Normal Modes of Soft-Sphere Packings: from High to Physical Dimensions

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1 Normal Modes of Soft-Sphere Packings: from High to Physical Dimensions Alexis Poncet École Normale Supérieure - University of Oregon Why do amorphous solids exhibit an excess of low-energy excitations compared with crystals? Only now are infinite dimensional mean-field predictions becoming available for glassy systems. Here we present simulations of packings of soft spheres in dimensions ranging from high (d = 7) to physical ones (d = 3). We attempt to connect the spectrum of jammed systems with these new predictions. The jamming phenomenon has been a much studied topic of statistical and soft-matter physics over the past ten years []. What happens to a packing of spheres when it is compressed until its particles have to overlap? In some cases the outcome might be a crystal but we often obtain an amorphous jammed system at a density that barely changes from one sample to another (φ J 0.64 in 3d) [2]. The geometric [3], rheologic [4], vibrational and elastic properties [2] of such a system are very different from what we would expect for a crystal. The behavior of jammed systems has been found to have universal properties usually summarized by a temperature-densitystress phase diagram [, 2, 4, 5]. Theoretical progress has especially been made by looking at hard-sphere packings within mean-field [5, 6], but some mean-field predictions for the force network have also been checked for softsphere packings [7, 8]. One should remember that the results of a mean-field theory are exact in the infinitedimensional limit. In the following we should use finitedimensional systems as a synonym for non-mean-field systems. In this article, we are especially interested in vibrational properties of jammed packings. It is now wellknown that amorphous systems exhibit an anomalous excess of low-frequency excitations compared to crystals [2, 9]. This is extremely interesting because the vibrational properties of solids are related to measurable properties such as the thermal conductivity. For instance, the acoustic modes in a crystal (linear dispersion relation) give a temperature-cube scaling for the thermal conductivity at low temperature [0]. The anomaly in the spectrum of amorphous solids is often referred to as the Boson peak. Several attempts were made to explain this by using a cutting argument [], effective medium theory [2] or random matrix theory [3]. Moreover, recently, analytical and numerical results were obtained for models known to share features with spheres jamming: the perceptron model [4] and the Heisenberg spin glasses model [5]. Such results contribute to an understanding of the jamming phenomenon. So far, most studies have dealt with the excitations of three-dimensional systems, thus probably encountering strong low-dimensional effects. Our goal is to study the alexis.poncet@ens.fr vibrational modes of soft-sphere packings both as a function of density and of dimension. In infinite dimension (mean-field), analytical results are available [4]. Thus a study of systems ranging from high dimension to low dimension gives hope to bridge the gap between those results and the physically interesting dimension three. Moreover, one should also be able to distinguish between features independent of dimension from features that strongly rely on dimension (low-dimensional effects). In addition to carrying out a study of the density of states, we also look at the localization of the normal modes and try to give an insight into the anharmonicity of the landscape. I. PACKINGS OF HARMONIC SPHERES In all this article, we are studying packings of frictionless harmonic spheres, that is to say particles (at positions r i with radii σ i ) in dimension d (ranging from 3 to 7) that interacts with the following potential: U = Θ [σ ij ρ ij ] (σ ij ρ ij ) 2 () 2 i,j where the sum is taken oven the pairs of particles. ρ ij = r i r j is the distance between particles i and j, σ ij = σ i + σ j is the sum of the radii and Θ is the Heaviside function (it allows to sum only on the contacts). We consider N particles (typically N = 892) in a periodic box of unit size. Note than we will focus only on monodisperse packings (σ i = σ) in dimensions 3 to 7. In dimension 2 it is impossible to obtain amorphous monodisperse packings: if we start from random positions for identical particles it is very likely that we will end up with a crystal. Studies of the physically interesting dimension 2 should deal with polydisperse packings. The relevant parameter is the packing fraction φ: the fraction of the space occupied by the particles. It has been found [2] that below a critical point φ = φ J, an energy minimization starting from random initial positions will lead to a zero energy, whereas for φ > φ J there are non-zero overlaps and thus a non-zero energy (Fig. ). This is the jamming transition (see [] for a review on jamming). Note that φ J is not really constant: it takes different values (in a small range) for different packings [2] and the typical value depends on the algorithm that

2 2 ϕ j Figure. Sketch of packings below jamming, at jamming and above jamming. ϕ is the packing fraction. ϕ J is the critical point. generates the packings [7]. However this was found not to affect some properties that are universal [7]. In the followinge focus on packings above jamming and we shall refer to the distance from jamming as φ = φ φ J > 0. Starting from random positions, we minimize the energy until we reach a local minimum: all the particles should be stable within a very low tolerance. This is done with an implementation by Eric Corwin [8] of the FIRE algorithm [6] using general purpose graphical processing units (GPGPU). The simulations are run on the University of Oregon ACISS supercomputer (56 NVDIA M2070) on nodes having up to three GPUs. When subtracting two floating-point numbers close to each other (imagine substracting and ), having a good numerical precision is important, otherwise the result might be wrong or imprecise. To be able to have a good resolution on the contacts double-double arithmetics was used: this method uses two double-precision numbers to get a precision nearly as good as quad-precision (which is not available on GPU). II. HESSIAN MATRIX OF SOFT SPHERES A. Expression of the Hessian matrix The Hessian matrix is defined as the matrix of small oscillations around a local minimum: H αβ ij = 2 U r α i rβ j i and j denote particles (i, j =... N) while α and β denote components of a vector (α, β =... d). The first derivative of () reads: U r α i = k i [ (ρ ik σ ik )(ri α rk α ) ] ρ ik ϕ (2) (3) where i stands for the contacting neighbors of particle i. Taking the second derivative, we obtain: H αβ ij = δ ij k i [ n α ikn β ik + ɛ ik ρ ik ( n α ikn β ik δαβ)] δ ij [ n α ijn β ij + ɛ ij ρ ij ( n α ijn β ij δαβ)] (4) where ɛ ij = σ ij ρ ij is the overlap, and n ij = rj ri ρ ij is the unit vector between particles i and j. δ ij and δ αβ are Kronecker deltas while δ ij indicates a contact. To get information about the vibrational modes of a packing, the diagonalization of the Hessian matrix needs to be done: we compute its eigenvalues λ k and its eigenvectors {u α i } k. Conventionally, the results are reported in term of the angular frequencies ω k = λ k. The density of states (abbreviated as DOS) is defined as D(ω) such that there are D(ω)dω modes between ω and ω + dω. B. Implementation The computation of the Hessian matrix and the diagonalization is done with a Mathematica script taking as input the output of the packing-generating program. The script was run on the ACISS cluster in order to process multiple systems on multiple nodes. There were a few choices to make concerning the diagonalization method. The first issue is precision: in order to allow Mathematica to use parallelism (multiple cores) we had to reduce the precision of our data to double precision. This should not be a problem for the computation of the density of states as we don t intend to use the results of the diagonalization for further computations needing enhanced precision. The second choice to make is the algorithm to use. The Arnoldi algorithm implemented in Mathematica seemed a good choice to compute the lowest eigenvalues of a sparse matrix such as the Hessian. Unfortunately, it turned out that it takes too much memory (and too much time) for big matrices. An attempt was made to use the LOBPCG algorithm [7] using the Hypre C library [8], and even if it seemed to work, there was little guarantee on the actual precision and stability of the algorithm when computing a lot of eigenvalues. Eventually, the entire spectrum was computed with the default Mathematica dense-matrix method: this has the advantage that there is no need for a threshold on the eigenvalues we want to consider. It should be noted that if a substantial number of eigenvalues (more that a few percent) needs to be computed, dense-matrix methods become competitive with sparse-matrix methods. The limitation on the size of the systems that were considered does not come from the generating program but from the time necessary to the diagonalization. With a reasonable time (less than 2 hours), we were never able to process packings of more than 6384 particles in

3 3 dimension 3 or 4. Most of the results below are given for 892 particles in dimension 3 to 7. While being efficient at finding only a few eigenvalues (eg the ground state of a system), sparse-matrix methods such as the Arnoldi algorithm struggles to commpute a lot of eigenvalues, probably because it needs to remember all the changes of bases. Information was found in the Arpack++ user s guide [9] for the Arnoldi algorithm: computing m eigenvalues of a matrix of size n has a memory usage (number of numbers in memory) of 2nm + 5n 2 + 5n + 2m. If m becomes proportional to n, the memory requirement scales as O(n 2 ). A numerical application for n = (0 4 particles in 3d), m = 0 3 and double precision (8 bytes per number) gives a need for 5Gb of RAM. This is still doable on our cluster, but bigger systems might not be possible. It should also be noted that some implementations of classical algorithms (eg in the Hypre library) are able to take advantage of special forms of matrices (2d laplacian, Hessian of a crystal,... ) but this doesn t apply in our case. Figure 2. Left: a rattler with no contact (in 2d). Right: a buckler and its buckling motion. (Figure from [8]) C. Low-contact particles and isostaticity criterion The first information immediately available is the number of modes of the Hessian matrix that have a zeroenergy (or zero-frequency). Obviously, as we are using periodic boundary conditions, there are d zero modes corresponding to the translations. Furthermore, some particles have fewer than the d + constraints (contacts) needed for local stability, we call them rattlers. There is at least one direction in which they can lead to a movement with no cost ie a zero-energy mode. In practice, as the energy is minimized (within a tiny tolerance), a rattler almost always has no contacting neighbor: it is not a part of the force network (see figure 2 left). The non-rattler particles should be stable, except unlikely cases for which all the contacts are on one side of a plane). We were able to verify that the nontranslational zero-modes are caused by rattlers: when removing the rattlers beforehand we obtained only d zero modes. On figure 3, we compare the number of rattlers to the number of zero-modes: this gives us the expected result: each rattler contributes to d zero-modes. The small difference between the two curves in 3d is due to the fact that, because of numerical tolerance, some rattlers do have one or two contacts: there are a few artificial nonzero modes. Looking at the inset in figure 3, we notice that the number of rattlers goes down (exponentially) as the dimension increases. Rattlers are a feature of lowdimensional systems: the higher the dimension is, the less likely are such particles. In the infinite-dimensional (mean-field) limit, there are no rattlers at all. Another kind of low-contact particles are the so-called bucklers [8]. These are particles with exactly d + contacts with one of them being weak (see figure 2 right). They are able to buckle easily: they are thought to give rise to localized low-energy excitations. Like rattlers, bucklers are a low-dimensional phenomena that is not present in mean-field. In fact, the main point of [8] is that if we remove bucklers from a packing, we find a mean-field scaling for the distribution of the forces, even in dimensions 2 and 3. A key notion to understand such phenomena is the isostaticity condition. A packing of spheres is stable if the number of constraints is greater than the number of degrees of freedom [8, ]. Letting N c be the number of contacts in the packing and taking into account the translations, if the packing is stable we need to have: N c dn d (5) (dn is the number of degrees of freedom of the particles, d is the number of global translations) This is known as the Maxwell criterion for rigidity. If we call z 2N c /N the average number of contacts per particle, we need to have z z c with ( ) N z c 2d 2d (6) N N A packing with z = z c is called isostatic, it has the minimum number of constraints necessary to be stable. At jamming, if the rattlers are removed, a packing is isostatic; and above jamming, the average number of contacts per particles is z = z c + z with z φ [2]. The facts that z = 2d at jamming and that we expect the distribution of number of contacts to become peaked upon increasing the dimension can be used as hand-waving arguments to justify that the rattlers (fewer than d+ contacts) and the bucklers (d+ contacts) disappear when the dimension increases. In high dimension a particle is less likely to be locally under-constrained.

4 4 Excess of zero modes / dimension (lines) e-07 e-06 e Distance to jamming Δφ/φ J Rattlers // Dimension d 4d 5d 6d 7d Number of rattlers (dots) 0. Δ φ/φ J = 9.4e-0 8.0e e e e e e-07 9.e Angular frequency ω Figure 3. Lines: Number of non-translational zero-modes divided by the dimension as a function of the distance to jamming, in dimensions 3 to 7. Dots: Number of rattlers. Inset: number of rattlers at jamming versus dimension III. RESULTS FOR THE DENSITY OF STATES A. Fitting function A typical plot of the density of states D(ω) (DOS) in 4d is given on figure 4 (top). As pointed out before, the DOS is obtained directly from the spectrum of the Hessian by making an histogram of the square root of the eigenvalues. The dots on the figure correspond to logarithmically spaced bins. Very close to jamming, the DOS goes to a constant as ω goes to 0. This is largely reported in the literature [2, 9, ] and is linked with the so-called Boson peak. As we move away from jamming, the curve peels off from this plateau. Above some ω ( φ), we still have the plateau but below ω, the DOS follows a scaling that we will investigate later. It has been found both numerically [9] and with a cutting argument [] that the scaling of ω with φ scales as: ω φ (7) Therefore, we attempt to scale ω with φ by taking ω/ φ/φ J as horizontal axis (Fig. 4 bottom). All the curves seem to collapse onto the same master curve, with an ω 2 behavior at low frequency and a plateau at high frequency. The analytical solution of the perceptron model, that belongs to the same universality class as mean-field packing of spheres, follows the same behavior [4] and the density of states is said to be (in the relevant phase) D(ω) = ω 2 ωmax 2 ω 2 π ω 2 + ω 2 (8) where ω max is the highest frequency and ω is the same kind of threshold frequency. Assuming ω ω max (low ω / (Δ φ/φ J ) Δ φ/φ J = 9.4e-0 8.0e e e e e e-07 9.e-08 Figure 4. Density of states for 892 particles in 4d. Top: without scaling. Bottom: with ω rescaled by φ/φ J. The black line corresponds to equation 9 frequency behavior) and acknowledging equation 7, we fit the curves of figure 4 with the following function (A and B are the fitting parameters): D(ω, φ) A + φ φ J B ω 2 (9) This function looks appropriate (Fig. 4 bottom). Furthermore, it is valid in every dimension from 3 to 7 with the same fitting parameters (A 0.4, B ), provided that the system remains rather close to jamming (Fig. 5 top). B. Discussion on the low frequency scaling The D(ω) ω 2 scaling at low-frequency is a meanfield scaling: it has been derived analytically for the per-

5 5 ceptron model [4] and also corresponds to the scaling expected in the Heisenberg spin glasses model in the absence of external field [5]. Moreover this ω 2 scaling (for ω ω ) was found within effective medium theory in 3d [2]. It is remarkable that the same scaling holds for soft spheres close to jamming in every dimension from 3 to 7. In a crystal, the low-frequency scaling would be the socalled Debye scaling [0] D(ω) ω d (d is the dimension). In fact, this scaling holds for the acoustic modes (linear dispersion relation) of any elastic medium. Note than in 3d, there is no distinction between the Debye scaling and the mean-field scaling. Going to higher dimension should enable us to distinguish between the two. This also has the consequence of minimizing the contribution of the phonons (ω d is smaller for high d). Even in an amorphous system, if we are rather far from jamming, we expect harmonic spheres to exhibit some elastic properties: phonons should be present. We tried to evaluate the frequency at which we should expect the lowest phonon using the relation ω phonon, min = c t k min (0) where k min = 2π/L (box of size L) and c t = G/ρ (G shear modulus, ρ density) is the speed of the tangential waves [20]. Unfortunately, our packings under periodic boundary conditions are not always stable with respect to shear. They exhibit a shear modulus that depends on the directions and can be negative: we were not able to obtain anything meaningful. Indeed, it has been shown [2] that packings under periodic boundary conditions made with basic compression algorithms can have a negative shear modulus (at least for one shear) while they have a well-defined bulk modulus and are mechanically stable within the fixed box. A shear in periodic boundary conditions is not a linear combination of translations of particles, instead it is a change in the space in which the particles are embedded. The remedy would be to change the energy-minimization algorithm to produce shear-stabilized packings. C. Going away from jamming One should notice that in figure 4 (bottom) the curve with the highest density deviates away from the fitting curve. In fact, when going away from jamming, in all dimensions 3 to 7, we see a deviation from the mean-field curve (Fig. 5, bottom). The scaling at low frequency is no longer ω 2. But, at least in dimensions 5 to 7 and for φ/φ J.5, it is not either a clear Debye scaling. Further investigation will be needed to understand what this scaling is and what it corresponds to. There have been proposals [2, 4], for a Debye scaling (ω d ) at very low frequency: the density of states is thought to scale as: ω / (Δ φ/φ J ) d Δ φ/φ J 3.3e e e e e ω / (Δ φ/φ J ) d Δ φ/φ J 3.4e e e e e+00 Figure 5. Density of states for 892 particles in 3 to 7d. Each curve corresponds to an accumulation of 30 to 50 systems. Top: rather close to jamming. Bottom: far from jamming. The black line corresponds to equation 9 ω d ω ω 0 D(ω) ω 2 /ω 2 ω 0 ω ω () flat ω ω where ω φ is the frequency we introduced before, and ω 0 is a new threshold frequency. In practice, there is a limitation on the size of our packings and the number of packings over which an average can be taken: it is unlikely that the regimes ω ω 0 and ω 0 ω ω of equation can be clearly distinguished. Another guess is that the density of states could, in analogy with equation 9, scale as: D(ω, φ) A + B φ + Debye scaling (2) ω 2 + C φ3 ω 4

6 6 IV. STUDY OF LOCALIZATION 0. d Δ φ / φ J = Fitting function e (Δ φ /φ J ) / ω 2 d Δ φ / φ J = Fitting function e (Δ φ /φ J ) / ω 2 + α (Δ φ /φ J ) 3 / ω 4 with α =.0e-02 A. Definition of the inverse participation ratio Let us now conduct a study of the localization of the eigenmodes, in analogy with similar studies [5]. Is a mode extended for instance a plane wave or localized for instance one particle buckling? The usual indicator of localization is the inverse participation ratio (IPR), Y, that is defined for a mode { u i } as: (i goes from to N) i Y = u i 4 ( i u i 2 ) 2 (3) For a completely localized mode (only one particle moving) Y =, while for an fully extended mode (all the N particles moving) Y N. An insight on the fraction of particles moving in one mode is given by the participation ratio P R = /(NY ). P R = for a fully extended mode, P R = N for only one particle moving. Whereas good results for the density of states can be obtained with only one system, the IPR gives only meaningful results when an average over several systems is taken. It was found empirically that the average gives sharper curves if it is done on the participation ratio instead of the IPR: we took the harmonic mean of the IPR for averaging. Whereas logarithmically spaced bins were used for the DOS, bins of same number of points were used for the IPR. B. General results for the IPR Figure 6. Density of states for 892 particles in 3 to 7d far from jamming. Top: the x-axis is φ/ω 2. Bottom: the x- axis is chosen so as to check equation 2. The fitting curve is f(x) = A/( + Bx) (A 0.4, B ) where A, B and C are fitting parameters. In high dimension (for instance d = 7), the Debye scaling should be negligible. We can try to plot D(ω) as a function of φ/ω 2 + α φ 3 /ω 4 (α = C/B needs to be adjusted by hand) and try to fit it with A/( + Bx). At low φ, the corrections are negligible so equation 2 works as well as equation 9. At high φ, equation 2 looks better than equation 9 (Fig. 6). The fitting curve (bottom) is closer to the low frequency scaling, and the higher the dimension is, the better it is. Note however we do not have conclusive evidence for this scaling to be valid. In both figures 5 and 6, the behavior of the 3d curve at high density is peculiar. We would expect the 3d data to always scale as ω 2 (both mean-field and Debye scalings). But it seems that something unexplained is happening at high density in 3d. Further investigation will be done in the next section. Figure 7 shows the IPR for the modes as a function of the frequencies in 4d at different packing fractions. As pointed out in [22], there are roughly three regions: (see figure 7, eg for φ/φ J =.6) At low frequency, we find quasi-localized modes ie modes that have large components in a small spatial region with a tiny plane-wave background. These modes have a relatively high IPR: up to 0 2 for 892 particles in 4d. We will try to account for them. At intermediate frequency, we have a plateau of extended modes: the IPR is roughly constant, around This should be compared to /N = if all the particles were moving. Extended modes are the most usual modes that one expects in the vibrations of a solid. At very high frequency, we have the so-called Anderson localized modes (high-frequency localized modes reported in many models of disordered media). Here, we are not interested in them. The fact that softer modes are more localized has also been reported for the Heisenberg model [5].

7 7 At high density, the low-frequency modes are rather localized (Fig. 7). But when going closer to jamming, these modes become more and more delocalized, until we see almost no sharp increase at low frequency. A guess is that the localized modes are somehow linked with lowcoordination particles: the particles that are rattlers or bucklers. These particles have little importance close to jamming (rattlers are not part of the force network), but if we compress the system the same particles, that are no longer rattlers nor bucklers, are still low coordination particles and may not behave like the others. The modes associated with those particles might be the localized low-frequency modes far from jamming. This still remains to be checked. Inverse participation ratio Y 0. Δφ/φ J.5e e-02.5e-0 5.e-0.6e+00 φ Angular frequency ω C. Variation of the IPR with the dimension On figure 9 is plotted the IPR as a function of the frequency in dimensions 4 to 7 both close to jamming and far from jamming. The behavior is roughly the same in all dimensions: fairly localized low-energy modes far from jamming and more delocalized modes close to jamming. Moreover, the IPR is going down with the dimension: the higher the dimension is, the more extended the modes are. This can be understood easily if our assertion that localized modes are linked with low-coordination particles is true. As pointed out earlier, the number of lowcoordination particles goes down with the dimension, and in the infinite dimensional limit this should vanish. In mean-field, there should be no reason for localized modes at low frequency. One might wonder why we didn t plot the IPR in 3d. It turns out, as we said earlier, that the data in 3d far from jamming exhibit a strange behavior. Close to jamming, the curves for the IPR (like those for the DOS) follow the general tendancy. But far from jamming, there are oscillations in the IPR (Fig. 8). This means that we have some precise frequencies for which the modes are localized. For the moment, this remains unexplained. Figure 7. Inverse participation ratio for 892 particles in 4d, as a function of the frequency. Each curve corresponds to an acummulation of 50 systems. In figure 0 is plotted the IPR versus the number of particles in 4d, rather close to jamming and far from jamming. We choose both low energy modes (modes, 2, 0 and 20) and typical low IPR modes (modes number 4N/4 and 4N/2). The IPR of the latter is roughly constant from one mode to another (Fig. 7). For both plots, the IPR of the typical low-ipr modes follows a N scaling: this confirms that these modes are extended. Then, the low-frequency modes far from jamming have a roughly constant IPR with respect to N: these modes are localized. Finally, there is the issue of the low-frequency modes at φ/φ J = : their IPR is decreasing with the number of particles. This confirms that they are rather extended modes. However, the scaling of the IPR with the number of particles is unclear: the slope is less steep than N but it is difficult to say if it is N log N. Note that we might not be close enough to jamming to match the prediction at jamming. D. Scaling of the IPR with the number of particles To check whether modes are localized or delocalized, it is interesting to monitor their evolution with the number of particles. As we said before, an extended mode (for instance a plane wave) will have an IPR going as Y N : the number of particles moving in the mode is proportional to the total number of particles. On the contrary, a localized mode will have a constant IPR: the number of particles moving is independent of the total number. There are claims [23] that at jamming the probabiliy distribution P ( u ) that a particle moves at a distance u in a low-energy mode goes as P ( u ) u 5 at large u. From it, a scaling for the IPR can be found: Y N log N (less steep than N ). V. STUDY OF ANHARMONICITY By computing and diagonalizing the Hessian matrix an harmonic approximation is made: the packing of spheres is replaced with a network of springs and only the small oscillations are studied. But if the packing is actually moved along some direction, some contacts between particles might be broken and others might be created: the initial computation of the Hessian matrix is no longer valid. For glassy systems the landscape is typically very rugged: there are a lot of minimas (a lot of wells ) in which the system can be trapped. Then, one might ask the following question: for a given direction (a given mode) { u i }, what is the maximum distance u max (the size of the catchment region of the well) at which

8 8 Inverse participation ratio Y Δ φ/φ J.7e e-02.3e-0 4.6e-0.8e+00 Inverse participation ratio Y 0. d Δφ/φ J 4.5e e e e d Angular frequency ω Angular frequencyω Figure 8. Inverse participation ratio for 892 particles in 3d, as a function of the frequency. The oscillations at high packing fraction are still unexplained. we can move the packing and still have a reminimization of energy bring it back to the same minimum? It should be noted that u max = ( i u i 2 ) /2 is a norm in a dn-dimensional space (d = dimension, N = number of particles). u i stands for the displacement of particle i. A similar study for spin glasses was done in [5] with the displacement of particles replaced by a forcing field. A distinction is made between forcing along a low-energy eigenvector (that is rather localized) and forcing along a random mode (that is delocalized). It is argued that moving along a random mode should be equivalent to moving along a bulk eigenvector. As for packings of sphere, [22] studied the size of wells along every normal mode at a given density. To start, we chose to study the size of the initial well along random directions, and its evolution with the packing fraction. It should be noted that the reminimization of energy (which is done with the GPU program) is computationally extensive: instead of looking for the exact size of a well, we probe logarithmically-spaced distances: we start at some distance u 0, if we end up in the initial well we stop here, else we probe a distance αu 0 (we took α = /2). This gives us logarithmically-sized bins. The difference between two wells can be seen by looking at their energies: if the two wells are the same one, the energy is unchanged under numerical precision (last digit of double-double precision), otherwise the change in energy is clear. By looking at different packings, and different random directions we are able to sketch an histogram for the sizes of wells at a given density. On figure, we show results in 4d for the catchment region of a well. It seems (for the four first curves) that the length of the catchment region of a well increases with the density. This agrees with our intuition: the further away from jamming (ie the denser we are), the more difficult it is to rearrange a packing by perturbing Inverse participation ratio Y 0. d Δφ/φ J d Angular frequencyω Figure 9. Inverse participation ratio for 892 particles in 4 to 7d. Each curve corresponds to an acummulation of 30 to 50 systems. Top: rather close to jamming. Bottom: far from jamming. it. However, the behavior far from jamming (last curve, φ/φ J =.6) is still unclear. Further studies should look at systems closer to jamming are perhaps average on more configurations. It would also be interesting to see if the behavior is different if we move along a (potentially localized) low-frequency eigenvector instead of a random mode. VI. CONCLUSION The density of states of soft-sphere packings in finite dimension close to jamming appear to follow a meanfield curve (by which we really mean the curve obtained for the perceptron model [4]) with a plateau at high frequency and a scaling D(ω) ω 2 at low frequency. This shows that the mean-field features are impressively resistant. Explaining the behavior far from jamming is harder

9 9 Inverse participation ratio Inverse participation ratio e-02 e-03 e-04 e-0 e-02 e-03 e-04 e-05 e+03 e+04 e+05 Number of particles Δφ =.5e-2 Mode Mode 2 Mode 0 Mode 20 Mode N Mode 2N log(x)/x /x e+03 e+04 e+05 Number of particles Δφ =.5 Mode Mode 2 Mode 0 Mode 20 Mode N Mode 2N constant /x Figure 0. Inverse participation ratio in 4d versus the number of particles. Top: rather close to jamming. Bottom: far from jamming and further theoretical studies might be needed to understand it. A curve steeper than ω 2 is seen at low frequency. The explanation might have to do either with modes due to low-coordination particles (see localization study) or with an increasing importance of phonons. Note however than even when going very far from jamming, a clear Debye scaling was never obtained. In low dimensions and rather far from jamming, we found localized low-frequency modes: those modes have a relatively high inverse participation ratio and involve a number of particles independent of the total number (constant IPR with respect to the number of particles). Our guess is that these modes are caused by lowcoordination particles. This would be consistent with the fact that close to jamming or in high dimension all modes become extended. Eventually, the direct probing of the landscape seems to show that the catchment region of a well becomes smaller as we approach jamming. However further stud- Density of probability 0 0. Δ φ =.5e e-02.5e-0 5.e-0.6e Size of the well Figure. Catchment region of a well along a random direction for 024 particles in 4d. Each curve corresponds to an accummulation of 625 points (25 systems, 25 random directions per system) ies will be needed in order to make a strong claim and have better insight on which information can be revealed by the landscape. We hope that these results will contribute to improve the insight of the jamming transition and that the data set will prove useful for further studies. AKNOWLEDGMENTS First, I would like to thank warmly Eric Corwin for the time he regularly dedicated to supervise my project and to make this work progress. Also warm thanks to Francesco Zamponi and Patrick Charbonneau for enlightening Skype meetings and discussions that kept this project moving forward, and for reading this report carefully. Thanks to Giorgio Parisi for theoretical guesses and insightful advice. And, last but not least, thanks to Robert Yelle, the ACISS cluster administrator, for help and advice on how to use the cluster. [] A. J. Liu, S. R. Nagel, W. van Saarloos, and M. Wyart, in Dynamical heterogeneities in glasses, colloids, and granular media (Oxford University Press, 200). [2] C. S. O Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel,

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