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$Edge excitations of paired fractional quantum Hall states$

498 S. Živković etal.302520(a) ξ = 3%t=2t=4t=8t=15t=20t=40t=70t=120302520(a) ξ = 3%t=2t=4t=8t=11t=20t=120P(ζ)15P(ζ)1510105501.1 1.15 1.2 1.25 1.3shape factor ζ01.1 1.15 1.2 1.25 1.3shape factor ζ302520(b) ξ = 0.7%t=2t=4t=8t=11t=15t=20t=40t=70t=120302520(b) ξ = 0.7%t=2t=4t=8t=20t=40t=70t=120P(ζ)15P(ζ)1510105501.1 1.15 1.2 1.25 1.3shape factor ζFig. 2 Temporal evolution of the probability distribution P(ζ ) of theshape factor ζ for the more dissipative disks (A) at tapping intensity. aξ = 3.0%, and b ξ = 0.7%01.1 1.15 1.2 1.25 1.3shape factor ζFig. 3 Temporal evolution of the probability distribution P(ζ ) of theshape factor ζ for the less dissipative disks (B) at tapping intensity.a ξ = 3.0%, and b ξ = 0.7%compaction is more pronounced for the larger tapping intensityξ, because the compaction dynamics gets faster whenthe tapping intensity ξ increases.The probability distribution P(ζ ) is also sensitive to thematerial properties of the grains. A typical distributions P(ζ )for the less dissipative disks (B) at two different tappingintensities ξ = 0.7%, 3% are shown in Fig. 3. Hereagain,the curves of distribution P(ζ ) are asymmetric with a quitelong tail on the right-hand side. The tails of such distributionsreduce during the compaction more quickly for the systemof less dissipative grains (B) than for the system of moredissipative grains (A). Indeed, for the less dissipative disks(B) there is a rapid approach of the system to the steady-statedensity [22]. Furthermore, for a given tapping intensity ξ,thesystem of disks (B) achieves a larger value of the asymptoticdensity than the system of disks (A) [22]. Consequently, inthe case of disks (B), collective rearrangements of the grainslead to the growth of hexagonal domains [5] and the systemends up in the steady-state configurations where large clus-ters of near-regular Voronoï cells (hexagonal domains) arefound. These large solid-like domains are clearly separatedby thin disordered regions because rupture occurs inside thecompact blocks of grains. We thus get a very narrow probabilitydistribution P(ζ ) centered at ζ ≈ 1.1 (the value forregular hexagons).3.1 Memory effectsHere we focus on the response of the granular system tosudden perturbation of the tapping intensity and discuss theaccompanying transformations of disk packings at a microscopiclevel. In the simplest compaction experiment [27], thetapping intensity was instantaneously changed from a valueƔ 1 to another Ɣ 2 ,aftert w taps. For a sudden decrease inƔ (Ɣ 1 >Ɣ 2 ) it was observed that on short-time scales thecompaction rate increases, while for a sudden increase in Ɣ(Ɣ 1
Structural characterization of two-dimensional granular systems 4990.873.0% → 0.5%0.5% → 3.0%50A 1 (t=30)A 2 (t=39)0.86400.85packing fraction0.840.830.820.81A 1 A 2P(ζ)3020100.5% → 3.0%0.810 100time (number of taps)Fig. 4 Time evolution of the packing fraction ρ(t) when the tappingintensity ξ is changed from 3 to 0.5% (dashed line) and from 0.5to3%(solid line)att w = 30. The points A 1 and A 2 correspond to the packingswith equal density ρ(t w ) = 0.828, at t w = t 1 = 30 and t 2 = 3901.1 1.15 1.2 1.25 1.3shape factor ζFig. 5 Distributions P(ζ ) of the shape factor ζ for the packings atpoints A 1 (solid line) andA 2 (dashed line) inFig.4. These distributionscorrespond to the packings having the same densitybehavior at constant Ɣ. This behavior is transient, and afterseveral taps the usual compaction rate is recovered.Figure 4 shows typical memory effects at short timesafter an abrupt change of the tapping intensity ξ. The tappingintensity ξ is switched from 3 to 0.5% and vice versaat t w = 30. We observe that after the transient regime the“anomalous” response ceases and there is a crossover tothe “normal” behavior, with compaction rate becoming thesame as in a constant forcing mode. These results indicatethat the system can be found in states, characterized by thesame packing fraction ρ, that evolve differently under furthertapping with the same tapping intensity ξ. Both pointsA 1 (t 1 = t w = 30) and A 2 (t 2 = 39) in Fig. 4 correspond tothe packings with equal density ρ(t w ) = 0.828, equal valueof ξ = 3%, but different further evolution. Their responses tothe same tapping intensity ξ = 3% are different: packing A 1becomes looser whereas packing A 2 pursues its compaction.In other words, the density evolution after the points A 1 andA 2 depends not only on the density ρ(t w ), but also on theprevious tapping history.The response in the evolution of the density to a changein the tapping intensity at a given time is accompanied bytransformation of the local configurations in the packing. Distributionsof the shape factor shown in Fig. 5 correspond topackings having the same density ρ(t w ) = 0.828. The behaviorof distributions P(ζ ) for A 1 (t w = 30) and A 2 (t 1 = 39)packings are globally similar, but the differences are perceptible.For 1.125 ζ , the distribution P(ζ ) for packingA 2 is slightly lower than for packing A 1 . However, for theζ less than ≈1.125 we observe an opposite effect on P(ζ ).Consequently, the probability distribution P(ζ ) is not unambiguouslydetermined by the density ρ(t w ) and the tappingintensity ξ. The memory of the history up to the density ρ(t w )is encoded in the arrangement of the grains in the packing.4 Comparison with experimental resultsA comparison between the numerical and experimentalresults provides insight into the physical accuracy of thenumerical model. In this paper we present an experiment thatenables the formation of mechanically stable granular packingsof various densities and allows the precise detectionof individual grains. To make a quantitative comparison, wemeasure the probability distribution P(ζ ) of the shape factorζ for various experimentally generated jammed disorderedpackings in two dimensions (2D).The experimental setup was already described in details inthe previous paper [29]. We shall therefore present it briefly.Experiments were carried out on a 2D granular medium, i.e.,the motion of the grains was confined to a plane. The granularpacking is constituted of metallic cylinders contained ina rectangular box made of two parallel glass plates, with aninner gap of thickness 3.4 mm, slightly larger than the heightof the cylinders, h = 3.00 ± 0.01 mm. The lateral walls ofthe box delimit a rectangular frame of height H = 340 mmand an adjustable width of typically L = 300 mm. The box issecured on a heavy plane able to be inclined at different ratesso that we could set an arbitrary inclination angle θ fromthe horizontal. The cylinders of diameter d = 4.00, 5.00,and 6.00 ± 0.05 mm were used to prepare the monodispersepackings containing about 4,500 grains.Disordered packings are prepared by pouring grains ontoan initially horizontal glass plate at once. They are thenspread until a flat layer is obtained, where the cylinders arerandomly deposited without contact between them. The angleof the plane is then slowly increased up to an angle θ = 90 ◦ ,at constant angular velocity of ≈5 ◦ s −1 . During the planerotation, grains therefore freely slide downward and reach amechanically stable state. The measured packing fractions123
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