PHYSICAL REVIEW X 8, 031050 (2018) · theories of the glass transition. In particular, these observations appear inconsistent with current formula-tions of theories based on a growing

Theory for Swap Acceleration near the Glass and Jamming Transitionsfor Continuously Polydisperse Particles

Carolina Brito,1 Edan Lerner,2 and Matthieu Wyart31Instituto de Fsica, UFRGS, 91501-970 Porto Alegre, Brazil2Institute for Theoretical Physics, University of Amsterdam,

Science Park 904, 1098 XH Amsterdam, Netherlands3Institute of Physics, EPFL, CH-1015 Lausanne, Switzerland

(Received 16 April 2018; revised manuscript received 28 June 2018; published 27 August 2018)

SWAP algorithms can shift the glass transition to lower temperatures, a recent unexplained observationconstraining the nature of this phenomenon. Here we show that SWAP dynamics is governed by aneffective potential describing both particle interactions as well as their ability to change size. Requiringits stability is more demanding than for the potential energy alone. This result implies that stableconfigurations appear at lower energies with SWAP dynamics, and thus at lower temperatures when theliquid is cooled. The magnitude of this effect is predicted to be proportional to the width of the radiidistribution, and to decrease with compression for finite-range purely repulsive interaction potentials. Wetest these predictions numerically and discuss the implications of our findings for the glass transition.These results are extended to the case of hard spheres where SWAP is argued to destroy metastable statesof the free energy coarse grained on vibrational timescales. Our analysis unravels the soft elastic modesresponsible for the speed-up induced by SWAP, and allows us to predict the structure and the vibrationalproperties of glass configurations reachable with SWAP. In particular, for continuously polydispersesystems we predict the jamming transition to be dramatically altered, as we confirm numerically. Asurprising practical outcome of our analysis is a new algorithm that generates ultrastable glasses by asimple descent in an appropriate effective potential.

DOI: 10.1103/PhysRevX.8.031050 Subject Areas: Soft Matter, Statistical Physics

I. INTRODUCTION

Understanding the mechanisms underlying the slow-ing-down of the dynamics near the glass transition is along-standing challenge in condensed matter [1,2].Unexpectedly, SWAP algorithms [3–5] (in which particlesof different radii can swap in addition to the usual movesof particle positions) were recently shown to allow forequilibration of liquids far below the glass transitiontemperature Tg [6–9]. This discovery has practicalimportance, as it allows one to reach quench rates similarto experiments in glasses, and to observe numericallyknown phenomena (including the brittleness of metallicglasses [10]) previously very hard to reproduce computa-tionally. For judicious choice of poly-dispersity, one findsthe following. (i) The glass transition is shifted to lowertemperatures: with swaps the α-relaxation time at Tg isonly 2 or 3 orders of magnitude slower that in the liquid,

instead of 15 orders of magnitude for regular dynamics.The slowing-down of the dynamics occurs at a lowertemperature, which we refer to as TSWAP

0 . (ii) The spatialextent of dynamical correlations, which are significantnear Tg, are greatly reduced with SWAP and occur only atTSWAP0 . (iii) The mean squared displacements of particles

on vibrational timescales increases significantly in thistemperature range [8]. These observations constraintheories of the glass transition. In particular, theseobservations appear inconsistent with current formula-tions of theories based on a growing thermodynamiclength scale [11]. A theory of the glass transition shouldexplain both SWAP and non-SWAP dynamics. Goldstein[12] proposed that the glass transition is initiated by atransition in the free-energy landscape: at high temper-ature, the system resides near saddles, whereas belowsome temperature T0, the dynamics can occur only byactivation (whose nature is debated), and is thus muchslower. In mean-field models of structural glasses such atransition in the landscape is predicted [13–16] andcorresponds to a mode coupling transition where therelaxation time diverges. It was suggested that themode coupling transition would be shifted to lowertemperature with SWAP dynamics in Ref. [11], as proven

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PHYSICAL REVIEW X 8, 031050 (2018)

2160-3308=18=8(3)=031050(13) 031050-1 Published by the American Physical Society

and confirmed numerically in a mean-field model ofglasses [17]. Yet, understanding the real-space mecha-nisms underlying the speed-up induced by SWAP in finitedimensions (where the relaxation time cannot diverge) aswell as the nature of the very stable glassy configurationsSWAP can reach remains a challenge.In this work, we tackle these questions by first

reviewing the equilibrium statistical mechanics theoryof polydisperse systems [18,19], to show that they areequivalent to a system of identical particles that canindividually deform according to a chemical potentialμðRÞ, where R is the particle radius. In the (practicallyimportant) case where polydispersity is continuous, μðRÞis smooth, allowing us to define normal modes of thegeneralized Hessian that includes radii as degrees offreedom. We prove that requiring its stability is strictlymore demanding than for the usual Hessian. Second, weshow that these results stringently constrain the glassystates generated by SWAP algorithms. We illustrate thispoint by studying the jamming transition in soft repulsiveparticles, which we prove must be profoundly altered:hyperstaticity is found with an excess number of contactsδz with respect of the Maxwell bound δz ∼ α1=2 > 0,where α characterizes the width of the radii distributionρðRÞ. Although we find that the vibrational spectrum ofthe generalized Hessian is marginally stable with respectto soft extended modes near jamming, these modes aregapped in the regular Hessian, unlike for packingsobtained with regular dynamics [20–23]. These resultsare verified numerically by introducing a novel algorithmperforming a steepest descent in the generalized potentialenergy that includes μðRÞ, which can generate extremelystable glasses without any activation. Third, we inves-tigate the glass transition. We show that the inherentstructures obtained after a rapid quench with the regulardynamics are unstable with respect to this new algorithm,which reaches significantly smaller energies. This resultindicates that metastable states appear at lower energieswith SWAP, and therefore at lower temperatures when theliquid is equilibrated. Thus the Goldstein transition mustbe shifted to a lower temperature with SWAP dynamics,suggesting a natural explanation for its speed-up, whichspecifies the collective modes facilitating the dynamicsfor TSWAP

0 < T < Tg. We predict this shift to be propor-tional to α in general, and to be inversely proportional tothe distance to jamming for sufficiently compressed softspheres. Lastly, we argue that these results apply to hardspheres as well, if the energy is replaced by a coarse-grained free-energy landscape as previously studied inRefs. [21,24–26]. We use this approach to provide asimple phase diagram where the Goldstein transitionand the emergence of marginality [20] (referred toas a Gardner transition in infinite dimension [26]) canbe related to structure for both SWAP and non-SWAP

dynamics.

II. GRAND-CANONICAL DESCRIPTION OFPOLYDISPERSE SYSTEMS

A. Effective potential

We next show that systems of polydisperse particlescan be described by an effective potential that includesparticles’ radii as degree of freedom, an idea first putforward in Refs. [18,19]. We consider a system of Nparticles with continuous polydispersity ρðRÞ, of widthα ¼ hðhR2i − hRi2Þ1=2i=hRi. Here fRg indicates the set ofparticle radii and frg their positions. In what follows wedenote hRi≡ R0, and Uðfrg; fRgÞ the total potentialenergy in the system. In the usual view point, frg aredegrees of freedom and fRg are fixed parameters. We nowshow that there is an equivalent formulation of the problemwhere both frg and fRg play a similar role.We define the partition function ZðfrgÞ annealed over

the particle radii:

ZðfrgÞ ¼X

PðfRgÞexp½−βUðfrg; fRgÞ�; ð1Þ

where the sum runs over all the permutations PðfRgÞ ofthe particle radii. In the thermodynamic limit, a grand-canonical formulation is equivalent, in which particles ofdifferent radii correspond to different species. The asso-ciated partition function is written as

ZGCðfrgÞ ¼XfRg

exp

�−β

�Uðfrg; fRgÞ þ

XNi

μðRiÞ��

;

ð2Þ

where μðRÞ is the chemical potential of particles withradius R. It is chosen such that in the thermodynamic limit,the distribution of radii that follows from Eq. (2) is

ρðRÞ≡ 1

Z

Xfrg;fRg

1

N

�Xi

δðR − RiÞ�

× exp

�−β

�Uðfrg; fRgÞ þ

XNi

μðRiÞ��

: ð3Þ

A key remark is that once Eq. (2) is integrated on particlepositions frg, one obtains the partition function for thecoupled degrees of freedom frg and fRg with an effectiveenergy functional:

Vðfrg; fRgÞ ¼ Uðfrg; fRgÞ þXNi

μðRiÞ: ð4Þ

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B. Grand-canonical dynamics

According to Eqs. (2) and (4), equilibration in poly-disperse systems can be reached by dynamics where bothpositions frg and radii fRg evolve in time, so as toequilibrate the effective energy V. We shall refer to suchprocedures as grand-canonical dynamics. We now arguethat SWAP belongs to that class in the thermodynamic limit.Indeed, if the system is very large, swapping two particlesis equivalent to swapping one particle with a bath ofparticles of different radii. This bath is described by achemical potential μðRÞ, which must precisely be such thatthe polydispersity ρðRÞ is maintained over time.We see below that the effective energy landscape

affecting grand-canonical dynamics (including SWAP) isprofoundly different from the usual potential energylandscape affecting the normal dynamics (while all thermo-dynamic quantities are by construction identical, independ-ently of the dynamics chosen). In particular, inherentstructures [i.e., minima of Vðfrg; fRgÞ] appear at lowerpotential energies for grand-canonical dynamics, as we nowshow using stability considerations.

C. Mechanical stability under SWAP

Let us consider inherent structures (our arguments for theenergy landscape are later extended to the free-energylandscape in the case of hard spheres). In the thermody-namic limit, mechanical stability under SWAP dynamicsrequires V to be at a minimum. Beyond the usual forcebalance condition, it implies:

∂U∂Ri

≡Xj

fij ¼ −∂μ∂R

����R¼R�

i

; ð5Þ

where fij are the contact forces between particle i and j(positive in our notations for repulsive forces), and (fr�g,fR�g) the particle positions and radii at the minimum.For unimodal distribution ρðRÞ, one expects μðRÞ to beunimodal, too. In an amorphous solid the fluctuations of theleft-hand side of Eq. (5) are of order pRd−1

0 , where p is thepressure and d the spatial dimension. To achieve adistribution of radii of width α, the stiffness kR actingon each particle radius must thus be of order

kR ≡ h∂2μ=∂2Rii ∼ pRd−20 =α; ð6Þ

where the average is taken over all particles i.

D. Generalised vibrational modes

Stability also requires the Hessian HSWAP (the matrix ofsecond derivatives of V) to be positive definite. Includingthe particles’ radii, we consider a total of Nðdþ 1Þ degreesof freedom; therefore, HSWAP is a Nðdþ 1Þ × Nðdþ 1Þsymmetric matrix, of eigenvalues ω2

SWAP. It contains a blockof size Nd × Nd, which is the regular Hessian Hij ¼

∂2U=∂ri∂rj. We denote by ω2 its eigenvalues. Becausehybridization with additional degrees of freedom can onlylower the minimal eigenvalues of the Hessian, HSWAP haslower eigenvalues than H (as quantified below), implyingthat mechanical stability is more stringent with SWAP

dynamics (and more generally with grand-canonicaldynamics).Let us illustrate this result perturbatively when kR ≫ k,

where k is the characteristic stiffness of the interactionpotential U. In general, the eigenvalues of H are functionsof the set of stiffnesses fkijg, but also of the interactionforces ffijg [27]. We first ignore the effects induced by thepresence of such interaction forces, referred to here and inwhat follows as prestress. Moving along a normal mode ofH by a distance x (while leaving the radii fixed) leads to anelastic energy ∼ω2x2 and changes forces by a characteristicamount δf satisfying δf2=k ∼ x2ω2. Because of suchchange, Eq. (5) is not satisfied anymore. Thus the potentialV can be reduced further by an amount of order δf2=kR ∼ω2x2k=kR if the radii are allowed to adapt. This reducedenergy can be approximatively written as x2ω2

SWAP, whereω2

SWAP is the eigenvalue associated to that mode in theeffective Hessian. We thus obtain:

ω2 − ω2SWAP ∼ ω2

kkR

; for kR ≫ k: ð7Þ

III. SOFT-SPHERE SYSTEMS

To illustrate these ideas, we consider soft spheres withhalf-sided harmonic interactions, so that

Uðfrg; fRgÞ ¼Xi;j

k2ðrij − Ri − RjÞ2ΘðRi þ Rj − rijÞ;

ð8Þ

where rij is the distance between particles i and j, andΘðxÞis the Heaviside step function.

A. Jamming transition for soft spheres under SWAP

When materials with such finite-range interactions arequenched to zero temperature, they can jam into a solid or not,depending on their packing fraction. At the jamming tran-sition separating these two regimes, vibrational properties aresingular [28,29], and the effects of SWAP are expected to beimportant, as we show next. The vibrational spectrum of theregular Hessian is strongly affected by the excess number ofconstraints δz with respect to the Maxwell threshold wherethe numbers of degrees of freedom and constraints match.Effective medium [30] or a variational argument [31] impliesthat in the absence of prestress, soft normal modes in theHessian must be present with eigenvalues:

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ω�2 ∼ k δz2: ð9Þ

For swap with a small polydispersity α ≪ Δ, where weintroduced the dimensionless particle overlapΔ≡ pRd−2

0 =k,then following Eq. (6) kR ≫ k, and Eq. (7) applies. It impliesthat soft normal modes will be present at lower eigenvaluesω� 2

SWAP ∼ δz2kð1 − C0α=ΔÞ,whereC0 is a numerical constant.Prestress can be shown to shift eigenvalues of the Hessian bysome amount ≈ − C1kΔ [20,32], leading to eigenvaluessatisfying ω0 2

SWAP ∼ δz2kð1 − C0α=ΔÞ − C1kΔ. Mechanicalstability requires positive eigenvalues, and we obtain

δz ≥

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1Δ

ð1 − C0αΔÞ

s; for α ≪ Δ: ð10Þ

Equation (10) indicates that away from jamming, the relativeeffects of SWAP on the structure are proportional to α=Δ,corresponding to the saturation of inequalities of the kindof Eq. (10). Below, we provide numerical evidence that it isalso the situation if SWAP is used, at least near jamming. Herethis assumption gives an expression for δzwhich is above (butvery close to, in the limit α ≪ Δ) the bound for non-SWAP

dynamics of Ref. [20], recovered by setting α ¼ 0. Thus, inthis limit,we expect very small change of structure in theglassphase between SWAP and non-SWAP dynamics.For SWAP with a large polydispersity α ≫ Δ, the sit-

uation is completely different. We then have kR ≪ k: in thisregime the strong interactions correspond to those betweenparticles in contact. As far as the low-frequency end of thespectrum is concerned, these interactions can be consideredto be hard constraints (i.e., k ¼ ∞), whose number isNz=2.The dimension of the vector space satisfying such hardconstraints is Nðdþ 1Þ − Nz=2 ¼ Nð1 − δz=2Þ. Thesemodes gain a finite frequency due to the presence of theweaker interactions of strength kR associated with thechange of radius. Importantly, the number of these weakerconstraints left is simply the number of particles N. If δzis small, the number of degrees of freedom Nð1 − δz=2Þ isjust below the number of constraints N: for this vectorspace we are close to the “isostatic” or Maxwell conditionwhere the number of constraints and degrees of freedommatch. Thus we can use the same results for the spectrumvalid near the jamming transition introduced above. Theyalso apply in that situation, with the only difference that thestiffness scale k is replaced by kR. In particular, if prestressis not accounted for, a plateau of soft modes must appearabove some frequency given by Eq. (9):

ω� 2SWAP ∼ kRδz2 ∼ k

Δαδz2; ð11Þ

This plateau survives up to the characteristic frequency:

ωi ∼ffiffiffiffiffikR

p∼

ffiffiffiffiffiffiffiffiffiΔ=α

p: ð12Þ

When prestress is accounted for, eigenvalues of the Hessianare again shifted by ∼ − kΔ. Mechanical stability thenimplies Δ=αδz2 > C2Δ and

δz ≥ C2

ffiffiffiα

p; for Δ ≪ α: ð13Þ

In this regime, marginal stability [the saturation of thestability bound of Eq. (13)] implies a pressure-independentcoordination, with δz ∼

ffiffiffiα

pand

ω�SWAP ∼

ffiffiffiffiffiffikΔ

p: ð14Þ

We thus predict that SWAP dynamics destroys isostaticity,and significantly affects structure and vibrations. Forsufficiently large α, this regime will include the entireglass phase, and vibrational properties and stability will beaffected in the vicinity of the glass transition (which sits at afinite distance from the jamming transition [33]) as well.Note that these predictions apply to algorithms that allow

for swap moves up to the jamming threshold. This is not thecase, e.g., in Ref. [34], where swaps are used to generatedense equilibrated liquids that are then quenched withoutswap toward jamming. We also expect isostaticity to berestored in algorithms for which the set of particle radii isstrictly fixed, but only below some pressure pN that vanishesas N → ∞, above which our predictions should apply.

B. Numerical model

As shown in Eq. (4), SWAP dynamics is equivalent to asystem of interacting particles which can individuallydeform. To test our predictions, we consider soft spheresas defined in Eq. (8), whose radii follow the internalpotential:

μðfRgÞ ¼ kR2

Xi

ðRi − Rð0Þi Þ2

�Rð0Þi

Ri

�2

; ð15Þ

where kR is a characteristic stiffness. We considered apotential diverging as Ri → 0 to avoid particles shrinking tozero size. To avoid crystallization we further considered

that particles are of two types: for 50% of them, Rð0Þi ¼ 0.5,

while for the others, Rð0Þi ¼ 0.7. This choice leads to a

bimodal distribution of size ρðRÞ, as shown in Fig. 1. Ourmodel corresponds to a SWAP dynamics where swap isallowed only between particles of the same type. Note thatbroad monomodal distributions can be optimized to makeSWAP more efficient while avoiding crystallization [8],which would be similar to having a very large α in ourtheoretical description. The spatial dimension is d ¼ 2 inour simulations, and k ¼ 1 is our unit stiffness, leading to asimple relation Δ ¼ p.To study the jamming transition, we consider a pressure-

controlled protocol at zero temperature described in theAppendix A 1. The chemical potential of Eq. (15) must

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evolve with pressure to maintain a fixed polydispersity. Asshown in Fig. 1(b), it can be achieved within great accuracysimply by imposing that kR ¼ p=α, where α is a parameterthat controls the width α, as shown in the inset of Fig. 1. Forthis bimodal distribution, α is defined as α ¼ ðα1 þ α2Þ=2,where α1, α2 are the relative width of each peak in ρðRÞ. Inthe limit where the non-SWAP dynamics is recovered—which happens when α → 0—α and α are proportional.

C. Structure and stability

Our central prediction is that for SWAP dynamics,materials must display a larger coordination to enforcestability. This prediction is verified in Fig. 2(a), whichshows δz versus Δ for various values of α. Isostaticity isindeed lost and the coordination converges to a plateauas Δ decreases. Strikingly, we find for the plateau valueδz ∼

ffiffiffiα

p, consistent with a saturation of the stability

bound of Eq. (13). This scaling behavior is implied bythe scaling collapse in Fig. 2(b), which also confirms thatthe characteristic overlap below which swap affects thedynamics scales as Δ ∼ α. Overall, these results support

that the numerical curves δzðΔÞ in Fig. 2(a) correspond tothe marginal stability lines under SWAP dynamics, shownfor different polydispersity (see more on that below).

D. Packing fraction

For traditional dynamics, polydispersity tends to havevery limited effects on the value of jamming packingfraction ϕJ. We confirm this result in the S.I., by showingthat although our model can generate very different dis-tributions ρðRÞ, the values we obtain for ϕJ cannot bedistinguished if jamming is investigated using non-SWAP

dynamics. However, for SWAP dynamics we expect thesituation to change dramatically: since stability requiresmuch more coordinated packings, they presumably needto be denser, too. We denote the jamming packing fractionfor SWAP ϕc ≡ limΔ→0ϕðΔÞ. The inset of Fig. 3(a) confirmsthat ϕc increases significantly as ρðRÞ broadens. Toquantify this effect we consider ϕðΔ; αÞ, as shown in themain panel. Assuming a scaling form for this quantity, andrequiring that it satisfies the known results for the jammingtransition for Δ ≫ α implies ϕðΔ; αÞ − ϕJ ¼ fðΔ=αÞαβ,where fðxÞ is some scaling function and β ¼ 1. Since thecoordination does not change for Δ ≪ α, we expect that itis true for the structure overall and for ϕ, implying thatfðxÞ ∼ x0 as x → 0. These predictions are essentiallyconfirmed in Fig. 3(b). Note, however, that the best scalingcollapse is found for β ¼ 0.83 < 1. These deviations arelikely caused by finite-size effects, known to be muchstronger for ϕ than for the coordination or vibrationalproperties [35], and which may thus be present for oursystems of N ¼ 484 particles.

E. Vibrational properties

We computed the Hessian HSWAP and diagonalized it (seedetails in SI) to extract the density of states DðωÞ, as shownin Fig. 4(a) for different pressures at fixed polydispersity. As

10-5

10-4

10-3

10-2

10-1

Δ

10-1

100

α=10−2

α=5x10-3

α=2x10-3

α=10−3

α=5x10-4

α=3x10-4

α= 0

10-3

10-2

10-1

100

101

102

103

Δ / α

101

102

δz δz / α 0.5

(a) (b)

FIG. 2. (a) Log-log plot of δz versus Δ for different values of αas indicated in legend. The black dashed line corresponds toδz ∼

ffiffiffiffiΔ

p. (b) All these curves collapse if the y axis is rescaled by

1=α0.5 and the x axis is rescaled by 1=α. Bars indicate standarddeviations.

0.3 0.4 0.5 0.6 0.7R

0

25

50

α=10−1

α=2x10-2

α=10−2

α=2x10-3

0.4 0.5 0.6 0.7R

0

10

20

Δ=10−2

Δ=10−3

Δ=10−4

10-4

10-2

10010

-4

10-2

100

ρ(R) ρ(R)

α

α

(a) (b)

FIG. 1. (a) Distribution of radii ρðRÞ for different α at fixedΔ ¼ 10−4 and for different Δ at fixed α ¼ 10−2. (b) Inset: αversus α compared to a linear relation in black. Data are averagedover 1000 systems with N ¼ 484 particles. For α ¼ 10−2, thewidths of the distributions are given by α1 ¼ 0.024 and α2 ¼0.018 for the smaller and bigger radii, respectively.

10-5

10-4

10-3

10-2

10-1

Δ

10-4

10-3

10-2

10-1

α=10−2

α=5x10-3

α=2x10-3

α=10−3

α=5x10-4

α=3x10-4

α=0

10-3

10-2

10-1

100

101

102

103

Δ / α

100

102

10-4

10-2

100

0.84

0.86

0.88

(φ−φJ) (φ−φ

J) / α 0.83

αφc

φc

(b)(a)

FIG. 3. (a) Log-log plot of ϕ − ϕJ versus Δ for differentvalues of α. ϕJ ¼ 0.8456 is defined as the packing fractionfor α ¼ 0, and for the smallest pressure we simulateΔ ¼ 10−5. Inset: ϕc versus α. ϕc is estimated by consideringthe packing fraction at Δ ¼ 10−5. (b) Scaling collapseshowing that ϕðΔÞ − ϕJ ≈ fðΔ=αÞα0.83.

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expected, at low particle overlap Δ two bands appear in thespectrum. The lowest-frequency band presents a plateauabove some frequency scale ω�

SWAP which satisfies ω�SWAP∼ffiffiffiffi

Δp

, as shown in the inset of Fig. 4(a), as expected if thestructure were marginally stable. As shown in the S.I., in theabsence of prestress the minimal eigenvalues of the Hessianincrease substantially, again a signature of marginal stability[20]. Further evidence appears in Fig. 4(b), showing DðωÞat fixed Δ ¼ 10−4 for varying polydispersity. ω�

SWAP

essentially does not depend on α, as shown in the insetof Fig. 4(b), as expected for marginal packings if the pressureis fixed. The cutoff frequency ωi of the low-frequencyplateau scales as ωi ∼

ffiffiffiffiffikR

p∼ 1=

ffiffiffiα

p, as predicted above.

IV. GLASS TRANSITION

We now turn to the glass transition, which alwaystakes place at a sizable distance form the jamming

transition [33]: for example, for hard disks, ϕg ≈ 0.78and ϕc ≈ 0.85. A similar difference of packing fractionoccurs by compressing soft spheres at overlap Δ ≈ 0.05, asillustrated in Fig. 5(d). From the arguments above, weexpect that if the polydispersity is sufficiently large,vibrational properties will be strongly affected even faraway from jamming, in particular, near the glass transition.The direct consequence of this fact is that the energy

landscape will be affected by SWAP, which will in turn affectthe glass transition. Configurations of high energy areunstable—they reside in the vicinity of saddles with manyunstable directions—whereas below some characteristicenergy, minima appear. However, since stability is strictlymore demanding with SWAP, this characteristic energy mustbe reduced when SWAP is allowed for. We prove this pointin Figs. 5(a) and 5(b), where inherent structures of energyU∞ are obtained after using a steepest descent for non-SWAP dynamics. These configurations are not stable forour generalized steepest descent that let particles deform,which leads to configurations of energy Uα < U∞. Thiseffect is stronger near jamming in relative terms, as shownin Fig. 5(c), but remains significant away from jamming if

10-4

10-2

10010

-2

10-1

100

D(ω

)Δ=10−2

Δ=5x10-3

Δ=10-3

Δ=5x10-4

Δ=10-4

Δ=5x10-5

Δ=10-5

10-5

10-4

10-3

10-2

10-3

10-2ω∗

Δ

α=10−3

ω∗

(a)

swap

swap

10-3

10-2

10-1

100

ω10

-2

10-1

100

D(ω

)

α=10−1

α=2x10-2

α=10−2

α=5x10-3

α=10−3

α=3x10-4

10-4

10-3

10-2

10-110

-4

10-2

100

ωi

ω∗

ωi

Δ=10-4

α

ω∗

(b)

swap

swap

FIG. 4. (a) DðωÞ for different Δ at fixed α ¼ 10−3. Inset: ω�SWAP

versus Δ, where ω�SWAP is extracted asDðω�

SWAPÞ ¼ 10−2. Dashedline corresponds to the marginality condition ω�

SWAP ∼ffiffiffiffiΔ

p.

(b) DðωÞ for different c and fixed Δ ¼ 10−4. Inset: ω�SWAP versus

Δ. Dashed line is the theoretical prediction ωi ∼ffiffiffiffiΔ

p=

ffiffiffiα

pand the

continuous line corresponds to ω�SWAP ¼ ffiffiffiffi

Δp ¼ 0.001.

(a)

(b)

(d)

(c)

FIG. 5. (a) Illustration of the effect of SWAP on the energylandscape. Systems rapidly quenched with non-SWAP dynamicsdisplay an energy U∞. These states are unstable to a steepestdescent with SWAP, leading to a lower energyUα: metastable statesappear at lower energies with SWAP. (b) Values for U∞ (fullsymbols) and Uα (open symbols) as a function of the dimension-less pressure Δ imposed during the quenches as a function of α asshown in the legend. The ratioU∞=Uα is shown in (c), it is strongernear jamming but the effect remains significant important even farfor jamming if the system is sufficiently polydisperse. As shown inthe inset, in relative terms the shift of the energy of inherentstructures ðU∞ − UαÞ=Uα is proportional to α and inverselyproportional to Δ when Δ is large enough. (d) The packingfraction ϕc obtained after SWAP is turned on is larger than ϕ∞obtained for non-SWAP dynamics, an effect that is stronger nearjamming.

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the polydispersity is broad enough. It corresponds, forexample, to a reduction of energy of 25% for Δ ¼ 0.05 forour α ¼ 0.06. We show in the inset of Fig. 5(c) that therelative shift of energy induced by SWAP ðU∞ − UαÞ=Uα isproportional to α and inversely proportional to Δwhen Δ islarge enough, consistent with what we found for thestructure in Eq. (10).Thus, as the temperature is lowered in these liquids, the

Goldstein temperature where activation sets in will besmaller when SWAP is allowed for. This analysis thuspredicts an entire temperature range in which the non-SWAP dynamics is slowed down by activation, whereas withSWAP dynamics the system can flow along unstable modes.More quantitatively, we predict the shift of glass transitiontemperature ΔTg=Tg induced by SWAP to be proportional toα, consistent with the observation that very broad distri-butions lead to large SWAP effects [8]. We also predict thatΔTg=Tg is inversely proportional to the distance to jam-ming Δ when this quantity is well defined (e.g., for softspheres, but also to some extent for Lennard-Jones poten-tials [29,36]) and large enough. In real space, the unstablemodes that render activation useless involve both transla-tional degrees of freedom as well as swelling and shrinkingof the particles. We show an example of such a mode inFig. 6, corresponding to the softest mode of the generalizedHessian we obtain with parameters α¼0.06 and Δ ¼ 10−2.It illustrates that the particle displacements are not neces-sarily divergence-free when SWAP is allowed, since thesystem can locally compress or expand by changing theparticle sizes.The interpretation of SWAP acceleration described above

is consistent with the observation that the dynamics is lesscollective with SWAP at the temperature where the non-SWAP dynamics is activated, since the system can rearrangelocally without jumping over barriers if there are enough

unstable modes. Collective dynamics is expected onlywhen these modes become less abundant at lower temper-atures. Likewise, we expect the Debye-Waller factor to belarger with SWAP, since the vibrational spectrum is softer.Note that these arguments are not restricted to finite rangeinteractions. We expect them to apply as well to Lennard-Jones potentials, for example, where the abundance ofdegrees of freedom versus the number of strong interactionsis also known to affect the vibrational spectrum [29,36].

V. HARD-SPHERE SYSTEMS

Our arguments above consider the potential energylandscape. For interaction potentials which are very sharp,nonlinearities induced by thermal fluctuations are impor-tant, and the vibrational properties of a glassy configurationat finite temperature T can differ significantly from those ofits inherent structure obtained by quenching it rapidly. Herewe consider the extreme case of hard spheres where theenergy is always zero, and cannot be used to definevibrational modes. Instead, by averaging on vibrationaltimescales within a glassy configuration, a local free energycan be defined [24–26], where particles that collide withinthat state interact with a logarithmic potential. Thisdescription is exact near jamming and systematic devia-tions are expected away from it [37]. However, in practice,the Hessian defined from this free energy captures wellthe fluctuations of particle positions and the vibrationaldynamics throughout the glass phase [25]. This procedurecan be pursued to include thermal effects in soft spheresas well [38].Stability and vibrational properties can be computed in

terms of this Hessian for non-SWAP dynamics [24,25].Salient results are shown in the simplified diagram ofFig. 7. Once again, two key determinants of stability arethe typical gap between interacting particles Δ≡ T=ðpRd

0Þrelative to the particle radius, and the excess coordinationδz, where the coordination is defined from the network ofparticles that are colliding within a glassy state. A marginalstability line separates stable and unstable configurations,as illustrated in Fig. 7, whose asymptotic behavior followsδz ∼ Δð2þ2θÞ=ð6þ2θÞ, where θ ≈ 0.41 [38]. (Strictly speaking,this line will depend slightly on the system preparation, butthis dependence is expected to be modest, and is irrelevantfor the present discussion.)Under a slow compression the system follows a line [in

red (Fig. 7)] in the ðΔ; δzÞ plane. Mechanical stability isreached only for some Δ < Δ0, a characteristic onset gapwhere the dynamic crosses over to an activated regimewhere vibrational modes become stable, consistent withGoldstein’s proposal. In these materials, deeper in the glassphase the system eventually returns to the stability line, andundergoes a sequence of buckling events that leave itmarginally stable [20,21,25]. Marginal stability implies thepresence of soft elastic modes (that differ from Goldstone

FIG. 6. Example of soft mode with ω ≈ 0.078 for α ¼ 1 andΔ ¼ 10−2. Blue disks indicate the initial particles radii, red disksthe new radii induced by motion along that mode. Arrowsrepresent the displacements δri of the particles multiplied by 4for visualization.

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modes) up to nearly zero frequency, and fixes the scalingproperties of both structure and vibrations as jamming isapproached [20,21]. These results, valid in finite dimen-sions, have been quantitatively confirmed in infinitedimension calculations [22,26,39]. In that case, the pointwhere buckling sets in was argued to be a sharp transition,coined Gardner, where the free-energy landscape fracturesin a hierarchical way [39], as supported by numericalstudies [40]. For very rapid quenches, it was argued that theentire glass phase should be marginal [25,39].How is this picture affected by SWAP? Our arguments for

the generalized Hessian of the soft-sphere system essen-tially go through unchanged for the generalized Hessian ofthe free energy in the hard-sphere system. Once again,stability becomes more demanding with SWAP, and themarginal stability line is shifted to higher coordination inthe ðΔ; δzÞ plane as represented in the right-hand panel ofFig. 7. Thus, the glass transition is shifted toward higherpacking fractions. At smaller gap Δ (corresponding to theapproach of jamming), stability implies δz ≥ α1=2, againimplying that isostaticity is lost with SWAP. We conjecturethat, just as for soft particles, marginal stability is reachedin the glass phase, which would correspond to a Gardnertransition in infinite dimensions.

VI. CONCLUSION

In SWAP algorithms, the dynamics is governed by aneffective potential Vðfrg; fRgÞ that describes both theparticles’ interaction and their ability to deform. As a result,we show that vibrational and elastic properties are softenedwhen swaps are allowed for, while thermodynamic quantitiesare strictly preserved (when thermal equilibrium is reached).This result supports that the crossover temperature T0, wheremechanical stability appears and dynamics becomes acti-vated, must be reduced with SWAP with TSWAP

0 < T0, leadingto a natural explanation as to why the glass transition occursthen at a lower temperature TSWAP

g < Tg. Secondly, SWAP

must strongly affect the structure of the glass phase. This isparticularly striking near the jamming transition that occurs inhard and soft spheres, where we predict that well-known keyproperties such as isostaticity must disappear. We confirmthese predictionsnumerically, and find that for rapid quenchesthe effective potential Vðfrg; fRgÞ appears to be marginallystable throughout the glass phase.Concerning the glass transition, our work does not

specify the mechanism by which activation occurs inglasses, but it does support that SWAP delays the temper-ature where activation is required for structural relaxation,which potentially explains several previous observations ofSWAP algorithms [6–9]. Possible theories to describe themechanism by which activation occurs in glasses includeelastic [41] and facilitation models [42]. We believe,however, that theories based on a growing thermodynamicbarrier (induced by a growing static length ξ) will be hard toreconcile with the notion that some collective modes do notsee any barriers at all [43].Our analysis also makes additional qualitative testable

predictions. By increasing continuously the width α ofthe radii distribution ρðRÞ, we predict that TSWAP

g ðαÞ willsmoothly decrease, while TgðαÞ should be essentiallyunchanged, with ½TgðαÞ − TSWAP

g ðαÞ�=TgðαÞ ∝ α and, morespecifically, ∝ α=Δ for soft spheres. Furthermore, manystudies have analyzed correlations between dynamics andvibrational modes, see, e.g., Refs. [14,15,25,46], which canbe repeated to relate the SWAP dynamics to the spectrum ofthe effective potential Vðfrg; fRgÞ. Near Tg, we predict thelatter to have more abundant modes at low or negativefrequencies than the much-studied Hessian of the potentialenergy, and its softest modes to be better predictors offurther relaxation processes. Lastly, the present analysissuggests that adding additional degrees of freedom (such aschanging the shape of the particles, and not only their size)will increase even further the difference between SWAP andnon-SWAP dynamics.Finally, we show that ultrastable glasses can be built

on the computer, simply by descending along the effectivepotential Vðfrg; fRgÞ. As illustrated in Fig. 7, theseconfigurations must sit strictly inside the stable region ofthe regular dynamics (i.e., at a finite distance from the blue

FIG. 7. Log-log representation of the stability diagram in thecoordination δz ¼ z − zc and Δ plane for continuously poly-disperse thermal hard spheres with non-SWAP (left) and SWAP

(right) dynamics. Note that for hard spheres Δ is independent oftemperature and vanishes at jamming. In the left-hand panel, theblue line separates mechanically stable and unstable configura-tions. The red line indicates the trajectory of a system under aslow compression. When Δ decreases toward the onset gap Δ0,metastable states appear and the dynamics becomes activated andspatially correlated. In the glass phase, the red trajectory willdepend on the compression rate, but will eventually reach the blueline at some (rate-dependent) ΔG. When this occurs, a buckling orGardner transition takes place where the material becomesmarginally stable, leading to a power-law relation between Δand δz. Right: For SWAP dynamics, stability is more demandingand is achieved only on the green line, which differs strictly fromthe blue one. Thus the onset gap decreases to some valueΔs

0 < Δ0: the dynamics become activated and correlated at largerdensities, shifting the position of the glass transition. Marginalityis still expected beyond some pressure Δs

G, but leads to plateauvalue for the coordination, indicating that isostaticity is lost.

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line in Fig. 7). As a consequence, the usual potential energylandscape UðfrgÞ around the obtained configurations doesnot display excess soft anomalous modes at very lowfrequency, even near the jamming transition: these modesare gapped. This result must hold for the ground state,too (which must be stable toward SWAP), and by continuityalso for low-temperature equilibrated states. It may explainwhy marginal stability (and the Gardner transition leadingto it) could not be observed in protocols where a thermalquench was used from SWAP-generated configurations [47].It would be very interesting to see if other well-knownexcitations of low-temperature glassy solids are alsogapped in these configurations, including two-level sys-tems, reported to be almost absent in experimental ultra-stable glasses [48].

ACKNOWLEDGMENTS

We thank L. Berthier, G. Biroli, M. Cates, M. Ediger, andF. Zamponi for discussions and E. DeGiuli for providinguseful comments on the manuscript. E. L. acknowledgessupport from the Netherlands Organisation for ScientificResearch (NWO) (Vidi Grant No. 680-47-554/3259).M.W. thanks the Swiss National Science Foundation forsupport under Grant No. 200021-165509 and the SimonsFoundation Grant (No. 454953 Matthieu Wyart).

APPENDIX: SUPPLEMENTAL MATERIAL

This appendix provides (i) descriptions of the numericalmodel, protocols, and methods used to generate athermalpackings under SWAP dynamics at different pressures, (ii) acomputation of the Hessian of the potential energy, togetherwith explanations about how prestress affects the vibra-tional modes, and (iii) a discussion about the effect of theradii distribution generated by SWAP dynamics on the valueof the packing fraction of our athermal packings obtainedwhile freezing the degrees of freedom associated withparticles’ radii.

1. Numerical model, protocols, and methods

We employ systems ofN ¼ 484 particles in a square boxin two dimensions. The total potential energy depends uponthe particles’ coordinates frg and radii fRg, as

Vðfrg; fRgÞ ¼ Uðfrg; fRgÞ þ μðfRgÞ: ðA1Þ

The pairwise potential term reads

Uðfrg; fRgÞ ¼ k2

Xij

½rij − ðRi þ RjÞ�2ΘðRi þ Rj − rijÞ;

ðA2Þ

where k is a stiffness, set to unity, rij is the distancebetween the ith and jth particles, and ΘðxÞ is the Heaviside

step function. The chemical potential associated with theradii is

μðfRgÞ ¼ kR2

Xi

ðRi − Rð0Þi Þ2

�Rð0Þi

Ri

�2 ≡X

i

μðRi; Rð0Þi Þ;

ðA3Þ

where kR is the stiffness of the potential associated with theradii fRg that serves as a parameter in our study, and is set

as described below. Rð0Þi denotes the intrinsic radius of the

ith particle. In each configuration we randomly assigned

Rð0Þi ¼ 0.5 for half of the particles and Rð0Þ

i ¼ 0.7 for theother half. The massm of particles, and that associated withtheir fluctuating radii, are all set to unity. Vibrationalfrequencies should be understood as expressed in termsof

ffiffiffiffiffiffiffiffiffik=m

p, and pressures in terms of k.

Configurations in mechanical equilibrium at zero tem-perature and at a desired target pressure p0 were generatedas follows. We start by initializing systems with randomparticle positions at packing fraction ϕ ¼ 1.2, and set the

initial radii to be Ri ¼ Rð0Þi . We then minimize the total

potential energy Vðfrg; fRgÞ at a target dimensionlesspressure Δ0 ¼ 10−1 using a combination of the FIRE

algorithm [49] and the Berendsen barostat [50]; see furtherdiscussion about the latter below. Each packing is then usedas the initial conditions for sequentially generating lower-pressure packings, as demonstrated in Fig. 8. Followingthis protocol, we generated 1000 independent packingsat each target dimensionless pressure, which ranges fromΔ0 ¼ 10−1 up to Δ0 ¼ 10−5. For each target pressure, weset the stiffness kR of the chemical potential of the radii

103

104

105

106

107

total number of iterations

10-4

10-3

10-2

10-1

Δ

α=1α=10−1

α=10−2

α=10−3

FIG. 8. Dimensionless pressure Δ as a function of iterationnumber for a packing-generating simulation starting from oneparticular initial condition. Each step of the staircase shape of thesignal corresponds to the production of a packing at some desiredtarget pressure. The criterion for convergence to mechanicalequilibrium at each pressure is explained in the text. We producedpackings ranging from Δ ¼ 10−1 to Δ ¼ 10−5.

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according to kR¼p0=α, and vary α systematically between3 × 10−4 and 1. During minimizations we calculate acharacteristic net force scale Ftyp ≡ ðPikFik2=NÞ1=2,where Fi ¼ −∂V=∂ri is the net force acting on the ithparticle, whose coordinates are denoted by ri. A packing isconsidered to be in mechanical equilibrium when Ftyp

drops below 10−8Δ0.

a. Berendsen barostat parameter

The FIRE algorithm [49] features equations of motionwhich are to be integrated as in conventional MD simu-lations. We exploit this feature and embed the Berendsenbarostat [50] in our Verlet integration scheme [51]. Thisamounts to scaling the simulation cell volume by a factor χ,calculated as

χ ¼ 1 − ξδtðΔ0 − ΔÞ; ðA4Þ

where δt is the (dynamical) integration time step, and ξ is aparameter that determines how quickly the instantaneousdimensionless pressure converges to the target dimension-less pressure [51]. Figure 9 shows the ξ dependence of theconvergence of the instantaneous dimensionless pressure Δto the target valueΔ0. Below ξ ¼ 0.01, the behavior ofΔ asa function of iteration number is similar. We therefore setξ ¼ 0.01 throughout this work.

2. Computation of HSWAP

The total potential energy Vðfrg; fRgÞ of our modelsystem is spelled out in Eqs. (A1)–(A3). We next work outthe expansion of V in terms of small displacements δri of

particle positions, and small fluctuations δRi of the radii,about a mechanical equilibrium configuration with energyV0, as

δV ≡ V − V0 ≃1

2

Xij

δri ·Hij · δrj þ1

2

Xij

δRiQijδRj

þXij

δRiTij · δrj; ðA5Þ

where Hij ≡ ∂2V=∂ri∂rj, Q≡ ∂2V=∂Ri∂Rj, and Tij≡∂2V=∂ri∂Rj. The expansion given by Eq. (A5) can bewritten using bra-ket notation as

δV ¼ 1

2hδljHSWAPjδli; ðA6Þ

where jδli is a ðdþ 1ÞN-dimensional vector which con-catenates the spatial displacements δri and the fluctuationsof the radii δRi: ðδr1; δr2;…; δrN; δR1;…; δRNÞ. Theoperator HSWAP can be written as

HSWAP ¼� ½HNd;Nd� ½TNd;N �½TT

N;Nd� ½QN;N ��:

The elements of the submatrix HNd;Nd can be written astensors of rank d ¼ 2 as

Hij ¼ δhiji

�kðrij − Ri − RjÞ

2rijn⊥ij ⊗ n⊥ij þ

k2nij ⊗ nij

�

þ δi;jXl

�kðril − Ri − RlÞ

2riln⊥il ⊗ n⊥il þ

k2nil ⊗ nil

�;

where nij is a unit vector connecting between the ith andjth particles, n⊥ij is a unit vector perpendicular to nij, ⊗is the outer product, δhiji ¼ 1 when particles i, j are incontact, δi;j is the Kronecker delta, and the sum is takenover all particles l in contact with particle i. The elements ofthe submatrix QN;N are scalars given by

Qij ¼ δhijikþ δi;j

�Xhli

kþ ∂2μðRi; Rð0Þi Þ

∂R2i

�: ðA7Þ

The matrix TN;Nd is not diagonal, and each element can beexpressed as a vector with two components given by

Tij ¼ −δhijiknij − δi;jXl

knil: ðA8Þ

The eigenvectors of HSWAP are the normal modes of thesystem, and the eigenvalues are the vibrational frequenciessquared ω2. The distribution of these frequencies is knownas the density of states DðωÞ.

100

102

104

number of iterations

0.001

0.002

0.003

Δ

ξ=1ξ=0.1ξ=0.01ξ=0.001p

0=10

-3

FIG. 9. Instantaneous dimensionless pressureΔ as a function ofthe number of iterations for one initial condition, taken to beΔ ¼ 0.025. Here we set α ¼ 10−3, fix the target pressure toΔ0 ¼ 10−3, and observe the form of the convergence of theinstantaneous dimensionless pressure to the target dimensionlesspressure, for different values of the Berendsen barostat parameterξ (see definition in text).

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a. Effect of the prestress on the vibrational modes

When a system of purely repulsive particles is atmechanical equilibrium, forces fij are exerted betweenparticles in contact. These forces give rise to a term in theexpansion of the energy, of the form

−1

2

Xhiji

fijrij

½ðδrj − δriÞ · n⊥ij�2; ðA9Þ

often referred to as the “prestress term.” For plane waves, itcan be shown that the energy contributed by this term isvery small. However, for the soft modes present when thesystem is close to the marginal stability limit, it can beshown that this term reduces the energy of the modesby a quantity proportional to the pressure [20]. Marginal

stability corresponds to a buckling transition where thedestabilizing effect of prestress exactly compensates thestabilizing effect of being overconstrained. In this scenario,where two effects compensate each other, the eigenvalue ofthe softest (non-Goldstone) modes of the Hessian in theabsence of prestress ω2 must be much larger than ω�2computed when prestress is present. To demonstrate this,we have calculated the density of states for systems whileincluding and excluding the prestress term. The results areshown in Fig. 10, where it is found that near jammingω�2=ω2 ≈ 5%, which is consistent with what was previ-ously found for the traditional jamming transition [21] andsupports that the system is very close to (but not exactly at)marginal stability.

3. Packing fraction

In the main text we show that the jamming packingfraction ϕc generated using the SWAP dynamics increaseswhen ρðRÞ broadens, i.e., for smaller values of theparameter α that controls the stiffness of the potentialenergy associated with the radii. Here we compare thedependence of the packing fraction on pressure as mea-sured for systems in which the radii are not allowed tofluctuate. In addition, in this test we borrow the distributionof radii ρðRÞ from SWAP packings generated at p ¼ 10−4,and at various values of the parameter α, varied between1 to ∞ (the latter corresponds to disallowing particleradii fluctuations). Packings were generated using the totalpotential energy as given by Eq. (A2) (with the radii Riconsidered to be fixed), and using the same protocol and

10-4

10-2

10010

-2

10-1

100

D(ω

)

Δ=10-2

Δ=5x10-3

Δ=10-3

Δ=5x10-4

Δ=10-4

Δ=5x10-5

Δ=10-5

10-5

10-4

10-3

10-2

10-3

10-2

10-1

ωω∗

ω

Δ

α=10−3

ω∗

10-3

10-2

10-1

100

ω10

-2

10-1

100

D(ω

)

α=10−1

α=2x10-3

α=10−2

α=2x10-3

α=10−3

α=3x10-4 10

-410

-310

-210

-110

0

10-3

10-2

ωω∗

ω

Δ=10−4

α

ω∗

FIG. 10. Top: Density of states DðωÞ for different dimension-less pressures Δ at fixed α ¼ 10−3, with the prestress termincluded (dashed lines) and excluded (continuous lines). Inset:Characteristic frequencies ω� and ω (as marked in the mainpanel) versus Δ. Dashed line correspond to the marginalitycondition ω� ∼

ffiffiffiffiΔ

pand ω ∼

ffiffiffiffiΔ

p. Bottom: DðωÞ for different

α and fixed Δ ¼ 10−4. Note that for these curves ω�2=ω2 ≈ 5%,indicating that the system is very close to marginal stability.

10-5

10-4

10-3

10-2

10-1

100

Δ

0.85

0.9

0.95

1

φ

α=1α=10−1

α=10−2

α=10−3

α=3x10-4

α=0

10-5

10-4

10-3

0.844

0.848

FIG. 11. Packing fraction ϕ as function of dimensionlesspressure Δ measured for packings in which radii are not allowedto fluctuate, and whose distribution of radii is borrowed fromSWAP packings generated at Δ ¼ 10−4 and at different values ofα, as indicated by the legend. The inset shows an enlargement ofvery small dimensionless pressures, demonstrating that the valueof ϕc depends very weakly on the borrowed distribution of radiiof the SWAP packings, as determined by the parameter α.

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numerical methods used to generate the SWAP packings.The results are shown in Fig. 11, where it can be seen thatthe value of ϕc is essentially the same for any borrowedρðRÞ from the SWAP packings.

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