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Next, we propose a generalisation of the by unit vectors uˆi along their principal axis, and for concrete- random perceptron model that mimics the BP and can be ness, we model them by the Gay-Berne potential (31, 46): solved analytically using the replica method. We confirm that h2 this model is in the same universality class of ellipsoids, BP, VN ( x , uˆ )= v(hij ) , v(h)= k θ( h) , [4] { } { } 2 − and other non-spherical particles that display hypostatic jam- i Here, rˆij =(xi xj )/ xi xj is the unit vector connecting Breathing particles model – The BP model (43) was originally − | − | introduced to understand the physics of the Swap Monte the i-th and j-th particles, εσ0 is the length of the principal axis, and χ = (ε2 1)/(ε2 + 1), where ε denotes the aspect Carlo algorithm (44), but here we will focus on its relation − with the jamming of ellipsoids. The model consists of N ratio. Because we are interested in the nearly spherical case, we expand the pair potential in small ∆ = ε 1 as spherical particles with positions xi in d-dimensions and ra- − dius Ri 0, interacting via the potential energy: (0) ∆ ′ (0) 2 2 ≥ v(hij )= v(h ) v (h ) (rˆij uˆi) +(rˆij uˆj ) ij − 2 ij · · VN ( x , R )= UN ( x , R )+ µN ( R ) , [1] 2 { } { } { } { } { } +∆ wij , [6] where, defining θ(x) as the Heaviside theta function, (0) 2 2 where h = rij /σ0 1 and ∆ wij denotes the O(∆ ) term ij − 2 that we do not need to write explicitly. Substituting this in hij UN = k θ( hij ) , hij = xi xj Ri Rj , [2] Eq. (4) and keeping terms up to ∆2, we obtain V U +µ , 2 − | − |− − N N N i 2 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Brito et al. 1/2 where zJ = 2d + c0∆ . This is compatible with previous nu- −0.35±0.1 merical results of ellipsoids, where z zJ ∆ p (48). − ∼ We can also study the response to shear deformation, which mainly excites the zero-modes (30). Applying the argument in Ref. (18) to the zero-modes and using Eq. (8), the shear 4 modulus G behaves as G δzkR δzki p/√∆, in perfectly − ∼ ∼ ∼ agreement with the numerical result (30). z Vibrational spectrum – The marginal stability argument sug- gests that 0 soft vibrational modes can be found in the fre- N ∗ ∗ quency range ω . ω . √kR, with ω 0 due to marginal ∼ stability and kR p/∆, while the remaining 3 modes have finite frequency at∼ jamming. We now refine theN argument to discuss in more details the vibrational density of states D(ω). ∆ It is convenient to define the Hessian matrix of the N ×N Fig. 1. Universal scaling of the contact number – Markers denote the BP model, with = N(d + dex), as the second derivative of 1 2 N numerical result, while the full line denotes the theoretical prediction δz ∼ ∆ / . the interaction potential VN w.r.t. xi and Ri/∆, in such a Data for non-spherical particles are reproduced from Ref. (40), and from the spheric- way that it has a similar scaling of the one of ellipsoids, where 1 2 ity , we defined / , which recovers the correct scaling relation A ∆ = c(A − 1) Ri/∆ is mapped onto the angular degrees of freedom uˆ. between the sphericity and aspect ratio of ellipses for small ∆. We set c = 1/6 to −6 Then, D(ω) near jamming can be separated into the fol- collapse all data. Data for the BP correspond to a pressure p = 10 . lowing three regions. (i) The lowest band corresponds to the 0 = N(dex δz/2) zero modes stabilized by µN . Their typ- N − 2 2 −1 2 2 ical frequency is ω ∂ µN /∂(∆ Ri) kR∆ ∆p. The 0 ∼ ∼ ∼ stiffness is kR ki k(p/∆) k, where k is the stiffness remaining 3 = 0 = Nz/2 modes can be split into ∼ ∼ ≪ N N −N associated to UN . Hence, the energy scale of these modes re- two bands: (ii) an intermediate band corresponding to the mains well separated from that of the 3 other modes, and extra (rotational or radial) degrees of freedom 1 = Nδz/2, N 2 2 −1 2N 2 we can restrict to the 0-dimensional subspace of the soft with typical frequency ω ∂ VN /∂(∆ Ri) ∆ , and N 1 ∼ ∼ modes. In this space, we have 0 = N(dex δz/2) degrees of (iii) the highest band corresponding to the 2 = Nd transla- N − N freedom, and µN provides Ndex constraints, hence the num- tional degree of freedom. For ∆ 1, the additional degrees ≪ ber of degrees of freedom is Nδz/2 less than the number of of freedom do not strongly affect these modes, and one can constraints. When δz 1, a variational argument was de- ≪ apply the standard variational argument of spherical parti- veloped in (17, 47) to describe the low-frequency spectrum. cles (17, 47), which predicts that their typical frequency is It shows that the soft modes are shifted above a character- 2 2 ω2 δz ∆. The resulting D(ω) differs significantly from 2 2 2 −1 2 istic frequency ω kiδz kRδz ∆ pδz , which is ∼ ∼ ∗ ∼ ∼ ∼ that of isostatic packings of spherical particles, which displays reduced by p by the so-called pre-stress terms, resulting a single translational band. 2 ∼ − −1 2 in ω∗(p) = c1∆ pδz c2p, where c1 and c2 are unknown − Numerical results for D(ω) of ellipsoids from (30), of the constants. Assuming that the system is marginally stable, BP from (43), and analytical results for the perceptron model ω∗(p) = 0, results in (43) to be introduced below, are reported in Fig. 2. Details about the simulations of the BP are explained in (43); here we show δz ∆1/2. [8] ∼ data for N = 484 particles, averaged over at least 1000 sam- This explains the universal square root singularity of the con- ples for each state point. As predicted by our theory, D(ω) tact number zJ observed in ellipsoids, BP andDRAFT several other consists of three separated bands with characteristic peak fre- models (9, 29, 43), as illustrated in Fig. 1. Eq.(8) holds when quencies ω0,1,2. Their scaling with ∆, also reported in Fig. 2 1/2 p ∆, because in the argument we assumed to be close to at fixed p, follows the theoretical predictions ω0 ∆ , ≪ 1/2 ∝1/2 jamming (p 0) at fixed ∆. On the contrary, when ∆ p, ω1 ∆ and ω2 ∆ . We also find that ω0 p for ∼ ≪ ∝ ∝ ∝ the contact number should have the same scaling of spherical small p, while ω1,2 do not change much with p, which is again particles: consistent with the theory. Finally, in Fig. 3 we report the fraction fi = i/ of modes in each band for the BP, which δz p1/2 . [9] N N ∼ also follow the theoretical prediction as a function of ∆ and Eqs. (8) and (9) imply that p and ∆ have the same scaling p. dimension and the following scaling holds: Mean field model – The universality class of isostatic jam- ming is well understood: it can be described analytically by δz =∆γ f (p/∆) . [10] particles in d (15) or, equivalently, by the perceptron → ∞ In the ∆ 0 limit, Eq. (10) reduces to Eq. (9), which re- model (24–26): both models reproduce the critical exponents → 1/2 quires γ = 1/2 and f(x) x for x 1. In the p 0 of isostatic jamming in all dimensions d, leading to the con- → ≫ → limit, we should recover Eq. (8), which requires f(x) const jecture that its lower critical dimension is d =2(49). → for x 1. For the BP, Eq. (10) is confirmed by numerical We now introduce a new mean field model which describes ≪ simulations (43). Assuming that f(x) is a regular function the universality class of hypostatic jamming in the BP, ellip- around x 0, one can expand it as f(x)= c0 + c1x + and soids and many other models of non-spherical particles. The ∼ · · · obtains model, which is a generalization of the perceptron, can be −1/2 solved analytically and, as we shall show, the solution repro- z zJ ∆ p, [11] − ∼ duces all the critical exponents of hypostatic jamming. It Brito et al. PNAS | November 6, 2018 | vol.XXX | no.XX | 3 ω1 ω ) 2 ) ) ω ω ω ( ( ( D D D ω0 ω ω ω ω0 ω ω1 ω ω ω2 ∆ ∆ ∆ Fig. 2. Universality of the density of states – (Top) Density of states for ellipses, breathing particles, and the perceptron. (Bottom) Evolution with ∆ of the −4 1/2 1/2 characteristic frequencies at p = 10 . Full lines denote the theoretical predictions, ω0 ∝ ∆ , ω1 ∝ ∆ and ω2 ∝ ∆ , respectively. Data of ellipses are reproduced from Ref. (32). unit variance. The dynamical variables are x and the Rµ, whose variance is controlled by the chemical potential µN . M 2 2 We fix the value of kR so that µ=1 Rµ = M∆ . In the ∆ 0 limit, the system reduces to the standard perceptron → model investigated in Ref. (26), whileP for ∆ > 0 the Rµ play the same role of the particle sizes in the BP model. f0 f0 Because the model can be solved by the same proce- f1 f1 f2 f2 dure of the standard perceptron model, here we just give a brief sketch of our calculation, which will be given in a longer publication. The free energy of the model at tem- p ∆ perature T = 1/β can be calculated by the replica method, DRAFT1 n N M −βVN βf = limn→0 log Z , where Z = d xd Re and Fig. 3. Weights of the density of states – Fraction of modes fi = Ni/N − nN in the three bands of D(ω) given in Fig. 2, plotted as functions of p at fixed ∆ = the overline denotes the averaging over the quenched ran- −1 −4 10 (left) and ∆ at fixed p = 10 (right) for breathing particles (with d = 2 domness ξµ. Here we are interestedR in the athermal limit and dex = 1). The theoretical predictions f0 = (1 − δz/2)/3, f1 = δz/6 and T 0. Using the saddle point method, the free energy can f2 = 2/3 are plotted as full lines, inferred from the measured δz. → a b be expressed as a function of the overlap qab = x x /N, · where xa and xb denote the positions of the tracer particles of the a-th and b-th replicas, respectively. In the n 0 consists of one tracer particle with coordinate x on the sur- → limit, qab is parametrized by a continuous variable x [0, 1], face of the N dimensional hypersphere of radius √N, and M ∈ qab q(x). The function q(x) plays the role of the order obstacles of coordinates ξµ and “size” σ+Rµ. The interaction → parameter and characterizes the hierarchical structure of the potential between the tracer particle and the obstacles is metastable states (50). We first calculate the phase diagram M M assuming a constant q(x) = q, which is the so-called replica kR 2 VN = UN + µN , UN = v(hµ) , µN = Rµ , [12] symmetric (RS) ansatz that describes an energy landscape 2 µ=1 µ=1 with a single minimum. The result for ∆ = 0.1 is shown X X 2 in Fig. 4. The control parameters are the obstacle density where v(h) = h θ( h)/2 and the gap variable hµ is defined − as α = M/N and size σ. If α is small, the tracer particle can easily find islands of configurations x that satisfy all the con- x ξµ hµ = · σ Rµ . [13] straints hµ > 0: the total potential energy UN and the pres- √N − − sure vanish and the system is unjammed. The overlap q < 1 The ξµ are frozen variables, and each of their components measures the typical distance between two zero-energy con- follows independently a normal distribution of zero mean and figurations. Upon increasing α, q increases and eventually 4 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Brito et al. reaches q = 1 at αJ , which is the jamming transition point (Fig. 4). Naturally, due to the additional degrees of freedom when ∆ > 0, we have αJ (∆) > αJ (0) for equal σ. For σ > 0, the RS ansatz is stable for all values of α and it describes the jamming transition. For σ < 0 instead, the jamming line Unjammed Jammed (RSB) is surrounded by a replica symmetry broken (RSB) region (RSB) where the RS ansatz is unstable. The jamming transition Jammed (RS) should thus be described by the RSB ansatz where q(x) is not α constant, corresponding to a rough energy landscape. The qualitative behavior of the phase diagram is independent of ∆, in particular the jamming line for σ < 0 is always sur- rounded by a RSB region. An important observable to characterize jamming is the Unjammed (RS) 1 M gap distribution ρ(h) N µ=1 δ(hµ h) that also gives ≡ 0 h − i the contact number z = dhρ(h). At jamming, z counts −∞P the gaps hµ that are exactly equal to zero. For comparison σ with numerical results, weR introduce the positive gap distribu- ∞ tion g(h) θ(h)ρ(h)/ dhρ(h), and the force distribution Fig. 4. The phase diagram of the perceptron model for ∆ = 0.1. The red line ≡ 0 denotes the jamming point. The blue lines denote the RSB instability. The jamming ∂h 0 ∂h P (f) θ( h)ρ(h) / ρ(h) df, where f = h/p (cor- line in the nonconvex region (σ < 0) is surrounded by the RSB lines. ≡ − ∂f R −∞ ∂f − responding to negative gaps), both normalized to 1. For the standard perceptron modelR with ∆ = 0 and σ < 0, jamming is isostatic with z =1(26), and both g(h) and P (f) exhibit a power law behavior (24–26). In the jammed phase and α & αJ , the system is described by a “regular” full RSB so- 2 −2 ) h ) lution where 1 q(x) yχx for q(x) 1, and g(h) and ( g h − ∼ ∼ ( P (f) are regular and finite functions. The prefactor yχ is g µγ predominantly controlled by the contact number z, and di- ∆ ∆=0 ∆=0.0005 ∆=0.0005 verges at isostaticity when z = 1(26) and the regular so- ∆=0.001 ∆=0.001 ∆=0.002 ∆=0.002 lution breaks down. At αJ , the model is described by the ∆=0.005 ∆=0.005 “jamming” solution where 1 q(x) x−κ, g(h) h−γ and θ − ∼ ∼ P (f) f , with critical exponents κ 1.42, γ = (2 κ)/κ − ∼ ≃ − h ∆ µh and θ = (3κ 4)/(2 κ) (15, 24–26). Near αJ , the regular − − solution should connect to the jamming solution. This match- Fig. 5. Gap distribution g(h) of breathing particles near the jamming point, p = −6 ing argument leads to z 1 p1/2, which is the same scaling 10 . (Left) Symbols denote the numerical result, while the full line denotes the − ∼ −0.413 behavior of spherical particles (6). theoretical prediction, g(h) ∝ h . (Right) Scaling plot of the same data according to Eq. (14). The situation is completely different if ∆ > 0. One can show that the contact number at jamming is zJ 1, meaning ≥ that the regular solution persists even at αJ . Consequently, g(h) and P (f) are finite and regular functions at jamming, density of states D(ω) = 2ωρ(ω2) can be expressed analyt- and the square-root behavior of the contactDRAFT number is re- ically in closed form as a function of z, kR and p. These placed by z zJ = c∆p. At αJ , the regular solution should quantities should be obtained by solving numerically the full − connect to the jamming solution in the limit of ∆ 0. Using RSB equations but for simplicity, because here we are inter- → the form of the scaling solution derived for ∆ 0 in (26) ested only in the scaling properties of D(ω), to obtain Fig. 2 → and z zJ p this matching argument leads to the scaling we used arbitrary functions z, k and p which are compatible − ∼ R behavior of g(h) and P (f) at αJ : with the analytical scaling derived from the full RSB equation. We find that D(ω) displays three separate bands (Fig. 2). As ∆−µγ p (h∆−µ) (h ∆µ) 0 in the standard perceptron (24), marginal stability in the full g(h) −γ ∼ , [14] ∼ h (h 1) RSB phase implies that the lowest band starts from ω = 0 ∼ 2 ∆θν p (f∆−ν ) (f ∆ν ) and for small ω, D(ω) ω . The lowest band terminates at 0 1/2 1/2 ∼ P (f) ∼ , [15] ω0 ∆ p near which D(ω) exhibits a sharp peak. At ∼ f θ (f 1) ∼ ∼ ω1 ∆ a delta peak is found, while the highest band starts ∼ 1/2 with new critical exponents µ = κ/(4κ 4) = 0.851, ν = from ω2 ∆ . The qualitative behavior of D(ω), and the − ∼ µ 1/2, and a universal scaling function p0(x). The scaling scaling of ω0, ω1 and ω2 are the same of all the models dis- − 1/2 −1/2 analysis also leads to zJ 1 ∆ and c ∆ , con- playing hypostatic jamming, such as ellipsoids (31, 32) and − ∼ ∆ ∼ sistently with the marginal stability argument, Eqs. (8), (11). BP (43). This confirms that the generalised perceptron can reproduce analytically all the critical properties of the hypo- The simplicity of the model allows us to derive the analyt- static jamming transition. ical form of the density of states D(ω). As before, we define As a final check of universality, we test the prediction for the Hessian matrix as the second derivatives of the interaction the ∆ dependence of the gap distribution function g(h) at the potential VN , Eq.(12) w.r.t xi and Rµ/∆. Using the Edwards- jamming point, Eq. (14). In Fig. 5, we show numerical results Jones formula for the eigenvalue density ρ(λ) (51, 52), the (obtained as in (43)) for g(h) of the BP model at p = 10−5, a Brito et al. PNAS | November 6, 2018 | vol.XXX | no.XX | 5 value small enough to observe the critical behavior. Here, as 17. Wyart M, Silbert LE, Nagel SR, Witten TA (2005) Effects of compression on the vibrational usual for particle systems, g(h) is normalized by g(h) 1 for modes of marginally jammed solids. Phys. Rev. E 72(5):051306. → larger h. When ∆ = 0, g(h) exhibits a power law divergence, 18. Wyart M (2005) On the rigidity of amorphous solids. arXiv preprint cond-mat/0512155. g(h) h−γ , where γ = 0.413, consistently with previous nu- 19. Maxwell JC (1864) L. on the calculation of the equilibrium and stiffness of frames. The ∼ London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 27(182):294– merical observation (6, 14, 15). For finite ∆, on the contrary, 299. the divergence of g(h) is cutoff (Fig. 5), consistently with the 20. Brito C, Wyart M (2006) On the rigidity of a hard-sphere glass near random close packing. EPL 76(1):149. theoretical prediction of Eq. (14). 21. Wyart M (2012) Marginal stability constrains force and pair distributions at random close packing. Phys. Rev. Lett. 109(12):125502. Conclusions – Using a marginal stability argument, we de- 22. Lerner E, Düring G, Wyart M (2013) Low-energy non-linear excitations in sphere packings. rived the scaling behavior of the contact number z and the Soft Matter 9(34):8252–8263. 23. DeGiuli E, Lerner E, Brito C, Wyart M (2014) Force distribution affects vibrational properties density of states D(ω) of ellipsoids and breathing particles. in hard-sphere glasses. Proceedings of the National Academy of Sciences 111(48):17054– Our theory predicts that the scaling behaviors of the two mod- 17059. els are identical, which we confirmed numerically. Many other 24. Franz S, Parisi G, Urbani P, Zamponi F (2015) Universal spectrum of normal modes in low- temperature glasses. PNAS 112(47):14539–14544. models of non-spherical particles display the same jamming 25. Franz S, Parisi G (2016) The simplest model of jamming. Journal of Physics A: Mathematical criticality (40), which defines a new universality class of hypo- and Theoretical 49(14):145001. static jamming. We introduced an analytically solvable model 26. Franz S, Parisi G, Sevelev M, Urbani P, Zamponi F (2017) Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems. SciPost Phys. which allows us to derive analytically the critical exponents 2(3):019. associated to the new universality class. 27. Man W, et al. (2005) Experiments on random packings of ellipsoids. Phys. Rev. Lett. One of the most surprising output of our theory is the 94(19):198001. 28. Delaney G, Weaire D, Hutzler* S, Murphy S (2005) Random packing of elliptical disks. Philo- universality of the density of states D(ω) (Fig. 2). This sophical Magazine Letters 85(2):89–96. might be relevant for some colloidal experiments where the 29. Donev A, Connelly R, Stillinger FH, Torquato S (2007) Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids. Phys. Rev. E 75(5):051304. constituents are non-spherical (53), in which the vibrational 30. Mailman M, Schreck CF, O’Hern CS, Chakraborty B (2009) Jamming in systems composed modes could be experimentally extracted from the fluctua- of frictionless ellipse-shaped particles. Phys. Rev. Lett. 102(25):255501. tions of positions (54, 55). Another relevant question is how 31. Zeravcic Z, Xu N, Liu A, Nagel S, van Saarloos W (2009) Excitations of ellipsoid packings near jamming. EPL (Europhysics Letters) 87(2):26001. non-spherical particles would flow under shear (30). The di- 32. 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Universality of Jamming of Non-Spherical Particles
