Jamming as a random first-order percolation transition
Antonio Piscitelli, Antonio Coniglio, Annalisa Fierro, Massimo Pica Ciamarra
aa r X i v : . [ c ond - m a t . s o f t ] F e b Jamming as a random first-order percolation transition
Antonio Piscitelli a,c , Antonio Coniglio a,b , Annalisa Fierro a , Massimo PicaCiamarra a,c a CNR-SPIN, c/o Complesso di Monte S. Angelo, via Cinthia - 80126 - Napoli, Italy b Physics Department, Universit`a degli Studi di Napoli “Federico II”, Napoli, Italy c Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore
Abstract
We determine the dimensional dependence of the percolative exponents ofthe jamming transition via numerical simulations in four and five spatialdimensions. These novel results complement literature ones, and establishjamming as a mixed first-order percolation transition, with critical exponents β = 0, γ = 2, α = 0 and the finite size scaling exponent ν ∗ = 2 /d for valuesof the spatial dimension d ≥
2. We argue that the upper critical dimensionis d u = 2 and the connectedness length exponent is ν = 1.
1. Introduction
Granular materials, emulsions, foams, and all systems of particles inter-acting via finite ranged repulsive interactions, for which thermal motion isnegligible, exhibit a jamming transition on increasing the volume fraction,signalling the onset of mechanical rigidity [1]. The transition has a first-ordercharacter, when approached from the unjammed phase, as the density ofjammed particles exhibits a jump, from zero to a macroscopic value. Equiv-alently, the average contact number per particle, Z , jumps discontinuouslyfrom Z = 0, in the unjammed phase, to the isostatic value Z iso = 2 d .The approach to jamming naturally lends itself to a percolative descrip-tion, with control parameter ǫ = | ( φ − φ j ) /φ j | , where φ is the fraction of thetotal volume occupied by the particles and φ j is the jamming threshold. Themixed character of the transition, however, made difficult the identificationof the percolative critical exponents, and the design of simple models [2, 3]able to capture the jamming phenomenology. In percolation theory [4, 5, 6],the density of the percolating cluster, which is the order parameter, scales as Preprint submitted to Physica A February 3, 2021 ∝ ǫ β for ǫ > P = 0.The mean cluster size S = P s n ( s ) / P sn ( s ), where n ( s ) is the number ofclusters of size s , scales as S ∝ | ǫ | − γ . The pair connected correlation func-tion in d spatial dimensions is g pc ( r ) ∝ r − d +2 − η f ( − r/ξ ), with f ( x ) beingan exponential decreasing function for large x and the connectedness lengthdiverging at the percolation transition as ξ ∝ | ǫ | − ν . The fractal dimension ofthe cluster is D = d − β/ν . Different set of exponents, which satisfy the scal-ing and hyperscaling law 2 β + γ = 2 − α = dν , identify different universalityclasses. More precisely hyperscaling relations contain the space dimension d . While scaling relations are verified for all dimensions d , hyperscaling donot hold for dimensions greater than the upper critical dimension d u , wheremean field exponents hold.To investigate the percolative character of the jamming transition, we con-sider a soft sphere packing at zero temperature prepared according to someprotocol. Since the number of touching spheres is zero below the thresholdand becomes finite at the threshold, the percolation order parameter P , de-fined as the density of touching spheres in the spanning cluster, jumps fromzero to a finite value P c at the transition [7, 8]. This discontinuity suggests β = 0. Finite size scaling investigations allow to extract an exponent ν ∗ which below or at the upper critical dimensionality d u coincides with theconnectedness length exponent ν . Conversely, for d > d u , while ν is givenby its mean field value independent on the dimensionality, ν ∗ depends onthe dimensionality [9, 10] in such a way that using ν ∗ instead of ν , the hy-perscaling relation are satisfied also above d u . The precise value of ν ∗ ishowever debated [1, 7, 8, 11, 12]. In particular, earlier numerical works gave ν ∗ ≃ . ± .
08, in both d = 2 and d = 3 spatial dimensions, suggestinga value ν ∗ = 2 / ν ∗ ≃ d = 2 and ν ∗ ≃ .
66 for d = 3,suggesting that the exponent ν ∗ might depend on the spatial dimensionalityas ν ∗ = 2 /d .To set this issue, here we consider the case for d = 4 and d = 5. We haveperformed large scale simulations and an accurate analysis, which stronglysupport ν ∗ = 2 /d also for d = 4 and d = 5. We present these results inSect. 2. Interestingly the value ν ∗ = 2 /d is that suggested for the idealglass transition by Kirkpatrick, Tirumalai and Wolyness within the RFOTtheory [14]. In Sect. 3 we describe jamming as random first-order percolationtransition. In Sect. 4 we show explicitly that jamming of hard spheres in 1 d is described by such percolative model although with different exponents.2 . Numerical results We have developed a molecular dynamics code to simulate systems ofHarmonic spheres in arbitrary dimensions. We consider monodisperse sys-tems of sphere of radius R and volume v d = π d/ R d Γ( d/ , only interacting if theirseparation r is smaller than 2 R , with energy given by V ( r ) = ǫ ( r − R ) .To investigate the jamming transition, we randomly place N spheres in ahypercube of volume L d , where L fixes the volume fraction φ = N v d /L d . Wethen minimize the energy of the system combining the conjugate-gradientminimization algorithm and a damped dynamics, and record properties ofthe final configuration. The final value of the energy per particle after theminimization, E , allows us to distinguish between jammed configuration, E > E t , and unjammed ones, E < E t , with E t a threshold value. Here,we fix E t = 10 − ǫ but have checked that the results are insensitive to thischoice.As in d = 2 and d = 3, also for d = 4 and d = 5 the order parameter P ,the fraction of particles of the percolating cluster made of touching spheres,jumps from zero to a finite value. Hence, the percolative exponent associatedwith the behaviour of the order parameter is β = 0, for all dimensions inves-tigated and is expected to be zero for any dimensions. At the same time, themean contact number is zero below the transition and jumps to the isostaticvalue at the transition Z iso . Specifically, we have found Z − Z iso ∝ p , with p the pressure.Repeating the minimization procedure 100 times, starting from differentrandom configurations, we associate to each system size and volume fractiona jamming probability P J ( φ, N ), which is the fraction of our minimizationsending in a jammed state. The jamming probability vanishes for small vol-ume fraction, while it approaches 1 for large volume fraction. As observedin both 2 d and 3 d [12, 15], also in 4 d and 5 d the volume fraction dependenceis well described by an error function, P J ( φ, N ) = 12 (cid:20) (cid:18) φ − φ J ( N ) σ J ( N ) √ (cid:19)(cid:21) . (1)Hence, each system size N is characterized by a typical jamming volumefraction, φ J ( N ), and standard deviation, σ J ( N ), we extract from a numericalfit of the P J ( φ, N ) data. 3 N -4 -3 -2 σ J ( N ) N -3 N -0.52 N -3/10 d=4 d=5(a) (b)N -0.52 N -3/8 Figure 1: Size dependence of the jamming probability standard deviation σ J (see Eq.(1))in d = 4 (a) and d = 5 (b). Data are compatible with N − Ω with Ω = 1 /
2, implying ν ∗ = 1 / Ω d = 2 /d . We expect [7, 8] σ J ( N ) = σ L − /ν ∗ , which gives σ J ( N ) ∝ N − Ω withΩ = 1 /dν ∗ , since the system size is L ∝ N /d . In Ref. [12], the value ofΩ = 0 . d = 2 and 3, giving ν ∗ = 1and 2 /
3, respectively. These findings suggest ν ∗ = 2 /d for all values of d .However, such predictions are non easily discernible from the value ν ∗ = 2 / d = 2 and 3.In Fig. 1a and Fig. 1b the standard deviation σ J ( N ) is plotted as functionof N for d = 4 and d = 5, respectively. The best fit of σ J ( N ) as N − Ω givesΩ = 0 . ± .
01 in both cases, whereas the values Ω ≃ / /
10, obtainedfrom ν ∗ = 2 /
3, do not well described our data, as we directly demonstratein figures.To further support the value ν ∗ = 2 /d , we show in Fig. 2(a) ( d = 4) and(b) ( d = 5) that data for the jamming probability for different N collapse ona master curve, when plotted as a function of ( φ − φ J ( N )) N Ω , with Ω = 1 / n c X N >N Z | P J ( x, N ) − P J ( x, N ) | dx, (2)where n c is the number of distinct N , N couples, and x (Ω) = ( φ − φ J ( N )) N Ω .Smaller values of the spread indicate a better data collapse. Fig. 2(c) and (d)4 ( φ - φ J (N))N P J ( φ , N ) N=200N=800N=1600N=3200N=12800 -0.2 0 0.2 ( φ - φ J (N))N N=1600N=3200N=6400N=9600N=12800 Ω s p r ea d Ω d=4 d=5d=4 d=5(a) (b)(c) (d) Figure 2: The jamming probabilities fall on the same master curve when plotted versus( φ − φ J ( N )) N Ω with Ω = 1 /
2, in both d = 4 (a) and d = 5 (b). The dependence of thespreading observed in the collapse on the exponent Ω, confirms that Ω = 1 / d = 4 (c) and d = 5 (d). illustrate the dependence of the spread on Ω, in d = 4 and d = 5. Regardlessof the dimensionality, we found the spread to have a minimum at Ω ≃ . / /
10 (red squares), suggested by ν ∗ = 2 /
3, provide a worse data collapse.Considering previous results [12, 15] in d = 2 and d = 3, and our presentones in d = 4 and d = 5, it appears that Ω ≃ .
5, regardless of the dimen-sionality. This suggests that the exponent ν ∗ depends on the dimensionalityas ν ∗ = 2 /d . 5 . Random first-order percolation transition We have found the approach to jamming to be a percolation transitioncharacterised by a jump in the order parameter with an exponent β = 0, a di-vergence in the finite size scaling length with an exponent ν ∗ = 2 /d .The criti-cal exponents α and γ can be obtained by scaling argument: α = 2 − dν ∗ = 0and γ = dν ∗ − β = 2 − α = 2. This transition has a mixed character, beingintermediate between a first- and a second-order transition, as the randomfirst-order transition introduced in the context of the glass transition. There-fore, we name it random first-order percolation transition [14]. Summarising,the exponents of this percolation transition are given by: β = 0 , γ = 2 , α = 0 , ν ∗ = 2 /d, (3)and satisfy the scaling and hyperscaling law:2 β + γ = 2 − α = dν ∗ . (4)Interestingly we have α = 0, β = 0, γ = 2, which are independent onthe dimensionality. The only exponent depending on the dimensionality is ν ∗ = 2 /d . Although ν ∗ = 2 /d has been verified only for d = 2 , , , d .We discuss now the question concerning the upper critical dimensionality d u . It has been suggested in the literature [7, 16], that for jamming d u = 2.Our result, that α , β , γ for d ≥ d u = 2. At d = d u = 2, ν = ν ∗ = 1 and, since the value of the exponentat the upper critical dimensionality coincides with its mean field value, wehave ν = 1 for any d ≥
2. In conclusion if d u = 2 is the upper criticaldimension, the critical exponents for this random first-order percolation are,for any d ≥ α = 0, β = 0, γ = 2, and ν = 1. These are mean fieldexponents, which obey scaling α + 2 β + γ = 2 for any d and hyperscaling2 β + γ = dν only for d = 2. Hyperscaling is restored for any d only if ν isreplaced by ν ∗ .We now show that the above percolative exponents are compatible withthe existence of clusters of typical size ξ . Specifically, while in the next sectionwe will give an appropriate definition of clusters for 1 d hard spheres, here wefocus on d = 2. We propose a definition reminiscent of the point-to-set proce-dure used in the context of glass transition by Biroli et al. [13] to characterize6 P r ob j a mm i ng R φ = P r ob j a mm i ng R ( φ c - φ ) ν ; ν = 2/d φ =0.820.8250.830.83250.8350.8380.839 Figure 3: If the particles enclosed in a circle of radius R of an unjammed configurationof volume fraction φ have their positions randomized within the circle, the subsequentenergy minimization conducted keeping frozen the particles outside the circle may leadto a jammed or to an unjammed configuration. The top panel illustrates the probabilitythat the resulting configuration is jammed as a function of R , for different values of thevolume fraction. Each data point is the average of 10 trials. The jamming probabilitiesfunction collapse on a master curve with R is scaled by R ∗ ∝ ( φ c − φ ) − ν and ν = 1. φ below jamming, we consider a configuration of un-jammed particles generated with the numerical protocol described in Sect.2, an energy minimization starting from a random arrangement of particles.We then focus on a set of particles enclosed in a circle of radius R , con-sidering frozen the particles outside this circle. We randomize the particleswithin the circle, ensuring that they remain within the circle, and then min-imize the energy, always keeping the outer particles frozen. As a result, theparticles inside the circle might be both in a jammed and in an unjammedconfiguration. We calculate the jamming probability of this set of particles,as a function of R and the volume fraction φ . Such probability will be a de-creasing function of R . Say R ∗ the value of R corresponding to the jammingprobability equal to 1 / R ∗ defines a typical cluster at volume fraction φ with linear dimension ξ = R ∗ .For large values of R the procedure is not much different from the finite-sizescaling approach. Therefore we expect that, as φ approaches the jammingthreshold, R ∗ diverges with the exponent ν = 1. Numerical simulations in2 d show indeed that this is the case (see Fig. 3). Moreover, at the threshold,the critical cluster coincides with the jammed percolating one. For dimension d >
4. Jamming in 1 d In d = 1, jamming corresponds to a percolation problem which can besolved exactly. We first consider the random percolation model on a lat-tice [21], where the percolation in d = 1 has the properties of a randomfirst-order percolation transition. Indeed the cluster size distribution is givenby n ( s ) = (1 − p ) p s , where p is the probability that a site is occupied, themean cluster size is given by S = (1 + p ) / (1 − p ) diverging at the percolationthreshold p c = 1 with an exponent γ = 1. The density of sites in the span-ning cluster is zero for p < p c = 1 with the critical exponent β = 0. The other critical exponents aregiven by α = 1, ν = 1 and τ = 2. 8e consider now jamming of hard sphere model in d = 1. In this case,the density of percolating jammed sites is 0 below the jamming transitionand then jumps to 1 at volume fraction φ j = 1, where a macroscopic numberof touching spheres appears. Below the jamming threshold, the spheres donot touch. So one may conclude naively that there are no finite clustersbelow jamming. However, we may define appropriate non-trivial clustersin the following way. Consider a configuration of particles distributed on aone-dimensional system of length L = M d , where d is the diameter of theparticles and M for simplicity is an integer number. Divide then the length L in M segments of size d . Each particle will overlap with part of two adjacentsegments, one overlap being smaller than the other. Then virtually shift theposition of each particle to the centre of the segment where the overlap islarger. This procedure will lead to configurations of particles which mapwith one-dimensional percolation on a lattice. The clusters so defined willpercolate at the jamming transition, with the same critical exponents of one-dimensional percolation on a lattice.In conclusion, this example shows how to define clusters in the unjammedphase, in 1 d . Extension in higher dimensions, as shown above, due to topo-logical randomness, needs a more elaborate definition.
5. Conclusions
Our numerical investigation of the jamming transition in d = 4 and d = 5spatial dimensions are consistent with previous results in d = 2 and d =3. Taken together, all these results indicate that the approach to jammingcan be described as random first-order percolation transition, with criticalexponents β = 0, γ = 2, α = 0 and finite size scaling exponent ν ∗ = 2 /d .for any d ≥
2. We have argued that the upper critical dimensionality forthis mixed-order random percolation model is d u = 2 consistent with thecommon idea in the literature that d u = 2 is the upper critical dimension forjamming transition. Consequently the connectedness length exponent ν = 1for any d ≥ cknowledgements A.F. acknowledges financial support of the MIURPRIN 2017WZFTZP “Stochastic forecasting in complex systems”. AP andMPC acknowledge support from the Singapore Ministry of Education throughthe Academic Research Fund (Tier 2) MOE2017-T2-1-066 (S) and from theNational Research Foundation Singapore, and are grateful to the NationalSupercomputing Centre (NSCC) of Singapore for providing computationalresources.