretical prediction. Com-pared to BMLJ65, the factor between equilibrium-likeand long-time diffusion is smaller and the long-time dif-fusivity appears earlier. This behavior indicates a nar-rowing of the underlying waiting time distribution ofBMLJ1560. One can rationalize this by regarding thelarge system as a composition of small elementary systemwhich are coupled so that relaxation processes of singlesubsystems induce further relaxation events in adjacentsystems. This concept has been discussed in [20, 24]. ∗ [email protected] † [email protected][1] M. D. Ediger and P. Harrowell, J. Chem. Phys. ,080901 (2012)[2] F. Fujara, B. Geil, H. Sillescu, and G. Fleischer, Z. Phys.B , 195 (1992)[3] L. Berthier, Phys. Rev. E , 020201 (2004)[4] D. Winter, J. Horbach, P. Virnau, and K. Binder, Phys.Rev. Lett. , 028303 (2012)[5] J.-P. Bouchaud and A. Georges, Phys. Rep. , 127(1990)[6] I. Gazuz, A. M. Puertas, T. Voigtmann, and M. Fuchs,Phys. Rev. Lett. , 248302 (2009)[7] M. V. Gnann, I. Gazuz, A. M. Puertas, M. Fuchs, andT. Voigtmann, Soft Matter , 1390 (2011)[8] C. J. Harrer, A. M. Puertas, Voigtmann, and M. Fuchs,Z. Phys. Chem. , 779 (2012)[9] C. J. Harrer, D. Winter, J. Horbach, M. Fuchs, andT. Voigtmann, J. Phys.: Condens. Matter , 428429(2012)[10] R. L. Jack, D. Kelsey, J. P. Garrahan, and D. Chandler,Phys. Rev. E , 011506 (2008)[11] S.-H. Lee and D. G. Grier, Phys. Rev. Lett. , 190601(2006)[12] W. Kob and H. C. Andersen, Phys. Rev. E , 4626(1995)[13] S. Nos´e, J. Chem. Phys. , 511 (1984)[14] B. Doliwa and A. Heuer, Phys. Rev. E , 030501 (2003)[15] O. Rubner and A. Heuer, Phys. Rev. E , 011504 (2008)[16] A. Heuer, J. Phys.: Condens. Mat. , 373101 (2008)[17] C. F. E. Schroer and A. Heuer, arXiv:1209.6526[18] C. Y. Liao and S.-H. Chen, Phys. Rev. E , 031202(2001)[19] D. Caprion, J. Matsui, and H. R. Schober, Phys. Rev.Lett. , 4293 (2000)[20] C. Rehwald and A. Heuer, arXiv:1209.5515[21] E. Flenner and G. Szamel, Phys. Rev. E , 031508(2005)[22] M. Fuchs, W. G¨otze, and M. R. Mayr, Phys. Rev. E ,3384 (1998)[23] T. Voigtmann, A. M. Puertas, and M. Fuchs, Phys. Rev.E , 061506 (2004)[24] C. Rehwald, O. Rubner, and A. Heuer, Phys. Rev. Lett.105