The response of jammed packings to thermal fluctuations

Qikai Wu,1 Thibault Bertrand,2 Mark D. Shattuck,3, 1 and Corey S. O’Hern1, 4, 5

1Department of Mechanical Engineering and Materials Science,Yale University, New Haven, Connecticut, 06520, USA

2Laboratoire Jean Perrin UMR 8237 CNRS/UPMC,Université Pierre et Marie Curie, 75255 Paris Cedex, France

3Department of Physics and Benjamin Levich Institute,The City College of the City University of New York, New York, 10031, USA

4Department of Physics, Yale University, New Haven, Connecticut, 06520, USA5Department of Applied Physics, Yale University, New Haven, Connecticut, 06520, USA

(Dated: October 4, 2017)

We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disksto thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties atfinite temperature. In contrast, numerous prior studies characterized the structural and mechanicalproperties of MS packings of frictionless spherical particles at zero temperature. Packings of diskswith purely repulsive contact interactions possess two main types of nonlinearities, one from theform of the interaction potential (e.g. either linear or Hertzian spring interactions) and one fromthe breaking (or forming) of interparticle contacts. To identify the temperature regime at which thecontact-breaking nonlinearities begin to contribute, we first calculated the minimum temperaturesTcb required to break a single contact in the MS packing for both single and multiple eigenmodeperturbations of the T = 0 MS packing. We find that the temperature required to break a singlecontact for equal velocity-amplitude perturbations involving all eigenmodes approaches the min-imum value obtained for a perturbation in the direction connecting disk pairs with the smallestoverlap. We then studied deviations in the constant volume specific heat CV and deviations of theaverage disk positions ∆r from their T = 0 values in the temperature regime Tcb < T < Tr, whereTr is the temperature beyond which the system samples the basin of a new MS packing. We find

that the deviation in the specific heat per particle ∆C0V /C

0V relative to the zero temperature value

C0V can grow rapidly above Tcb, however, the deviation ∆C

0V /C

0V decreases as N

−1 with increasingsystem size. To characterize the relative strength of contact-breaking versus form nonlinearities, wemeasured the ratio of the average position deviations ∆rss/∆rds for single- and double-sided linearand nonlinear spring interactions. We find that ∆rss/∆rds > 100 for linear spring interactions is in-dependent of system size. This result emphasizes that contact-breaking nonlinearities are dominantover form nonlinearities in the low temperature range Tcb < T < Tr for model jammed systems.

PACS numbers: 45.70.–n, 63.50.–x, 64.70.pv

I. INTRODUCTION

Static packings of frictionless disks and spheres are in-formative model systems for studying jamming in gran-ular media [1] and dense colloidal suspensions [2]. Me-chanically stable (MS) packings of frictionless disks intwo spatial dimensions (2D) are isostatic at jamming on-set [3] and possess N0c = 2N

′− 1 contacts (with periodicboundary conditions), where N ′ = N−Nr is the numberof disks in the force-bearing contact network, N is thetotal number of disks, and Nr is the number of “rattler”disks with fewer than three contacts per disk [4]. (SeeFig. 1 (a) and (b).) Mechanically stable disk packingspossess a full spectrum of 2N ′− 2 nonzero eigenvalues ofthe dynamical matrix (i.e. the Hessian of the interactionpotential [5]), which represent the vibrational frequen-cies of the zero-temperature packings in the harmonicapproximation. The structural and mechanical proper-ties of isostatic disk and sphere packings near jamming atzero temperature have been reviewed extensively [6–8], in-cluding the pressure scaling of the bulk and shear moduli,excess contact number, and low-frequency plateau in the

density of vibrational modes near jamming onset. Morerecently, several groups have investigated how the scalingbehavior of these quantities is affected by thermal fluctu-ations using computer simulations [9–12], and mechanicalvibrations in experiments of granular media [13–15].

In particular, several authors have used computer sim-ulations of soft disks that interact via purely repulsivelinear spring potentials to study how the density of vi-brational modes of mechanically stable packings at zerotemperature and finite overcompression (with potentialenergy per particle U > 0) changes with increasing tem-perature. This work has shown that there is a character-istic temperature T ∗ ∼ U ∼ ∆φ2, where ∆φ = φ− φJ isthe deviation in the packing fraction above jamming on-set at φJ , above which the density of vibrational modesbegins to deviate strongly from that at zero tempera-ture [10, 11, 16]. In addition, they showed that T ∗ cor-responds to the temperature above which an extensivenumber of the contacts in the T = 0 contact network hasbroken.

These prior studies emphasized that an extensive num-ber of broken contacts (or more [12]) were required to

arX

iv:s

ubm

it/20

2709

0 [

cond

-mat

.sof

t] 4

Oct

201

7

2

(a) (b)

UN 410-10 10-5 100 105 1010

Nm(U

)=N

tot

0

0.2

0.4

0.6

0.8

1

(c)

U10-10 10-8 10-6 10-4 10-2

hm=N

i

10-4

10-2

100

102

(d)

FIG. 1: Examples of isostatic mechanically stable bidisperse disk packings at zero temperature with (a) Nc = N0c = 127

contacts and Nr = 0 rattler particles and (b) Nc = N0c = 115 contacts and Nr = 6 rattler particles. The non-rattler (rattler)

disks are outlined in black (red). For both (a) and (b), the total number of disks N = 64, the potential energy per particleU = 10−12, and the solid black lines connecting disks centers indicate force-bearing interparticle contacts. (c) The fraction ofmechanically stable packings Nm(U)/Ntot that possess m = Nc −N0c = 0 (circles), 2 (exes), and 4 (plus signs) excess contactsas a function of UN4 for three system sizes N = 32 (solid lines), 128 (dashed lines), and 256 (dotted lines). (d) The numberof excess contacts m normalized by N averaged over 5000 MS packings and plotted versus U for three system sizes N = 32(circles), 128 (exes), and 256 (plus signs). The dashed line has slope 0.25.

significantly change the binned density of vibrationalmodes. However, do any important physical quantitieschange when a single contact or sub-extensive number ofcontacts in the zero-temperature contact network is bro-ken by thermal fluctuations? The answer to this ques-tion may depend on the number of excess contacts inthe T = 0 contact network m = Nc − N0c . For exam-ple, if a zero-temperature packing has zero excess con-tacts (m = 0), the breaking of a single contact wouldcause the system to become unjammed. In Fig. 1 (c), weshow the fraction of MS packings with m excess contacts,Nm(U)/Ntot, can be collapsed for each m and differentsystem sizes by plotting Nm(U)/Ntot as a function ofUN4. We find that the average number of excess contactsscales as 〈m〉/N ∼ U1/4 (Fig. 1 (d)), which is consistentwith previous studies at zero temperature [6]. Thus, inthe large-system limit isostatic packings with m = 0 existonly at U = 0.

In this article, we will first characterize the minimumtemperature required to break a single contact as a func-tion of the protocol used to add thermal fluctuations. Wefocus on this quantity because it can be determined ex-actly in the low-temperature limit from the eigenvaluesand eigenmodes of the dynamical matrix for the T = 0MS packings. In particular, we will measure the mini-mum temperature T1(m,m − 1) above which a T = 0MS packing with m excess contacts changes to a packingwith m − 1 excess contacts in response to a perturba-tion along a single eigenmode. Thermal fluctuations canalso be added to the zero-temperature MS packing byperturbing the system along a superposition of n eigen-modes of the dynamical matrix, and we can measure theminimum temperature, Tn(m,m − 1), required to breaka single contact. We will show that that the minimumtemperature required to break a single contact over allsingle mode excitations scales as T1(m,m− 1) ∼ U/Nα,

3

where α ≈ 2.6 ± 0.1, which is consistent with pre-vious measurements [17]. For multi-mode excitations,Tn(m,m−1) decreases as the number of eigenmodes n in-volved in the perturbation increases, reaching a minimumfor perturbations with equipartition of all 2N ′ eigen-modes. The minimum temperature required to breaka single contact for a perturbation with equipartition ofall eigenmodes of the T = 0 dynamical matrix scales asT2N ′(m,m−1) ∼ N−β , where β ≈ 2.9±0.1. This system-size size dependence is stronger than that for single-modeperturbations.

We also measured the temperature required to breakmultiple contacts. In this case, we employed moleculardynamics simulations to determine the temperature atwhich a given fraction of simulation snapshots possess aspecified number of contacts. This information cannotbe obtained from the T = 0 dynamical matrix, since theeigenmodes change after contacts begin breaking. Wefind a power-law scaling relation between the tempera-ture, number of broken contacts Nbc = N

0c + m − Nc,

system size, and potential energy per particle U .

After investigating the temperatures at which a givennumber of zero-temperature contacts break, we searchfor physical quantities that may be sensitive to changesin the interparticle contact networks. We focused ontwo quantities: 1) the deviation in the specific heat∆CV /C

0V = (CV − C0V )/C0V from the zero temperature

value C0V and 2) the deviation of the average positionsof the disks xi and yi in a packing at a given tempera-

ture, ∆r =√∑N

i=1[(x̄i − x0i )2 + (ȳi − y0i )2]/N , from theT = 0 disk positions ~R0 = {x01, y01 , . . . , x0N ′ , y0N ′}. Cal-culating ∆r is important for understanding how far theinitial packing can move in configuration space beforetransitioning to the basin of a new MS packing. We com-pare ∆rss for systems with purely repulsive (single-sided)linear and nonlinear spring interactions to ∆rds obtainedfor systems with double-sided linear and nonlinear springinteractions, which allows us to quantify the additionalnonlinearities that arise from contact breaking. We findthat both quantities, ∆CV /C

0V and ∆r

ss/∆rds are sensi-tive to the breaking of a single contact. However, the de-viation ∆CV /C

0V decreases with increasing system size.

In contrast, ∆rss/∆rds > 100 for purely repulsive lin-ear springs and does not depend strongly on system size.We also quantify ∆rss/∆rds for packings with Hertzianspring interactions and show that contact breaking in-creases the magnitude of the nonlinearities at finite tem-perature, but not as much as for linear repulsive springinteractions.

There are several important temperature scales to con-sider when studying the response of MS packings to ther-mal fluctuations. In Fig. 2, we show four temperatureregimes: 0 < T < Tcb, Tcb < T < Tr, Tr < T < Tg, andT > Tg. For 0 < T < Tcb, where Tcb is the minimumtemperature at which a single contact breaks, the sys-tem is weakly nonlinear with “form” nonlinearities thatarise when the interaction potential cannot be expressed

FIG. 2: Schematic of four important temperature regimeswhen studying the response of MS packings to thermal fluc-tuations. For T < Tcb, the T = 0 contact network remains in-tact. In this regime, “form” nonlinearities occur when the in-terparticle potential cannot be written exactly as a harmonicfunction of the disk positions. For Tcb < T < Tr, the T = 0contact network changes, both form and contact-breakingnonlinearities occur, and the system remains in the basin ofattraction of the original MS packing. For Tr < T < Tg,the system can move to the basins of attraction of other MSpackings, but the relaxation times are sufficiently long thatstructural relaxation is not complete. For T > Tg, the sys-tem is liquid-like with finite structural relaxation times. Thisarticle focuses on the (unshaded) temperature regimes thatoccur for 0 < T ≤ Tr.

exactly as a harmonic function of the disk positions.(We use the notation Tcb for the temperature requiredto break a single contact when we do not specify thetype of initial perturbation.) In the temperature regimeTcb < T < Tr, contacts begin breaking, both form andcontact-breaking nonlinearities occur, and the system re-mains in the basin of attraction of the original MS pack-ing. In this regime ∆rss can be much larger than ∆rds

due to contact-breaking nonlinearities. At larger temper-atures, Tr < T < Tg, the system rearranges and movesbeyond the basin of attraction of the original T = 0 MSpacking, but the timescales are prohibitively long to al-low complete structural relaxation. Finally for T > Tg,the system is liquid-like with a finite structural relaxationtime.

We emphasize that a number of studies have char-acterized the structural and mechanical properties ofMS packings at T = 0 [18–20]. Further, many studieshave tracked the growth of the dynamical heterogeneitiesand the structural relaxation times as T → Tg fromabove [21–23]. However, few studies have focused on thelow-temperature regimes 0 < T < Tcb and Tcb < T < Tr,where the contribution of contact breaking to the mag-nitude of the nonlinearities can be quantified at finitetemperature. In future work, we will focus on the tem-perature regime Tr < T < Tg to understand the con-nection between the geometry of the high-dimensionalenergy landscape and slow structural relaxation.

The remainder of this article will be organized as fol-lows. In Sec. II, we describe the methods we employto generate zero-temperature MS packings, the proto-cols used to add thermal fluctuations to the MS pack-ings, and the measurements of the changes in the specificheat ∆C0V /C

0V and average particle positions ∆r of the

packings from their T = 0 values as a function of tem-

4

perature. In Sec. III, we present our results for ∆CV /C0V

and ∆r. We show that ∆CV /C0V increases more strongly

when a single contact in the T = 0 MS packing changes.We find that ∆CV /C

0V decreases with increasing sys-

tem size, however, the quantity ∆rss/∆rds, which iden-tifies the distinct contribution of contact breaking to thenonlinear response, does not depend strongly on systemsize. In Sec. IV, we summarize our results and high-light promising future research directions that stem fromthis work. We also provide several Appendices that in-clude additional details of the methods and calculationswe implement. In Appendix A, we provide additionaldetails concerning the method we used to calculate theminimum temperature required to break a single con-tact with perturbations that involve n eigenmodes of theT = 0 dynamical matrix with equal velocity amplitudes.In Appendix B, we discuss the additional nonlinearitiesthat arise from rattlers in MS packings and affect ∆r(T )at finite temperatures. In Appendix C, we describe themethods that we employed to measure the rearrangementTr and glass transition Tg temperatures. Finally, in Ap-pendix D, we show that the leading order term in thechange in the average position scales linearly with tem-perature, ∆r ∼ T , for a particle in a one-dimensionalcubic potential well.

II. METHODS

Our computational studies focus on measuring the re-sponse of MS packings composed of N bidisperse friction-less disks (N/2 large and N/2 small disks with diameterratio σL/σS = 1.4) to thermal fluctuations with systemsizes in the range from N = 16 to 1024 disks using peri-odic boundaries in square simulation cells. The disks (allwith mass m) interact via the pairwise, purely repulsivepotential,

U(rij) =�

α

(1− rij

σij

)αΘ

(1− rij

σij

), (1)

where rij is the separation between the centers of disksi and j, σij = (σi + σj)/2 is the average disk diameter,� is the energy scale of the repulsive interaction, Θ(x)is the Heaviside step function, and α = 2 (5/2) corre-sponds to linear (Hertzian) repulsive spring interactions.We also consider disk packings that interact via double-sided spring potentials with a similar form to that inEq. 1:

Uds(rij) =�

α

∣∣∣∣1− rijσij∣∣∣∣α . (2)

For studies involving interactions in Eq. 2, the interpar-ticle contact network is fixed to that in the T = 0 MSpacking for all temperatures [24]. Comparison of the re-sults from single- versus double-sided interactions allowsus to determine the strength of the nonlinearities thatarise from contact breaking alone.

We generate MS packings as function of the total po-tential energy per particle U = Σi>jU(rij)/N using aprotocol that successively compresses or decompressesthe system in small packing fraction steps ∆φ followedby conjugate gradient energy minimization [4]. The com-pression/decompression protocol is terminated when thetotal potential energy per particle satisfies |Uc−U |/U <10−16, where Uc is the current and U is the target po-tential energy per particle.

The initial perturbations will be applied along oneor more of the eigenmodes of the dynamical matrix ofthe T = 0 MS packings. We denote the 2N ′ − 2non-zero eigenfrequencies of the dynamical matrix as{ω1, . . . , ω2N ′−2}. Each eigenfrequency ωi has an associ-ated eigenvector Êi = {eix1, eiy1, eix2, eiy2, . . . , eixN ′ , eiyN ′}that satisfies (Êi)2 = 1. The disk velocities ~V 0 ={v0x1, v0y1, . . . , v0xN ′ , v0xN ′} corresponding to the initial per-turbation can be expressed as a linear combination of theeigenmodes of the dynamical matrix:

~V 0 =

2N ′−2∑i=1

AiωiÊi. (3)

We will use the notation that upper case vectors, e.g. ~R

and ~V , include both the particle and spatial dimensions,while lower case vectors, e.g. ~r and ~v, only include thespatial dimensions.

For sufficiently small amplitude perturbations, thetime evolution of the multi-particle velocities and posi-tions are given in the harmonic approximation by

~V (t) =

2N ′−2∑i=1

AiωiÊi cos(ωit), (4)

and

~R(t) = ~R0 +

2N ′−2∑i=1

AiÊi sin(ωit), (5)

where ~R0 gives the disk positions in the T = 0 MS pack-ing. We calculate the temperature of the system usingthe average kinetic energy per particle K/N [25].

For sufficiently large temperatures, when multiple T =0 contacts break and new contacts form, we cannot usethe T = 0 eigenmodes of the dynamical matrix to de-termine the properties of the contact networks. Thus,we will characterize the relation between the tempera-ture, number of contacts, system size, and potential en-ergy per particle using molecular dynamics simulationsat constant number of disks, area, and total energy E.For the MD simulations, we use the velocity Verlet in-tegration scheme with a time step ∆t ∼ tcol/40, wheretcol = σS

√�/m is a typical interparticle collision time

scale, which provides total energy conservation with rel-ative standard deviation δE/E < 10−13.

To investigate the effects of contact breaking, we willmeasure two physical quantities as a function of the am-plitude (or temperature) of the thermal fluctuations. We

5

will first study the change in the constant volume specificheat ∆CV from its zero-temperature value C

0V = 2N

′kb:

∆CV (T )

C0V=CV (T )− C0V

C0V, (6)

where CV = dE/dT and kb is the Boltzmann constant.In the low-temperature limit, Eqs. 4 and 5 can be usedto calculate the total energy:

E = K(t)+U(t) = U0+m

2

2N∑i=1

A2iω2i = U0+2N ′kbT, (7)

where U0 is the initial total potential energy. We willmeasure the specific heat per particle CV in the molec-ular dynamics simulations by taking the temperaturederivative numerically,

CV (T ) = (8)

1

N ′E(T + dT )− E(T )

dT= kb

E(T + dT )− E(T )K(T + dT )−K(T )

.

From Eq. 8, the deviation in the specific heat per particlecan be written as

∆CV (T )/C0

V =1

2

E(T + dT )− E(T )K(T + dT )−K(T )

− 1. (9)

We will also quantify the changes in the average posi-tions of the disks, ∆r, as a function of temperature. Wedefine ∆r as

∆r(T ) =

√√√√ 1N ′

N ′∑i=1

[(x̄i(T )− x0i )2 + (ȳi(T )− y0i )2],

(10)where (x0i , y

0i ) are the x- and y-coordinates of the ith disk

in the T = 0 MS packing and (x̄i(T ), ȳi(T )) are the time-averaged x- and y-coordinates of disk i at temperature T .∆r(T ) can be interpreted as the average distance in the2N ′-dimensional configuration space between the T = 0MS packing and the packing at finite T . Note that rattlerdisks are not included in the calculations of ∆r.

III. RESULTS

We organize our results into two main sections. InSec. III A, we discuss the results for the minimum temper-atures required to break one or more contacts for single-and multi-mode perturbations. In Sec. III B, we show ourresults for the temperature dependence of the deviationin the specific heat per particle ∆CV and deviation inthe average disk positions ∆r from those in the T = 0MS packing as a function of temperature. For T < Tcb,form nonlinearities give rise to non-zero values of ∆CVand ∆r. For T > Tcb, both form and contact-breakingnonlinearities are present. By comparing ∆CV and ∆rfor single- and double-sided spring interactions, we can

N101 101.5 102 102.5 103

hT1(m

;m!

1)i

=U

10-10

10-8

10-6

10-4

10-2

FIG. 3: The minimum temperature T1(m,m − 1) requiredto break a single contact when perturbing an MS packingalong one of the eigenmodes of the dynamical matrix averagedover 5000 MS packings, normalized by the potential energyper particle U , and plotted as a function of system size N .T1(m,m−1) was obtained by minimizing over all single modeperturbations. We include results for U = 10−12 (circles),10−8 (exes), and 10−4 (pluses). The slope of the dashed lineis −2.6. Rattler disks are removed from the packings prior toperforming these calculations.

isolate the effects of the contact-breaking nonlinearities.We find that for Tcb < T < Tr the specific heat deviation∆CV scales as N

−1, whereas ∆r is roughly independentof system size.

A. Temperatures required to break single andmultiple contacts

In this section, we study the minimum temperaturerequired to break a given number of contacts in the T = 0MS packing. We first focus on the breaking of a singlecontact and then study the breaking of multiple contacts.We will show that the temperature required to break thefirst contact depends strongly on the form of the initialperturbation. For example, the minimum temperatureis smaller for perturbations along multiple eigenmodescompared to the minimum temperature for perturbationsalong a single eigenmode.

At sufficiently low temperatures, we can use the har-monic approximation for the disk positions given in Eq. 5to calculate exactly the minimum temperature requiredto break a single contact. If we introduce a perturbationalong a single eigenmode k, the minimum temperaturerequired to break a single contact T k1 (m,m − 1) can becalculated by first solving r2ij = σ

2ij for all contacting disk

pairs i and j and then finding the minimum perturbation

6

n1 2 3 4 5 6

Tn(m

;m!

1)=

U

#10-3

0

0.2

0.4

0.6

0.8

1

(a)

N101 101.5 102 102.5 103

hTm

ini=

U

10-10

10-8

10-6

10-4

10-2

3 #10-3-3 -2 -1 0 1 2 3

U0 !

U

10-8

10-7

10-6

10-5

(b)

(c)

3

(d)

FIG. 4: (a) The minimum temperature 〈Tn(m,m−1)〉 (normalized by U and averaged over 5000 MS packings) required to breaka single contact in response to perturbations that include n = 1, 2, . . . , 6 eigenmodes of the dynamical matrix. 〈Tn(m,m− 1)〉is obtained by minimizing over all possible n-mode combinations of the 2N ′− 2 eigenmodes for each MS packing at U = 10−12(dashed line), 10−8 (solid line), and 10−4 (dotted line). The horizontal lines give the minimum temperature 〈Tmin/U〉 requiredto remove the smallest overlap between a pair of contacting disks at each U (averaged over 500 MS packings). The inset showsthe scaling of 〈Tmin/U〉 with system size N for the same values of U as in the main panel. The slope of the dashed line is −2.9.(b) Difference in the potential energy per particle between MS packings before (U) and after (U ′) separating the pair of diskswith the smallest interparticle overlap as a function of the angle θ between the old and new separation vectors between thetwo disks. (c) and (d) Schematic of the process to measure Tmin/U . In panel (c), the disk pairs with the smallest overlap areshaded in blue. In panel (d), this pair of disks is shifted so that rij = σij . The original positions are indicated by the dashedcircles. The new separation vector makes an angle θ with the old separation vector (as indicated by the dotted lines). Aftershifting disks i and j, potential energy minimization is performed allowing all disks to move except i and j. In both panels,the contact networks of the blue-shaded disks are indicated by solid lines.

amplitude (or temperature) over all disk pairs:

T k1 (m,m− 1) = (11)

mini>j

|~δkij · ~r0ij ||~δijk |2

√√√√1 + (σ2ij − |~r0ij |2)|~δijk |2

|~δijk · ~r0ij |2− 1

2 ,

where ~δkij = ~ekij sin(ω

kt)/ωk and ~ekij = (ekxi−ekxj , ekyi−ekyj).

To calculate the minimum T k1 (m,m − 1) over all eigen-modes, we set | sin(ωkt)| = 1 and find T1(m,m − 1) =mink T

k1 (m,m − 1). (See additional details in Ap-

pendix A.)In Fig. 3, we show 〈T1(m,m−1)〉/U averaged over 5000

MS packings as a function of system size N for three val-ues of U . We find that 〈T1(m,m− 1)〉 normalized by Ucollapses the data and 〈T1(m,m− 1)〉/U displays power-law scaling with system-size, 〈T1(m,m − 1)〉/U ∼ N−α,where α ≈ 2.6±0.1. Thus, 〈T1(m,m−1)〉 tends to zero inthe large system limit [17], which stems from the increas-ing probability for MS packings to possess anomalouslysmall overlaps as N →∞.

We now consider multi-mode perturbations and mea-sure the minimum temperature required to break a singlecontact in T = 0 MS packings. If we include n eigen-modes in the perturbation, in the low-temperature limit,

7

the disk positions and velocities are given by

~V (t) =

n∑k=1

AkωkÊk cos(ωkt), (12)

and

~R(t) = ~R0 +

n∑k=1

AkÊk sin(ωkt). (13)

As for the single eigenmode perturbations, we can use

the harmonic expression for ~R(t) (Eq. 13) to determinethe minimum temperature required to break a single con-tact for multi-mode perturbations. Setting r2ij = σ

2ij for

each pair of disks in the force-bearing backbone yields

an expression similar to that in Eq. 11, except ~δijk is re-

placed by ~δij =∑nk=1 ~e

kij sin(ω

kt)/ωk. The minimumtemperature required to break a single contact is ob-tained by evaluating the extrema of the sine functions,| sin(ω1t)| = | sin(ω2t)| = . . . = | sin(ωnt)| = 1, wherewe must check all combinations of sin(ωkt) = ±1, andby minimizing over all contacting disk pairs. For smalln, we discretize all of the possible eigenmode amplituderatios between 10−2 and 102 and identify the amplituderatio combination that yields the minimum temperatureTn(m,m − 1) to break a single contact. For n = 2and 3, we explicitly showed that Tn(m,m − 1) is mini-mized (over all possible perturbations) for equal velocity-amplitude perturbations. For n > 3, we assumed thatA1ω

1 = A2ω2 = . . . = Akω

k perturbations give the mini-mum Tn(m,m− 1). (Additional details concerning thesecalculations are included in Appendix A.)

In Fig. 4 (a), we plot Tn(m,m − 1)/U for single MSpackings using multi-mode perturbations as a function ofthe number of eigenmodes n = 1, 2, . . . , 6 for three valuesof U , 10−12, 10−8, and 10−4. For all U , we find thatTn(m,m − 1)/U decreases with increasing n and thenbegins to saturate for n & 6. In general, the minimumtemperature required to break a single contact decreaseswith an increasing number of eigenmodes in the pertur-bation because the perturbation is more likely to havea signficant projection onto the separation vector corre-sponding to the smallest overlap between disks. Satu-ration of Tn(m,m − 1) with increasing n is interestingbecause it implies that the probability to obtain a pair ofdisks in the force-bearing backbone with vanishing over-lap is zero in any finite-sized system with U > 0.

We also developed a method to estimate Tn(m,m− 1)in the large-n limit, which is illustrated in Fig. 4 (c) and(d). We first identify the pair of disks i and j in theforce-bearing backbone with the smallest overlap. Weseparate disks i and j so that rij = σij , while main-taining the center of mass of the two disks and fixing allof the positions of the other disks in the MS packing.We then minimize the total potential energy, allowingall disks to move except disks i and j, as a function ofthe angle θ between the old and new separation vectorsbefore minimization. In Fig. 4 (b), we plot the differ-ence U ′ − U in the potential energy per particle before

(U) and after (U ′) shifting disks i and j and minimiz-ing the potential energy as a function of θ. We findthat the θ = 0 direction gives rise to the smallest en-ergy barrier, and thus we define the temperature scaleTmin = U

′(θ = 0) − U . Tmin/U provides an accurateestimate of the large-n plateau value of Tn(m,m− 1)/U .(See Fig. 4 (a).) In the inset of Fig. 4 (a), we show〈Tmin〉/U averaged over 500 MS packings as a functionof system size. 〈Tmin〉/U ∼ 〈Tn(m,m − 1)〉/U ∼ N−β ,where β ≈ 2.95± 0.05, and displays stronger system-sizedependence than 〈T1(m,m− 1)〉/U .

(a)

Nbc

0 10 20 30 40lo

g 10T

-10

-9

-8

-7

-6

-5

log 1

0(f

(T;N

bc))

-7

-6

-5

-4

-3

-2

-1

0

Nbc

0 1 2 3 4 5 6

T$(N

bc)

10-14

10-12

10-10

10-8

10-6

10-4

(b)

N.bcN/U 1

10-10 10-8 10-6 10-4

T$(N

bc)

10-10

10-8

10-6

10-4

FIG. 5: (a) The fraction of time f(T,Nbc) that the system(with N = 64 and U = 10−4) possesses a given number ofbroken contacts Nbc = N

0c +m−Nc at temperature T . The

color scale from yellow to blue represents decreasing f on alog10 scale. The horizontal line indicates the rearrangementtemperature Tr. The solid curve with exes gives the character-istic temperature T ∗(Nbc) < Tr for multiple contact breakingfor which the fraction f = 0.1. (b) The characteristic tem-perature T ∗(Nbc) for three system sizes, N = 32 (solid lines),64 (dashed lines), and 128 (dotted lines), and three values ofU , 10−5 (circles), 10−4 (exes), and 10−3 (pluses), for each N .The inset shows the same data as in the main panel, but T ∗

is plotted as a function of NγbcNδUζ , where γ ≈ 2.2 ± 0.3,

δ ≈ −2.2 ± 0.2, and ζ ≈ 1.0 ± 0.1. The slope of the dashedline is 1.

Thus far, we have focused on the minimum tempera-ture Tcb required to break a single contact in T = 0 MS

8

packings for different forms of the initial perturbations.For these calculations, we used the harmonic approxi-mation to determine the time-dependent disk positionsfollowing the perturbation. We now consider tempera-tures beyond which multiple T = 0 contacts can breakand new contacts can form. As discussed previously inRef. [11, 17], the eigenmodes and associated eigenvectorscan change significantly from those at T = 0 for T > Tcb,where new contacts can form and contacts at T = 0 canbreak. Thus, for multiple contact breaking, we use con-stant energy molecular dynamics simulations to directlymeasure the number of contacts as a function of time fol-lowing equal velocity-amplitude perturbations. For thesestudies, we remove rattler disks prior to starting the sim-ulations and focus on the temperature range T < Tr.

During long trajectories, we measure the fraction oftime f(T,Nbc) at each temperature T that the systempossesses a given number of broken contacts Nbc =N0c + m − Nc. We show f(T,Nbc) for a system withN = 64 and U = 10−4 in Fig. 5 (a). At low tem-peratures T < 10−9, f is large only for Nbc = 0. AsT increases, more configurations possess an increasingnumber of broken contacts. We can define a character-istic temperature T ∗(Nbc) for multiple contact breakingby setting f(T,Nbc) = 0.1.

In Fig. 5 (b), we show the characteristic tempera-ture T ∗(Nbc) for three system sizes N and three valuesof the initial potential energy per particle, U , for eachN . We find that T ∗ obeys the following scaling form:T ∗ ∼ NγbcNδUζ , where the exponents γ ≈ 2.2 ± 0.3,δ ≈ −2.2 ± 0.2, and ζ ≈ 1.0 ± 0.1. (Other thresh-olds 0 < f < 0.1 give similar values for the expo-nents γ, δ, and ζ.) The scaling form suggests thatT ∗/U ∼ N2bc/N2 ∼ (∆m/N)2, where ∆m is differencein the excess number of contacts at T = 0 and finite T ∗.This result shows that the temperature required to breakan extensive number of contacts scales quadratically withthe change in the number of contacts per particle in therange 0 < T < Tr, which is consistent with prior re-sults [10, 11, 16].

B. Measurement of the deviation of the specificheat and average positions

In this section, we investigate the effects of form andcontact-breaking nonlinearities on two physical quanti-ties: 1) the deviation in the specific heat per particle atconstant volume, ∆CV , from the value at T = 0 and 2)the deviation in the average disk positions ∆r from theirpositions at T = 0. We will measure both quantities us-ing constant energy MD simulations with equal velocity-amplitude perturbations involving all eigenmodes (i.e.A1ω

1 = A2ω2 = . . . = Akω

k). In general, nonlineari-ties will cause ∆CV > 0 and ∆r > 0 for temperaturesT > 0. Form nonlinearities can occur for T < Tcb, whileboth form and contact-breaking nonlinearities occur forT > Tcb.

T=Tcb

10-4 10-2 100 102 104

"C

V=C

0 V

10-5

10-4

10-3

10-2

10-1

100(a)

N

10 101.5 102 102.5

"C

V=C

0 V

10-510-410-310-210-1

T=Tcb

10-4 10-2 100 102 104

"C

V=C

0 V10-5

10-4

10-3

10-2

10-1

100(b)

FIG. 6: (a) The normalized deviation in the specific heatper particle at constant volume from the value at T = 0,

∆CV /C0V , at U = 10

−5 for purely repulsive linear springinteractions as a function of temperature T normalized byTcb, where the first contact breaks. The data is obtained fromMD simulations at constant energy following equal velocity-amplitude perturbations applied to 50 T = 0 MS packingswith N = 16 (circles), 32 (exes), 64 (pluses), and 128 (stars).

The inset shows ∆CV /C0V versus system size N for 10 values

of T/Tcb from 1 to 102 (from bottom to top). The dotted line

has slope −1. (b) ∆CV /C0V as a function of T/Tcb for MS

packings with N = 32 disks that interact via purely repulsivelinear (circles) and Hertzian spring interactions (exes) at U =10−5.

We show ∆CV /C0

V (defined in Eq. 9) as a functionof temperature T/Tcb (normalized by the temperatureTcb required to break a single contact) for several systemsizes for purely repulsive linear springs (α = 2 in Eq. 1)in Fig. 6 (a). For purely repulsive linear springs, the devi-

ation ∆CV /C0

V is set by the noise floor for T < Tcb, andthus deviations in the specific heat per particle from formnonlinearities for T < Tcb are below the noise floor. In

Fig. 6 (b), we compare ∆CV /C0

V for purely repulsive lin-ear and Hertzian springs (α = 5/2 in Eq. 1) as a functionof T/Tcb. As expected, the form nonlinearities are largerfor Hertzian interactions. In particular, the deviation in

∆CV /C0

V is above the noise floor for T < Tcb.

9

T10-10 10-8 10-6 10-4 10-2 100

"r

10-10

10-8

10-6

10-4

10-2

100

Tr

Tcb

Tg

(a) (b)

(c) (d)

FIG. 7: (a) The deviation ∆r in the disk positions from their T = 0 values as a function of temperature T for a MS packingwith N = 64 and U = 10−5. The data is obtained from constant energy MD simulations with equal velocity-amplitude initialperturbations involving all eigenmodes. We consider both purely repulsive linear spring interactions (circles; α = 2 in Eq. 1)and double-sided linear spring interactions (exes; α = 2 in Eq. 2). The dashed line has slope 1. The three dotted verticallines indicate 1) the measured temperature Tcb at which the first contact breaks, 2) the temperature Tr at which the systemtransitions to the basin of a new MS packing, and 3) the temperature Tg at which the structural relaxation time (from the selfpart of the intermediate scattering function) appears to diverge. (b) The average disk positions at a temperature T < Tcb (gray-shaded disks). White solid lines indicate contacts between disks in the backbone. The arrows represent the displacement of thedisks relative to their positions at T = 0, where the length of each arrow is proportional to the logarithm of the displacement ofthe disk . (c) Same as in (b) except for the average disk positions at a temperature Tcb < T < Tr. Gray-shaded disks withoutedges are rattlers, circular outlines with dashed edges indicate the initial positions of rattler disks, and white-dotted lines showcontacts that include rattlers. (d) Same as (c) except for the average disk positions at a temperature Tr < T < Tg.

For purely repulsive linear springs, ∆CV /C0

V increasesstrongly above the noise floor for temperatures near Tcb.

∆CV /C0

V for purely repulsive Hertzian springs also in-creases rapidly, but the onset of the rapid increase is notas sharp and occurs for T < Tcb. However, the rate ofincrease of ∆CV /C

0V slows for increasing system sizes. In

the inset to Fig. 6 (a), we plot ∆CV /C0

V for 10 values ofT/Tcb in the range from 1 to 10

2 as a function of systemsize for purely repulsive linear springs. We find that the

deviation scales as ∆CV /C0

V ∼ N−1 for a wide rangeof T/Tcb, which implies that the effect of both form andcontact-breaking nonlinearities on the specific heat van-ishes in the large-system limit in this temperature range.

We also study the change in the average disk positions∆r (defined in Eq. 10) as a function of temperature us-ing constant energy MD simulations with equal velocity-amplitude initial perturbations involving all eigenmodes.We consider both purely repulsive (single-sided) anddouble-sided linear and nonlinear spring interactions. InFig. 7 (a) and 8, we show ∆rds(T ) (double-sided) and∆rss(T ) (single-sided) for disk packings with N = 64,U = 10−5, and linear and Hertzian spring interactions.For double-sided linear and Hertzian spring interactions,with no contact breaking, ∆rds ∼ T over a wide range ofT .

This scaling behavior for ∆rds(T ) stems from formnonlinearities in the total potential energy U , which when

10

T10-10 10-8 10-6 10-4 10-2

"r

10-8

10-6

10-4

10-2

100

Tr

Tcb

Tg

FIG. 8: The deviation ∆r in the disk positions from theirT = 0 values as a function of temperature T for a MS packingwith N = 32, U = 10−5, and single- (circles) and double-sided(exes) Hertzian spring interactions. The three dotted verticallines indicate Tcb, Tr, and Tg (from left to right).

expanded gives:

U = U0 −∑i

F 0i ∆Ri + (14)

1

2!

∑i,j

D0ij∆Ri∆Rj +1

3!

∑i,j,k

G0ijk∆Ri∆Rj∆Rk + . . . ,

where ∆~R = ~R − ~R0, F 0i = −∂V/∂Ri|∆~R=0,D0ij = ∂

2V/(∂Ri∂Rj)|∆~R=0, and G0ijk =

∂3V/(∂Ri∂Rj∂Rk)|∆~R=0. For T < Tcb, when thecontact network does not change, the third-order termin the expansion of U gives rise to the scaling ∆r = CT ,where C is set by G0. (See Appendix D for the calcu-lation of ∆r for a potential with cubic terms in 1D.)Rattler disks are excluded from the measurement of ∆rbecause collisions between backbone and rattler diskswill introduce additional nonlinearities. (∆rss for an MSpacking with rattlers is shown in Appendix B.)

As expected, for T < Tcb, ∆rss = ∆rds ∼ T , be-

fore contact breaking occurs for both linear and Hertzianspring interactions. The disk displacements in thisregime are small and randomly oriented (Fig. 7 (b)). Forpurely repulsive linear spring interactions in the tempera-ture regime T > Tcb, ∆r

ss begins to grow rapidly, reach-ing values that are several orders of magnitude above∆rds. In the temperature regime Tcb < T < Tr, somecollective motion occurs and disks can disconnect fromthe force-bearing backbone and become rattlers (Fig. 7(c)). At T = Tr, ∆r

ss jumps discontinuously when thesystem switches to the basin of a new MS packing. (SeeAppendix C for a discussion of the method that we usedto measure Tr.) In the temperature regime Tr < T < Tg,strong collective motion can occur and all of the disks candisconnect from the force-bearing backbone when rattlerdisks are identified recursively (Fig. 7 (d)). Similar be-havior occurs for the deviations in the average positions

for purely repulsive Hertzian spring interactions (Fig. 8),i.e. ∆rss increases above ∆rds in the temperature regimeTcb < T < Tr, but the increase is more modest than thatfor repulsive linear springs. Comparing the disk positionsat temperatures T > Tg and zero is not meaningful.

In Fig. 9 (a), we plot the displacement ratio ∆rss/∆rds

for single- and double-sided linear spring interactions asa function of T/Tcb below Tr for three values of U (10

−5

(circles), 10−4 (exes), and 10−3 (pluses)) and N = 128.We find that the ratio begins growing for T > Tcb reach-ing an approximate plateau value ≈ 100 that increasesweakly with decreasing U . Thus, contact-breaking non-linearities are much larger than form nonlinearities in thetemperature range Tcb < T < Tr for linear spring inter-actions. In Fig. 9 (b), we compare the ratio ∆rss/∆rds

for linear and Hertzian springs for Tcb < T < Tr. Thecontact breaking nonlinearities have a much stronger ef-fect on ∆r for linear compared to Hertzian spring in-teractions. This result likely stems from the fact thatform nonlinearities are much weaker for linear spring in-teractions compared to Hertzian spring interactions. InFig. 9 (c), we plot 〈∆rss/∆rds〉 averaged over the tem-perature range Tcb < T < Tr for linear spring interac-tions as a function of system size N for each U . We findthat 〈∆rss/∆rds〉 shows no sign of decreasing with sys-tem size. Thus, contact-breaking nonlinearities are dom-inant for MS packings with purely repulsive linear springinteractions in the temperature regime Tcb < T < Tr.

IV. CONCLUSIONS AND FUTUREDIRECTIONS

In this article, we studied the effects of thermal fluctu-ations on MS packings composed of bidisperse, friction-less disks generated at different values of the potentialenergy per particle U or excess number of contacts m/Nin two spatial dimensions. We consider disks that in-teract via single- and double-sided linear and nonlinearspring interactions to disentangle the effects of form andcontact-breaking nonlinearities. To identify the temper-ature range where contact-breaking nonlinearities occur,we first focused on calculating the minimum temperaturerequired to break a single contact in T = 0 MS packingsfor both single- and multi-mode perturbations. Beforecontact breaking and for weak form nonlinearities (e.g.purely repulsive linear springs), the minimum tempera-ture required to break a single contact can be calculatedexactly using the eigenmodes of the dynamical matrixat T = 0. Above the contact breaking temperature orfor interactions that possess strong form nonlinearities,the eigenvalues and eigenmodes change significantly fromthose at T = 0, and thus the T = 0 eigenvalues and eigen-modes cannot be used to calculate the contact breakingtemperature accurately.

For single eigenmode perturbations, we find that theminimum temperature (over all single-mode excitations)required to break the first contact, T1(m,m−1) ∼ U/Nα,

11

T=Tcb

100 101 102 103 104 105

"rs

s="

rds

100

101

102

103(a)

T=Tcb

100 101 102 103 104 105

"rs

s="

rds

100

101

102

103(b)

N101 101.5 102 102.5

h"rs

s="

rdsi

101

102

103(c)

FIG. 9: (a) The ratio ∆rss/∆rds between the deviations in positions for single- and double-sided linear spring interactionsas a function of temperature normalized by contact-breaking temperature T/Tcb for packings with N = 128 and three valuesof U (10−5 (circles), 10−4 (exes), and 10−3 (pluses)). Each curve is averaged over 50 packings in the temperature range1 < T/Tcb < 10

4. The three vertical lines indicate 〈Tr〉 for these packings, U = 10−5 (solid line), 10−4 (dashed line), and10−3 (dotted line). (b) The ratio of position deviations ∆rss/∆rds from single- and double-sided linear (circles) and Hertzian(exes) spring interactions as a function of T/Tcb for MS packings with N = 32 and U = 10

−5. Each curve is averaged over50 packings in the temperature range 1 < T/Tcb < 10

4. The two vertical lines indicate 〈Tr〉 for MS packings with purelyrepulsive linear (solid line) and Hertzian spring interactions (dotted line). (c) 〈∆rss/∆rds〉 averaged over the temperaturerange 1 < T/Tcb < 10

4 for linear spring interactions as a function of system size N for U = 10−5 (circles), 10−4 (exes), and10−3 (pluses).

where α ≈ 2.6. This strong system-size dependence em-phasizes that weak overlaps between disks in MS pack-ings near jamming onset can break at any finite tem-perature in the large-system limit. We also showed thatthe form of the initial perturbation affects the minimumtemperature required to break a single contact. The tem-perature required to break a single contact is minimalfor equal velocity-amplitude perturbations involving alleigenmodes of the T = 0 dynamical matrix and scales asTn(m,m− 1) ∼ U/Nβ , where β ∼ 2.9. Tn(m,m− 1) canbe estimated by identifying the smallest pair of overlap-ping disks i and j at a given U , shifting them so thattheir separation satisfies rij = σij , and then minimizingthe potential energy with i and j held fixed, allowingthe other disks to move. The difference in the poten-tial energy per particle before (U) and after (U ′) mini-mization U ′ − U ∼ Tn(m,m − 1) determines the mini-mum temperature required for breaking a single contactfor equal-velocity amplitude perturbations involving alleigenmodes.

To study multiple contact breaking, we employed con-stant energy MD simulations for initial packings at U(and excess number of contacts m) over a range of tem-peratures T < Tr. We measure the fraction of time dur-ing the simulations at a given temperature T and systemsize N that the system possesses Nbc = N

0c + m − Nc

broken contacts. We identify a characteristic temper-ature T ∗(Nbc) at which a finite fraction f of the time(i.e. f = 0.1) the system possesses a given number ofbroken contacts Nbc. By studying a range of U andN , we obtain the following power-law scaling relation:T ∗ ∼ NγbcNδUζ , where γ ≈ 2.2 ± 0.3, δ = −2.2 ± 0.2,and ζ ≈ 1.0 ± 0.1. The scaling relation involving inte-ger exponents, T ∗/U ∼ (Nbc/N)2, is within error of thenumerical data. These results support prior studies that

find that the temperature required to break an extensivenumber of contacts scales quadratically with the numberof contact changes per particle.

We also investigated the effects of form and contact-breaking nonlinearities on the specific heat (at constantvolume) and the average disk positions as a function oftemperature. We employed both single- and double-sidedlinear and nonlinear spring interactions, which allowed usto compare the strength of the form and contact-breakingnonlinearities. For the specific heat per particle, we findthat the deviation ∆CV from the zero-temperature value,

C0

V , is below the noise threshold for T < Tcb for purely re-pulsive linear spring interactions, and begins to increaserapidly for T > Tcb. For Hertzian interactions, the form

nonlinearities give rise to measurable deviations C0

V /C0V

for T < Tcb, and the strong increase in C0

V /C0V with

increasing temperature occurs over a larger range. How-

ever, we find that ∆CV /C0

V ∼ N−1 decreases with in-creasing system size (for purely repulsive spring interac-tions) in the temperature range Tcb < T < Tr. Thus, weexpect that form and contact-breaking nonlinearities donot have strong effects on the specific heat for T < Tr.

We also characterized the change in the average diskpositions ∆r from their T = 0 values arising from formand contact-breaking nonlinearities as a function of tem-perature. ∆r is more sensitive to form and contact-breaking nonlinearities than ∆CV . We first showed that∆rds ∼ T for double-sided linear and Hertzian spring in-teractions over the full range of temperature 0 < T < Trdue to form nonlinearities. The linear scaling with tem-perature arises from third-order terms in the expansionof the total potential energy in terms of the disk po-sitions. As expected, ∆rss = ∆rds ∼ T for T < Tcbsince there is no contact breaking. Near T = Tcb, ∆r

ss

12

begins increasing rapidly above ∆rds for linear springsdue to contact-breaking nonlinearities. We show thatthe ratio ∆rss/∆rds can increase by a factor of 100 forTcb < T < Tr. In contrast, ∆r

ss/∆rds < 10 for Hertzianinteractions, presumably because the form nonlinearitiesare much stronger. We show that ∆rss/∆rds for lin-ear springs displays very weak system size dependence.This result emphasizes that contact-breaking nonlineari-ties are much stronger than form nonlinearities for linearspring interactions in this low-temperature regime.

Topics of future studies will include rattler disks, sys-tem rearrangements, and nonlinearities induced by non-spherical particle shapes. In most of the current work,we excluded rattler disks by removing them from the MSpacking before adding thermal fluctuations. As shownin Appendix B, additional nonlinearities (e.g. collisionsbetween disks in the T = 0 force-bearing backbone andrattlers at T > 0) are present when rattlers are includedin the system. Second, in the current study, we focusedon the low-temperature regime T < Tr, below which thesystem remains in the basin of the original T = 0 MSpacking. In future studies, we will characterize changesin key physical quantities (such as the shear modulus)as the system moves among a series of related basins forT < Tg, where the system is prevented from undergoingcomplete structural relaxation [26]. The current workwas important in this context, since we characterized themagnitude of changes in the disk positions that arise fromnonlinearities before rearrangements.

At low temperatures T < Tcb and for systems withweak nonlinearities, the eigenvalues and associated eigen-modes from the dynamical matrix at T = 0 agree withthose from S = VC−1, where Vij = 〈vivj〉 is the time-averaged velocity correlation matrix and

Cij = 〈(Ri −R0i )(Rj −R0j )〉 (15)

is the time-averaged position correlation matrix [11, 27,28]. An important future direction is to characterize howthe eigenmodes of S change as a function of increasingtemperature, e.g. do the modes become more or lesslocalized at a given frequency?

Another interesting research direction is to character-ize the nonlinearities that arise at finite temperature inMS packings composed of non-spherical particles suchas ellipsoids, sphero-cylinders, or other elongated parti-cles. Several studies have shown that packings of ellip-soids possess quartic modes near jamming onset [29–31],i.e. directions along which the potential energy increasesas the fourth power of the amplitude in that direction.These results point out that MS packings of non-sphericalparticles possess form, contact-breaking, and shape non-linearities at finite temperature. Determining the relativestrength of these nonlinearities and how they affect thestructural and mechanical properties of MS packings atfinite temperature is an important, open question.

Acknowledgments

The authors acknowledge financial support NSF grantnos. CMMI-1462439 (C.O. and Q.W.), CMMI-1463455(M.S.), and CBET-1605178 (C.O. and Q.W.). This workwas also supported by the High Performance Computingfacilities operated by, and the staff of, the Yale Centerfor Research Computing.

Appendix A: Calculation of minimum temperaturerequired to break a single contact for equal

velocity-amplitude perturbations

In this Appendix, we provide additional details con-cerning the calculation of the minimum temperature re-quired to break a single contact for perturbations in-volving multiple T = 0 eigenmodes with equal velocity-amplitude excitations. (See Sec. III A.) In Eq. 11, wederived the expression for the minimum temperature re-quired to break a single contact (for T < Tcb and systemswith weak nonlinearities) by setting r2ij = σ

2ij and using

Eq. 13 for the time-dependent disk positions. Here, wewill justify why the the maximum of r2ij is obtained when

| sin(ω1t)| = | sin(ω2t)| = . . . = | sin(ωnt)| = 1, where nis the number of eigenmodes in the initial perturbation.The pair separations satisfy r2ij = x

2ij + y

2ij , where

xij = ∆0x +

n∑p=1

∆px sin(ωpt) (A1)

yij = ∆0y +

n∑p=1

∆py sin(ωpt), (A2)

the parameters ∆0x, ∆1x,. . . ,∆

nx , and ∆

0y, ∆

1y,. . . ,∆

ny are

constants determined by the initial perturbation and po-sitions of disks i and j. We define Imij = (x

mij )

2 + (ymij )2,

where xmij = ∆0x +

∑mp=1 ∆

px sin(ω

pt), and ymij = ∆0y +∑m

p=1 ∆py sin(ω

pt). When m = 0, Im = (∆0x)

2 + (∆0y)2

and when m = n, Im = r2ij . Suppose that when m = q,

Iqij = (xqij)

2 + (yqij)2 is maximal. For m = q + 1,

Iq+1ij = (A3)

(xqij + ∆q+1x sin(ω

q+1t))2 + (yqij + ∆q+1y sin(ω

q+1t))2.

The maximum of Iq+1ij is obtained when dIq+1ij /dt =

0, which is satisfied when cos(ωq+1t) = 0 and| sin(ωq+1t)| = 1. When the proof by induction is re-peated, the maximum r2ij is obtained if and only if

| sin(ω1t)| = | sin(ω2t)| = . . . = | sin(ωnt)| = 1. We thenstudy all possible combinations of ±1 for each of the sineterms and and disk pairs i and j and choose those thatgive the smallest perturbation temperature.

13

T10-10 10-8 10-6 10-4 10-2 100

"r

10-10

10-8

10-6

10-4

10-2

100

Tr

Tcb

FIG. 10: ∆r versus temperature T for an initial MS packingwith purely repulsive linear spring interactions, N = 64, andU = 10−5 with the two rattlers kept in the system (triangles)and the two rattlers removed (circles).

Appendix B: Measurement of ∆r in MS packingswith rattlers

In Fig. 7, we showed results for ∆r (the deviation of theaverage positions of the disks from their T = 0 values) us-ing constant energy MD simulations as a function of tem-perature for MS packings with rattlers removed from thesystem before the perturbations were applied. In this Ap-pendix, we show that rattlers can have a strong effect on∆r by introducing new nonlinearities into the system. InFig. 10, we compare ∆r(T ) for an MS disk packing withthe same force-bearing backbone at T = 0 (with purelyrepulsive linear spring interactions) with and without rat-tlers removed. (Note that the rattlers do not directlyreceive thermal excitations.) For sufficiently low temper-atures when the rattlers are not excited by fluctuations inthe force-bearing backbone, ∆r(T ) is the same for bothsystems with and without rattlers. For the MS packingstudied in Fig. 10, the force-bearing backbone comes intocontact with the rattlers at a temperature slightly aboveTcb (defined using the force-bearing backbone at T = 0)and ∆r jumps discontinuously for the system with rat-tlers. (Note that the jump in ∆r can occur over a rangeof temperatures depending on the placement of the rat-tlers.) Above this temperature, the evolution of ∆r isdifferent for the systems with and without rattlers, un-til the system without rattlers switches to the basin ofa new MS packing. Since this article focused on quan-tifying form and contact-breaking nonlinearities in thetemperature regime Tcb < T < Tr, we mainly performedMD simulations of MS packings with rattlers removed.

Appendix C: Measurement of the rearrangementand glass transition temperatures, Tr and Tg

Our constant energy MD simulations mainly focusedon the temperature regime Tcb < T < Tr, where Tcb isthe temperature at which the first contact breaks duringthe simulations and Tr is the temperature below whichthe system remains in the basin of the T = 0 MS pack-ing. To calculate Tr, we first simulate a long trajectoryat a temperature T for a given initial perturbation andtotal time ttot. For each time step of the simulation, weuse the current configuration as the initial condition forfinding the nearest MS packing (at a given U) using thepacking-generation protocol described in Sec. II. We thencalculate the fraction fi of time that the system spends inthe basin of MS packing i. The MS packings are distin-guished using the eigenvalues of the dynamical matrix.

T10-10 10-8 10-6 10-4

f i

0

0.2

0.4

0.6

0.8

1

Tr

FIG. 11: The fraction fi of time that three particular MSpackings occur in the temperature range 10−10 < T < 10−4

for systems with N = 64 and U = 10−5. The dashed verti-cal line indicates the rearrangement temperature Tr for theT = 0 MS packing. At the lowest temperatures, the systemonly populates the T = 0 MS packing (circles). At intermedi-ate temperatures a different MS packing (exes) becomes mostfrequent. At the highest temperatures, the system spends allof the time in a third MS packing (pluses).

In Fig. 11, we plot fi as a function of temperature Tafter perturbing a given T = 0 MS packing with equalvelocity-amplitude excitations involving all eigenmodesat each T . We find that three particular MS packingsoccur most frequently over this range of T and for thisinitial condition. At the lowest T , only the T = 0 MSpacking (circles) occurs. At Tr, the fraction of time thatthe system spends in the T = 0 MS packing tends to zero,and the fraction of time that the system spends in a newMS packing (exes) increases to one. At T ≈ 10−6, thesystem begins spending time in several MS packings, andat T ≈ 10−5, the system spends all of its time in a thirdMS packing (pluses). In most cases, the behavior of fi(T )mimics that shown in Fig. 11 for the first rearrangement,i.e. there is a rapid drop in occupancy of the T = 0 MSpacking and a rapid increase in the occupancy of another

14

MS packing at a well-defined temperature. Thus, Tr canbe measured accurately for each T = 0 MS packing. Wealso find strong agreement when we measure Tr using fiand when we define Tr as the temperature at which thefirst discontinuous jump in ∆r occurs for systems whererattlers have been removed. (See Fig. 7 (a).)

T #10-42 4 6 8 10 12

=

100

101

102

103

104

t10-4 10-2 100 102 104

Fs(~q

;t)

00.20.40.60.8

1

FIG. 12: Structural relaxation time τ (from the decay of theself-part of the intermediate scattering function) as a functionof temperature T for a system with N = 64 and U = 10−5.The dashed line gives τ(T ) = C exp[ATg/(T − Tg)], whereC = 1.1, A = 15, and Tg = 1.3 × 10−4. The inset shows theself-part of the intermediate scattering function at qσS = 2π,Fs(q, t), for several temperatures from T = 10

−4 to 10−3 fromtop to bottom. The horizontal line indicates Fs(q, τ) = e

−1.

To emphasize that our measurements focus on theextremely low-temperature regime, we also calculatedthe structural relaxation time from the self-part of theintermediate scattering function (ISF) versus tempera-ture [32]:

Fs(~q, t) =1

N

N∑j=1

〈exp(−i~q · [~rj(t)− ~rj(0)])〉, (C1)

where ~q is the wave number and 〈·〉 indicates an aver-age over time origins and directions of the wavevector.Near the glass transition temperature, the ISF developsa plateau, whose length increases dramatically with de-creasing T . At the longest timescales and for T > Tg,the ISF decays as a stretched exponential with stretch-ing exponents that depend on q and T [33]. (See the insetto Fig. 12.) We define a structural relaxation time τ asFs(q, τ) = e

−1 for qσS = 2π.For fragile glasses, the structural relaxation time obeys

super-Arrhenius scaling with temperature [34]. As arough estimate of the glass transition temperature Tg,we use the Vogel-Fulcher-Tammann form [35] for τ(T ):

τ ∼ exp[ATg/(T − Tg)], (C2)

where A is a constant and Tg is glass transition tem-perature at which the structural relaxation time appearsto diverge. In Fig. 12, we show that for N = 64 and

U = 10−5, Tg ≈ 10−4, which is several orders of magni-tude larger than Tcb and Tr for this system.

r-0.5 0 0.5

V(r

)

0

0.05

0.1

0.15

(a)

T0.01 0.02 0.03 0.04 0.05

jhrij

0

0.02

0.04

0.06

0.08

(b)

FIG. 13: (a) The potential energy U(r) as a function of posi-tion r for a quadratic form, Uq(r) = Ar2/2 (solid line), and acubic form, V c(r) = Ar2/2 +Br3/6 (dashed line). The verti-cal dotted line indicates r = 0. Note that the cubic potentialis asymmetric about r = 0. (b) The absolute value of theaverage position |〈r〉| versus temperature T for the quadratic(circles) and cubic (exes) potentials.

Appendix D: The temperature dependence of ∆r inmodel 1D systems

To better understand the temperature scaling of theaverage position deviation, ∆r ∼ T , for MS packingsat non-zero temperatures, we studied a model systemconsisting of a particle in a one-dimensional (1D) poten-tial well. We considered two forms for the potential: aquadratic potential, Uq(r) = Ar2/2, and a cubic poten-tial, U c(r) = Ar2/2 +Br3/6 as shown in Fig. 13 (a).

The average position 〈r〉 as a function of temperaturecan be calculated using

〈r〉 =∫∞

0rf(r)dr∫∞

0f(r)dr

, (D1)

15

where the position distribution function in 1D is

f(r) =1√

2(2T − U(r)). (D2)

For the quadratic potential, the average particle posi-tion 〈r〉 = 0 for all T . In contrast, for the cubic potential,|〈r〉| = BT/A2 increases linearly with T with a slope thatscales with the coefficient of the cubic term. A similaranalysis can be applied to MS packings of disks. Beforecontact breaking, the system lies in a high-dimensional

potential energy well. All of the potentials that we stud-ied (i.e. Eqs. 1 and 2 with α = 2 and 5/2) possess “form”nonlinearities with nonzero values for the third deriva-tives of the total potential energy with respect to thedisk positions (Eq. 14). Thus, similar to the model 1Dsystem, ∆r in MS packings before contact breaking isproportional to the temperature T with a slope that isdetermined by the third-derivative of the potential energywith respect to the particle coordinates in the directionof the initial perturbation.

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I IntroductionII MethodsIII ResultsA Temperatures required to break single and multiple contactsB Measurement of the deviation of the specific heat and average positions

IV Conclusions and Future Directions AcknowledgmentsA Calculation of minimum temperature required to break a single contact for equal velocity-amplitude perturbationsB Measurement of r in MS packings with rattlersC Measurement of the rearrangement and glass transition temperatures, Tr and TgD The temperature dependence of r in model 1D systems References