Probing Ergodicity in Granular Matter

Fabien Paillusson and Daan Frenkel

Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom(Received 23 July 2012; revised manuscript received 27 September 2012; published 13 November 2012)

When a granular system is tapped, its volume changes. Here, using a well-defined macroscopic

protocol, we prepare an ensemble of granular systems and track the statistics of volume changes as a

function of the number of taps. This is in contrast to previous studies, which have focused on single

trajectories and assumed ergodicity. We devise a new method to assess the convergence properties of a

sequence of ensemble volume histograms and introduce a reasonable approximate version of an invariant

histogram. We then compare these invariant histograms with histograms generated by sampling a long

trajectory for one system and observe nonergodicity, which we quantify. Finally, we use the overlapping

histogram method to assess potential compatibility with Edwards’ canonical assumption. Our histograms

are incompatible with this assumption.

DOI: 10.1103/PhysRevLett.109.208001 PACS numbers: 45.70.�n, 05.90.+m

Introduction.—In 1989 Edwards and Oakeshott [1] pro-posed that the statistical properties of staticpowders could be described by a formalism analogous toGibbsian statistical mechanics. In the Edwards formalism,the volume of allowed stable packings plays a role analo-gous to that of energy in Gibbs statistical mechanics forsystems in thermodynamic equilibrium. Reference [1]introduced a temperaturelike quantity called the compac-tivity X for packing ensembles where the packing volumeV is not fixed. Each possible static configuration occursthen with a frequency weighted by the factor expð�V=XÞ.

Over the past two decades, Edwards’ theory has beenmuch debated: analytical [2–4], numerical [5–7], andexperimental [6,8] studies have been done without reach-ing a clear consensus. While some of these studies observespecific features that support Edwards’ assumption of acanonical prior, others do not. One reason why these resultsappear to disagree is that they typically employ differentexperimental or numerical protocols to get a set of powderpackings: Edwards’ conjecture may apply to some but notto others.

A typical choice for the protocol to generate a granularpacking is a vertical tapping of a collection of hard,spherical particles in a container. Provided the successivetappings are not too closely spaced, the tapping protocol ischaracterized only by the tapping amplitude [9]. A numberof ‘‘tapping’’ studies have obtained measures of the com-pactivity by probing global volume fluctuations over timefor single system trajectories [6,10,11].

Recently, McNamara et al. [12] carried out a carefulanalysis of volume histograms sampled over time.However, if the systems under study are not ergodic—ana priori assumption in most of these studies—then timeaverages over fluctuations for one system need not be thesame as averages over an ensemble of systems. In fact,while systems in thermal equilibrium do satisfy ergodicty,it is not clear that granular systems do the same over the

course of a tapping process. To study this question, oneshould ideally look at an ensemble of systems preparedin the same macroscopic way and follow the evolution oftheir volume with time. To our knowledge, such a study isstill lacking.In this Letter, we use numerical simulations to study the

evolution of an ensemble of granular systems preparedwith the same macroscopic protocol. We analyze the rateof convergence of these histograms and find that, undercertain conditions, individual trajectories are nonergodic.Finally, we find that our results are incompatible withEdwards’ canonical hypothesis.‘‘Equilibrium’’ distribution.—Numerical simulation:

The shaking experiment is simulated using a dissipative,event-driven molecular dynamics simulation of frictionlesshard spheres subject to gravity. We use lateral periodicboundary conditions and place a wall at the bottom ofthe simulation box.A known problem with event-driven schemes applied to

dissipative systems is that a linear restitution coefficientmakes the simulation ‘‘freeze’’ locally in higher-densityregions [13]. To avoid this problem, we use a nonlinearrestitution coefficient, so that the spheres do not lose alltheir kinetic energy, even after many collisions. When thesystem relaxes after an external perturbation (e.g., a singletap), the final state will have a well-defined structure, butthe spheres will still move locally-albeit with a typicaldisplacement amplitude that is much smaller than theirsize. We assume that the packings thus obtained are rep-resentative of the typical stable configurations that areobtained in real experiments at sufficiently high densities.We start every simulation by generating an equilibrated

fluid configuration of 1000 hard spheres at packing fraction� ¼ 0:2 in a cubic simulation box. We then switch ongravity and let the system reach mechanical equilibrium,as explained above. This is the preparation stage of thesimulation. Once the packing has settled, we ‘‘tap’’ the

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system by accelerating the bottom layer of the spheresimpulsively. Concretely, we add to these spheres the verti-cal velocity A

ffiffiffiffiffiffi

gap

, where A is a dimensionless tapping

amplitude, g is the acceleration of gravity, and a is thediameter of the spheres. We repeat this process at regulartime intervals, which are longer than the typical time ittakes the packing to settle after a perturbation.

Volume histogram evolution: To study the ensemblestatistics of our system, we prepare about 50 000 stableinitial conditions as described above. The correspondingdistribution of the initial volumes is denoted �0ðVÞ. For agiven amplitude A, the volume histogram after i tappingsteps is denoted �A

i ðVÞ. The volume of a packing is definedas the smallest axis-aligned cuboidal volume that com-pletely contains all the spheres, and whose bottom facelies in the plane of the wall. The results discussed below donot depend qualitatively on the choice of the definition ofvolume.

In analogy with the Gibbs approach in statistical me-chanics, we define the statistical ‘‘equilibrium’’ ensembledistribution as the asymptotic volume distribution obtainedwhen the number of taps tends to infinity. If there is awell-defined time scale for approach to the steady state,then this procedure should yield a good approximation ofthe invariant distribution.

In Fig. 1 we show the evolution of a typical volumehistogram as a function of the number of tapping steps.Starting from a very broad volume distribution, theensemble histograms become narrower, and eventuallyreach an invariant shape.

Convergence analysis: To quantify the convergenceobserved for the volume histograms in Fig. 1, we want toassess to what extent a sequence f�A

i ðVÞgi¼1;2;... becomes

independent of i for large i. For this purpose, we introducethe two-sample Kolmogorov-Smirnov (KS) statisticD½�1; �2� between two one-dimensional histograms �1

and �2 [14]:

D½�1; �2� � supVfjF1ðVÞ � F2ðVÞjg; (1)

where F1 and F2 are the respective cumulative distributionfunctions associated with histograms �1 and �2. This‘‘distance’’ is commonly used as a test for the nullhypothesis that the two histograms �1 and �2 are differentrealizations of the same underlying distribution. Thishypothesis can be rejected with 99% of certainty if [15]

D½�1; �2�> 1:63

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n1 þ n2n1n2

s

� ��ðn1; n2Þ; (2)

where n1 and n2 are the respective number of samplesused to build �1 and �2. From Eq. (2), we expect that theKS statistic will decrease as the accuracy of each histogramincreases [i.e., ��ðn1; n2Þ decreases], unless the twohistograms sample different underlying distributions.In practice, we seek the existence of an equilibrium step,

denoted by IðAÞ, past which the deviation from the invari-ant histogram is within the statistical noise. In Fig. 2, weplot the KS distances D½�A

i ; �Aiþk� as a function of both

i (x axis) and k (color code). We also display the criticalvalue �� ¼ 0:0109 corresponding to histograms built from50 000 samples [dashed line, Eq. (2)]. We identify IðAÞ asthe first step for which all colored points (from black fork ¼ 1 to bright yellow for k ¼ kmax ¼ 25) are below thedashed line. We emphasize that it is not enough forD½�A

i ; �Aiþ1� to be less than ��. For example, as shown in

Fig. 2, in the case of A ¼ 7:5, this weaker condition issatisfied (black points) as early as the 30th tap, whereasour convergence criterion is fulfilled only after the stepIð7:5Þ ¼ 164.We also notice that the relaxation time increases when

going from A ¼ 6:0 to A ¼ 7:5, whereas it decreases whengoing from A ¼ 7:5 to A ¼ 8:0.In Fig. 3, we plot the mean invariant volumes hVi

and their corresponding equilibrium step IðAÞ for theamplitudes we have tested. We notice that hVi increaseswith tapping amplitude, as found in previous studies[9,10,12,16,17]. The values of IðAÞ are consistent with thenumber of steps required for the ensemble average volumeto reach a plateau value for a given amplitude A (data notshown). As stressed previously, we note that IðAÞ is notmonotonically decreasing with A. To our knowledge, this isthe first time that a nonmonotonic dependence of relaxa-tion time scales on tapping amplitude has been observed.Normally, one would expect a decrease of the relaxationtime as the tapping amplitude is increased [9,10,12,16,17],although a nonmonotonic dependence on A was alreadysuggested in Ref. [16] but in a different context. Ourcurrent understanding of this dependence is the following:At sufficiently low A, the ensemble behaves nonergodi-cally such that each system is trapped in a local region ofthe configuration space. In this regime, higher amplitudesresult in the system exploring a larger local basin, whichincreases the time to sample it and, hence, the relaxation

FIG. 1 (color online). Typical volume histogram evolutionfor an ensemble of about 50 000 replicas of a system of 1000spheres. The picture shown corresponds to an amplitude A,equal to 6.

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time. Another time scale beyond which any system canprobe the whole configuration space may exist but isnot accessible in our simulations. On the other hand, ifwe were to tap on the system strongly enough so that each

shaking step is akin to our preparation protocol, then weshould not observe any change in the ensemble histogrambetween successive taps, which would yield a vanishingrelaxation time. These two limits can only be reconciled ifthe relaxation time depends nonmonotonically on tappingamplitude, as shown in Figs. 2 and 3.Tests on the invariant histograms.—Ergodicity

analysis: To explain the behavior observed in Figs. 2and 3, we have proposed in the preceding section thatsome conditions lead to nonergodicity. In this section, wetest this proposal quantitatively.Denote by �A

n ðVÞ a volume histogram built from nuncorrelated volume values taken from a very long trajec-tory of one system tapped with an amplitude A. In practice,we consider a single trajectory for a system tapped about105 times and we compute the corresponding volumecorrelation function. We then use the latter to select nuncorrelated values belonging to this trajectory. Under theergodic hypothesis, this ‘‘time’’ histogram and the invari-ant ensemble histogram, �A

i>IðAÞðVÞ, both sample the same

underlying distribution. To test this idea, we again use theKS statistic. Let DAðnÞ be the quantity D½�A

n ðVÞ;�Ai>IðAÞðVÞ�. If DAðnÞ is bigger than the rejection value

��ðnÞ calculated from Eq. (2), with n1 ¼ n and n2 ¼50 000, then the ergodic hypothesis is rejected. Recallthat the KS statistic decreases with increasing n if andonly if the null hypothesis, ergodicity, is true. Otherwise,DAðnÞ saturates at a finite value when n is large enough.Figure 4 shows that for high values of n, DAðnÞ indeedtends to saturate. In that case, the difference between thesaturation value of DAðnÞ and the curve ��ðnÞ at a fixed nserves as a useful measure of nonergodicity that we willcall the ergodicity gap. Figure 4 shows that for tappingamplitudes A ¼ 4:5, 6.5, and 7.5, trajectories are nonergo-dic, with ergodicty gaps that increase with decreasingtapping amplitude. In contrast, for A ¼ 7:9 and 8.0, trajec-tories are ergodic according to our criterion. These findingsare consistent with our current understanding of the non-monotonic trend for IðAÞ observed in Figs. 2 and 3.

FIG. 2 (color online). Convergence analysis. Plot of the KSdistances D½�A

i ; �Aiþk� (k running from 1 to 25 being represented

by the different colors) for different typical amplitudes corre-sponding to different relaxation regimes: A ¼ 6:0 (top), A ¼ 7:5(middle), and A ¼ 8:0 (bottom). The dashed line in each plotcorresponds to the critical value �� for histograms built from50 000 data points.

FIG. 3 (color online). Invariant properties. Red curve (tri-angles): Relaxation number of steps IðAÞ (for kmax ¼ 25) vstapping amplitude, A. The line is a cubic spline interpolation andis meant to guide the eye. Blue curve (circles): Invariant averagevolume hVi vs A. Increasing the value of kmax above 25 does notchange these results (cf. Supplemental Material [19]).

FIG. 4 (color online). Ergodicity analysis. Plot of the distanceDAðnÞ as a function of the number of uncorrelated volume valuesn for different amplitudes A. The plain black line corresponds tothe threshold value ��ðnÞ calculated from Eq. (2). The samplingis done after having tapped 200 times on the system.

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Compatibility with Edwards’ prior: In their study,McNamara et al. tested Edwards’ prior by generatingtime-sampled volume histograms for different tappingamplitudes and looking directly at ratios between pairs ofhistograms. This procedure is known as the overlappinghistogram method [18]: if the logarithm of the ratiobetween two histograms is linear, then there is a potentialcompatibility with Edwards’ canonical assumption.Otherwise, this assumption should be rejected, at least forthe tested protocol. McNamara et al. found a linear behav-ior for the log ratios for the amplitudes that they tested bothnumerically and experimentally. Here, we apply the sametest on our own invariant ensemble histograms.

Figure 5 shows the log ratio of two invariant ensemblehistograms (A ¼ 7:5 with A ¼ 8:0 for the main plot andA ¼ 4:5 with A ¼ 6:5 in the inset). We find that over alimited range of volumes where the histograms overlapsignificantly, there is indeed a linear decrease in the logratio of the two invariant histograms, consistent with thefindings in Ref. [12]. However, we also find that for highvolumes, this ratio tends to saturate, whereas for lowvolumes, it rapidly decreases. These trends are presentfor all pairs of invariant volume histograms that we havetested, including those where the two histograms areapparently ‘‘close by.’’

It is worth noting that the authors of Ref. [12] havealready suggested the possibility that the observed linearityis only true locally. Moreover, a close look at their ownfigures reveals deviations from linearity for very highand very low volumes that are consistent with our ownobservations (Fig. 5).

Overall, Edwards’ canonical hypothesis as a globalproperty is not compatible with the protocol we are testing.Although this is not the first time that a strong disagree-ment with Edwards’ theory has been found [2], to ourknowledge, this is the first time that a full histogramanalysis reports incompatibility with Edwards’ canonicalassumption for vibrated granular matter. Our study doesnot rule out the possibility of a local compatibility withEdwards’ assumption, but this is already different fromEdwards’ original theory.Conclusion.—In this Letter, we introduced an ensemble

volume statistic for a simulated vertically vibrated granularsystem. To quantitatively assess the properties of the gen-erated sequences of ensemble histograms, we used the KStest. This allowed us to devise a convergence criterion for asequence of histograms. Subsequently, we tested the ergo-dicity of a tapped system as a function of the tappingamplitude A and found clear evidence for nonergodicitywhen the tapping amplitude is low. Finally, we tested thecompatibility of our invariant histograms with Edwards’hypothesis and concluded that it is not compatible with oursimulation protocol.We should point out that the results found in this Letter

depend a priori on the chosen tapping protocol and alsoon the preparation stage. Although the dependence of ourfindings on the preparation protocol is hard to predict, thetools that we introduced to characterize our ensemblehistograms can be used to test any numerical or experi-mental protocol.F. P. is grateful to Nicolas Dorsaz, Frank Smallenburg,

and Patrick Varilly for very helpful discussions. This workhas been supported by the EPSRC Grant No. EP/I000844/1. D. F. acknowledges support from ERC Advanced GrantNo. 227758 and Wolfson Merit Grant No. 2007/R3 of theRoyal Society of London.

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FIG. 5 (color online). Overlapping histograms. The red pointscorrespond to the plot of the logarithm of the ratios �7:5

i>I7:5=�8:0

i>I8:0

(main figure) and �4:5i>I4:5

=�6:5i>I6:5

(inset). We superimposed the

corresponding invariant histograms in both the main figure andthe inset. The histogram in light gray (green) always correspondsto the smallest amplitude in a given pair. The plain lines arethe best linear fits through the midpoints within the crossoverregions.

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[12] S. McNamara, P. Richard, S. K. de Richter, G. Le Caer,and R. Delannay, Phys. Rev. E 80, 031301 (2009).

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[18] C. Bennett, J. Comput. Phys. 22, 245 (1976).[19] See Supplemental Material at http://link.aps.org/

supplemental/10.1103/PhysRevLett.109.208001 for an ad-ditional figure regarding the dependence of the equilib-rium step IðAÞ on the window size kmax such that twohistograms separated by less than kmax after the step IðAÞare statistically indistinguishable.

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