The study of the properties of glass-forming liquids is difficult for many reasons. Analytic solutions of mean-field models are usually available only for systems embedded in a space with an unphysically high number of spatial dimensions; on the experimental and numerical side, the study of the properties of metastable glassy states requires thermalizing the system in the supercooled liquid phase, where the thermalization time may be extremely large. We consider here a hard-sphere mean-field model that is solvable in any number of spatial dimensions; moreover, we easily obtain thermalized configurations even in the glass phase. We study the 3D version of this model and we perform Monte Carlo simulations that mimic heating and cooling experiments performed on ultrastable glasses. The numerical findings are in good agreement with the analytical results and qualitatively capture the features of ultrastable glasses observed in experiments

Mariani, Manuel Sebastian

Parisi, Giorgio

Rainone, Corrado

arXiv

PARISI, Giorgio

RAINONE, CORRADO

Archivio della ricerca- Università di Roma La Sapienza

Calorimetric glass transition in a mean-field theory approach

Significance
          Understanding the properties of glasses is one of the major open challenges of theoretical physics. Making analytical predictions is usually very difficult for the known glassy models. Moreover, in experiments and numerical simulations thermalization of glasses cannot be achieved without sophisticated procedures, such as the vapor deposition technique. In this work we study a glassy model that is simple enough to be analytically solved and that can be thermalized in the glassy phase with a simple numerical method, opening the door to an intensive comparison between replica theory predictions and numerical outcomes.</jats:p

Manuel Sebastian Mariani

Giorgio Parisi

Corrado Rainone

Crossref

Proceedings of the National Academy of Sciences

RERO DOC Digital Library

Supporting InformationMariani et al. 10.1073/pnas.1500125112Simulation DetailsPlanting Method.We generate the initial configuration as follows:• We generate randomly the positions x of the N spheres, witha uniform probability distribution over the simulation box.• For each sphere i, we generate its shifts lij with uniform dis-tribution in the simulation box. Each shift lij is accepted if andonly ifjxi − xj + lijj> 1; [S1]otherwise it is generated again until condition S1 is satisfied.It can be shown (1) that in the infinite volume limit this pro-cedure generates a thermalized initial configuration where theannealed average of entropy is equal to the quenched one (i.e.,a configuration in the liquid phase) (1).We have tested that the procedure works, that the config-urations that we generate are at equilibrium and their propertiesare independent from time (as long as the density remainsconstant). The result for the observable gðr; 0Þ for plantingdensity φ0 = 2:5 is shown in Fig. S1. We can see in the low rregion a behavior compatible with the equilibrium one:gðx− yÞ= expð−vðx− yÞÞ=0 if  jx− yj<D1 if  jx− yj>D [S2]within 1 or 2 SDs. For example, for the contact point we havegð1Þ= 0:987± 0:019. We do not study the large r behavior ofgðr; 0Þ, which shows a decay caused by the finite size of thesimulation box and cannot be compared with the infinite-volumeanalytical result (Eq. S2).Monte Carlo Evolution Algorithm and Verlet Lists.We used a MonteCarlo evolution algorithm. At each step t we propose a dis-placement ΔxiðtÞ to each sphere i. The proposed displacement isgenerated uniformly in a 3D sphere of radius δ, where δ= 0:2 isa fixed parameter. The proposed displacement is accepted if andonly if the conditionjxi +Δxi − xj + lijj> 1 [S3]is satisfied for all other spheres j≠ i; otherwise it is rejected. Thisstochastic dynamics satisfies the detailed balance property, thusimplying relaxation toward equilibrium. To reduce computa-tional time we use Verlet lists (see, e.g., ref. 2).Radial Distribution Function Computation. We denote by ΩΔ;lðr; tÞthe number of sphere couples such thatjxiðtÞ− xjðtÞ+ lijj∈ ½r; r+Δand we define a fixed time radial distribution functiongΔ; lðr; tÞ= 14NφΩΔ; lðr; tÞðr+ΔrÞ3 − r3:The parameter Δ is fixed and corresponds to the histogram binlength. We choose Δ= 0:05. To gain central processing unit timewe perform measurements at equispaced intervals in time (typ-ically every 20 Montecarlo sweeps). In our simulation we furtheraverage over the different starting configuration. The number ofconfigurations is M = 6, a reasonable value for a self-averagingquantity. The statistical error is estimated from sample-to-sample fluctuations.Decompression Protocol. In the following we denote by k thelogarithm in base 2 of the number of Monte Carlo steps per-formed for each density value. We are interested in a decom-pression protocol that mimics the physical heating of a glass. Westart from a planted initial configuration at a density φ0 in theglassy region (φ0 >φd) in the liquid phase (φ0 <φk), equivalentto the supercooled liquid region in real glass formers. To be safewe choose φ0 values between 2 and 3. After planting, we de-compress the system changing the box size, leaving the integersphere positions unchanged, causing a jump φ0→φ1 of density,with φ1 <φ0. The system evolves for 2k Monte Carlo steps atdensity φ1, then density jumps again to a lower value φ2 <φ1, thesystem evolves for 2k steps at density φ2, and so on.Compression Protocol. To compress the system, we increase theparticle radius until the particles touch. When this happens, MonteCarlo steps are performed to separate the particles; afterward, theradius is increased again, until the final density is reached.The procedure is slow and therefore the final system is nearlythermalized. After the final density is reached, we run a longsimulation for final thermalization and we take measurementsonly in the second half of the run.Mean Square DisplacementFor each density value φ we measured the relaxation time by fittingthe plateau escape region of the MSD ΔðtÞ with the power lawΔðtÞ= a1+ ctb:We discarded the fast relaxation region. We considered for eachdensity value only points with t> 211. We defined the relaxationtime using the relation ΔðτRÞ= 1:5  Δð211Þ. The value 1.5 is some-what arbitrary: It should be neither too small, to reduce noiseeffects, nor too large, to allow us to obtain relaxation time valuesnot too large compared with the typical time scales of our sim-ulations. Using this procedure we obtained the value of relaxa-tion time for each value of density. For the highest studied valuesof density (i.e., φ= 1:7; 1:68; 1:66), we obtained τR > 222, meaningthat 222 steps were not sufficient to observe relaxation: These areextrapolated points and we discarded them.Computation of Isocomplexity LinesThe first step is the computation of the replicated entropy Sðm;φÞas a functional of the replicated density. Denoting by x = fxðaÞgthe set of the positions of the m replicas, by ρðxÞ the replicateddensity, for the mean-field MK model we haveSðm;φÞ½ρ=−Zdx  ρxlogρx+12Zdx   y  ρxρyfx − y+N logðNÞ;[S4]wherefx − y= exp Xma=1vxðaÞ − yðaÞ + l!−11Published in 3URFHHGLQJVRIWKH1DWLRQDO$FDGHP\RI6FLHQFHV±which should be cited to refer to this work.http://doc.rero.chis the replicated Mayer function. In practice, the replicated den-sity is usually parameterized asρx=NVZdX Yma=1gAxðaÞ −X; [S5]where xðaÞ is the position of replica a and gA is the Gaussianfunction with variance A. The parameter A represents the aver-age cage radius and can be also interpreted as the plateau valueof the mean square displacement of particles in the caging re-gime. For the mean-field MK model of hard spheres, combiningresults presented in appendix A of ref. 3 and in section VI ofref. 4, putting parameterization S5 in S4 we obtainSðm;φ;AÞ= logðNÞ− logφVdð1Þ+ Sharmðm; AÞ− 2d−1φð1−Gðm; AÞÞ;[S6]where Gðm;AÞ and Sharmðm;AÞ are defined in ref. 4. We stressthat Eq. S6 is the same of a pure hard-sphere system (that is,without shifts) in infinite dimension. We must then optimizethis with respect to A (4), getting the equation1φ^=Fðm; Aðm;φÞÞ; [S7]where φ^= 2dφ=d andFðm;AÞ= A1−m∂Gðm;AÞ∂A: [S8]Eq. S16 and the form of the function F (4) imply a first-ordertransition at the endpoint of metastable curves, both in the ðm;φÞplane and in the pressure-density plane. We discuss this result inSingularity of Cage Radius and Pressure at the Clustering Line.We plug the solution Aðm;φÞ of Eq. S16 in Eq. S6, obtainingSðm;φÞ. Using then the replica relationsseqðm;φÞ= ∂Sðm;φÞ∂m ; [S9]Σeqðm;φÞ=m2∂	m−1Sðm;φÞ∂m; [S10]on Eq. S6 we get the following expression for the complexity:Σðm;φ;AÞ= Sðm;φ;AÞ−d2ð1+m+m logð2πAÞÞ+ 2d−1φHðm;AÞ;[S11]whereHðm;AÞ=−m ∂Gðm;AÞ∂m:The only remaining task is now to solve the equationΣðm;φÞ= Σ0 = Σð1;φ0Þwith respect to m, for various values of φ. Because in the clus-tering region Σ is a decreasing function of m at fixed φ, thesolution mΣ0ðφÞ of the isocomplexity condition can be found witha simple bisection algorithm. We start at φ0, then we change thedensity of a small amount Δφ, and we use bisection to find thesolution mΣ0 of the equationΣðmΣ0;φ0 +ΔφÞ=Σ0: [S12]Once it has been found, we change the density again and the pro-cedure is repeated until the clustering line is reached and the so-lution for A disappears.In-State PressureIn principle the ratio pðφÞ between physical pressure PðφÞ of astate of complexity Σ0 and density φ can be computed usingthe relationpðφÞ= ρ−1PðφÞ=−φ ddφseqðmðφÞ;φÞ; [S13]where mðφÞ solves Eq. S12 for a given complexity value Σ0.Eq. S13 is uncomfortable because it involves also the partial de-rivative with respect to m. Instead of directly using Eq. S13 wedefine a modified replicated entropy for each complexity value Σ0:~Sðm;φÞ= Sðm;φÞ−Σ0 =mseqðm;φÞ+Σeqðm;φÞ−Σ0:Isocomplexity Eq. S12 is then equivalent to the equationm2∂∂mm−1~Sðm;φÞm=mðφÞ= 0: [S14]Therefore, the pressure of a metastable state can be expressed interms of total entropy S:pðφÞ=− φmðφÞ∂∂φ~Sðm;φÞm=mðφÞ=−φmðφÞ∂∂φðSðm;φÞÞm=mðφÞ:[S15]Eq. S15 is all we need to pass from ðm;φÞ plane to the pressure-density plane. It is also easy to pass to the contact value of radialdistribution function through the relation (5) gð1Þ= ðp− 1Þ=ð4φÞ.Singularity of Cage Radius and Pressure at the Clustering LineFor each metastable curvemðφÞ the clustering point φd is definedas the lowest φ value for which equation1φ^=Fðm;Aðm;φÞÞ [S16]admits a finite solution AðmðφÞ;φÞ. The corresponding cage ra-dius Amax is the value of A for which FðmðφdÞ;AÞ has a maximum(4). Expanding Fðm;AÞ in Taylor power series near φd and re-arranging terms we obtain from Eq. S16AðφÞ− Amax =Cﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃφ^− φ^dp+ Oðφ−φdÞ; [S17]where Amax =AðmðφdÞ;φdÞ, φ^≡ 2dd φ and the constant C is given byC=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ−12∂2FðmðφdÞ; AmaxÞ∂A2 1cφd2 + ∂FðmðφdÞ; AmaxÞ∂m ∂mðφdÞ∂φ^!vuut :Eq. S17 implies thatdAðφÞdφ=−C=2ðφ^− φ^dÞ1=2+ regular  terms; [S18]that is, A′ðφÞ has a square-root singularity at φ=φd.We show now that this square-root singularity is transmitted tocompressibility. Expanding the expression for the pressure2http://doc.rero.chpðφÞ= 1mðφÞ1+ 2d−1φð1−Gðmd;Aðmðφ;Σ0Þ;φÞÞÞ; [S19]in φ=φd we obtainpðφ^Þ= pðφ^dÞ−Bðφ^Þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃφ^− φ^dp+Oðφ^− φ^dÞ; [S20]where we defined the (positive) constantBðφ^Þ=−dð1−mðφ^ÞÞ2mðφ^ÞAmaxφ^φ^dC:Deriving Eq. S20 we obtaindpðφ^Þdφ^=−Bðφ^dÞ=2ðφ^− φ^dÞ1=2+ regular  terms; [S21]that is, the derivative p′ðφ^Þ of the pressure has a singularity inφ^= φ^d with the same critical exponent of A′ðφ^Þ. This fact impliesan overshoot in the pressure as the system escapes from themetastable state. Indeed, the same overshoot can be seen alsoin the state-following method (6), not only in the pressure vs.density plane, but also in the shear stress vs. shear strain plane.1. Krzakala F, Zdeborová L (2009) Hiding quiet solutions in random constraint satisfactionproblems. Phys Rev Lett 102(23):238701.2. Frenkel D, Smith B (2002) Understanding Molecular Simulation (Academic, New York).3. Mari R, Kurchan J (2011) Dynamical transition of glasses: From exact to approximate.J Chem Phys 135(12):124504.4. Parisi G, Zamponi F (2010) Mean field theory of hard sphere glasses and jamming. RevMod Phys 82(1):789.5. Hansen JP, McDonald IR (2006) Theory of Simple Liquids (Academic, New York).6. Rainone C, Urbani P, Yoshino H, Zamponi F (2015) Following the evolution of hardsphere glasses in infinite dimensions under external perturbations: Compression andshear strain. Phys Rev Lett 114(1):015701. 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1  1.2  1.4  1.6  1.8  2  2.2  2.4  2.6g(r)phi=2.5equilibriumFig. S1. Radial distribution function gðrÞ of the planted configuration at density φ0 = 2:5.3http://doc.rero.ch

http://dx.doi.org/10.1073/pnas.1500125112

Calorimetric glass transition in a mean-field theory approach

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