Commonalities between Edwards ensemble and glasses

Transcription

1 Commonalities between Edwards ensemble and glasses Hernan Makse City College of New York jamlab.org Adrian Baule Lin Bo Max Danisch Yuliang Jin Romain Mari Louis Portal Chaoming Song Workshop: Physics of glassy and granular materials July 16-19, Kyoto

2 Random packings of hard spheres Physics Granular matter Random close packing (RCP) Bernal packings (1960) Mathematics Information theory Shannon (1948) Applications Glasses Signals High dimensional spheres 2. High-dimensional packings 1. Edwards ensemble for grains and glass theory 3. Non-spherical shapes Volume fraction Force distribution

3 Theoretical approach I: Statistical mechanics (Edwards ensemble) Edwards and Oakeshott, Physica A (1989) Z(X, T) = Z Constraint optimization problem exp[ W(~x )/X]exp[ H(~x, ~ f)/t ]D~x D ~ f Minimize volume (X=0) with constraint of force balance (T=0) and non-overlaping. OPTIMIZATION STATISTICAL PHYS EDWARDS instance sample packing cost function energy volume optimal configuration ground state RCP at X=0 minimal cost ground state energy minimal volume 3

4 Theoretical approach II: Mean field theory of jammed hard-sphere (remnant of RSB solution from replica theory) Parisi and Zamponi (2010) Jammed states: J-line liquid state splits s! 0 max{ +s} Replica theory: jammed states are the infinite pressure limit of metastable hard sphere glasses Approach jamming from the liquid phase. Predict a range of RCP densities [ th, GCP] [0.64, 0.68] Mean field theory based on RSB solution in the glass phase.

5 (un)commonalities between Edwards ensemble and RT: 3d max{s} 8 densities in 0.64 ± % th =0.64 GCP =0.68 RLP =4/(4 + 2 p 3) edw =6/(6 + 2 p 3) MRJ c fcc =0.74 max{ +s} s! 0 onset RCP = J d =0.58 K =0.62

6 Very difficult in practice: very small range for 3d equal-size spheres 3d ~ φj fluid coexistence crystal Equilibrium * * * * *10-6 φ = φ j PY EOS φ φj φ = φ j Pade EOS Glass fit φ 4d FIG. 1 (From (Skoge et al., 2006))Evolutionofthepressureduringcompressionatrate γ in d =3(left)andd =4(right). The density ϕ is increased at rate γ and the reduced pressure p(ϕ) =βp/ρ is measured during the process. See (Skoge et al., 2006) for details. The quantity ϕ j(ϕ) = p(ϕ)ϕ is plotted as a function of ϕ. If the system jams at density ϕj, p and p(ϕ) d ϕ j ϕ j. Thus the final jamming density is the point where ϕ j(ϕ) intersectsthedot-dashedline ϕ j = ϕ. (Left) The dotted line is the liquid (Percus-Yevick) equation of state. The curves at high γ follow the liquid branch at low density; when they leave it, the pressure increases faster and diverges at ϕ j.thecurvesforlowerγ show first a drop in the pressure, which signals crystallization. (Right) All the curves follow the liquid equation of state (obtained from Eq.(9) of (Bishop and Whitlock, 2005)) and leave it at a density that depends on γ. In this case no crystallization is observed. For γ =10 5 the dot-dashed line is a fit to the high-density part of the pressure (glass branch). The arrow marks the region where the pressure crosses over from the liquid to the glass branch. RLP =4/(4 + 2 p 3) edw =6/(6 + 2 p 3) MRJ GCP =0.68 RCP onset c fcc =0.74 th =0.64

7 1. Full solution: Constraint optimization problem Z Z(X, T) = exp[ W(~x )/X]exp[ H(~x, f)/t ~ ]D~x D f ~ T=0 and X=0 optimization problem 2. Approximation: Decouple forces from geometry. Z Z Z(X, T) = exp[ W(~x )/X]D~x exp[ H( f)/t ~ ]D f ~ 3. Edwards for volume ensemble + Isostaticity Song, Wang, Jin, Makse, Physica A (2010) 4. Cavity method for force ensemble 20 NX 6 H = 4@ X A23 7 f ab ˆn ab 5 a=1 b,(ab)2e Bo, Mari, Song, Makse (2013) 1

8 The Volume function is the Voronoi volume contact network consist of all points closer to the center of the grain than to any other second coordination shell

9 Easily generalizable to other systems equal size spheres polydisperse system ellipsoids, spherocylinders, non-convex particles, rods, sphere/ellipsoids mixtures, etc. any dimension

10 Easily generalizable to other systems equal size spheres polydisperse system ellipsoids, spherocylinders, non-convex particles, rods, sphere/ellipsoids mixtures, etc. any dimension

11 Analytical formula for Voronoi boundary Voronoi particle j r ij i θ ij r ij/ cos θ ij ŝ W i = 1 3 ( 1 2R min j ) r 3ds ij cos θ ij Important: global minimization. Reduce to one-dimension

12 Analytical formula for Voronoi boundary Voronoi particle j r ij i θ ij r ij/ cos θ ij ŝ W i = 1 3 ( 1 2R min j ) r 3ds ij cos θ ij Important: global minimization. Reduce to one-dimension

13 Analytical formula for Voronoi boundary Voronoi particle j r ij i θ ij r ij/ cos θ ij ŝ W i = 1 3 ( 1 2R min j ) r 3ds ij cos θ ij Important: global minimization. Reduce to one-dimension

14 Analytical formula for Voronoi boundary Voronoi particle j r ij i θ ij r ij/ cos θ ij ŝ W i = 1 3 ( 1 2R min j ) r 3ds ij cos θ ij Important: global minimization. Reduce to one-dimension

15 Average free-volume per particle Voronoi boundary w = Z 1 1 (c 3 1)p(c)dc = c/2 Z 1 0 (c 3 1)dP > (c) Geometrical interpretation of cumulative > Probability to find all particles outside excluded volume and surface: V (c) P > (c) = p(c) S (c)

16 Mean-field approximation analogous to decorrelation principle particles belong to bulk or in contact: Particle gas Particle gas V P (V )=(1 V (c) Similar to car parking problem (Renyi, 1960). Probability to find a spot with V (c) in a volume V V /V ) N! e V

17 Calculation of P>(c) Particles are in contact and in the bulk: r V (c) P > (c) =P B (c) P c (c) 2R R θ c = r/ cos θ ŝ Bulk term: S (c) Contact term: P B (c) =e ρv (c) ρ(w) = 1 w mean free volume density P C (c) =e ρ ss (c) z = geometrical coordination number ρ s (z) = 1 S = 3 4π z mean free surface density

18 Average Voronoi volume Self-consistent equation: equal to zero represent the average free-volume of a single particle

19 Prediction: volume fraction vs z free volume volume function w = 2 3 z φ = z z+2 3 RCP z =6 w = 1 p 3 Equation of state agrees well with simulations and experiments Aste, JSTAT 2006 X-ray tomography 300,000 grains = p 3 =.634 Theory

20 Definition of jammed state: geometric coordination z bounded by mechanical coordination Z effectively excludes the ordered states positions geometrical constraints

21 Edwards phase diagram for hard spheres Isostatic plane φ RLP (Z) = Z Z+2 3 Forbidden zone no disordered jammed packings can exist Disordered Packings Decreasing compactivity X φ RLP = =0.536 φ RCP =

22 Jammed packings in high dimensions Rigorous bounds Minkowsky lower bound: Kabatiansky-Levenshtein upper bound: Question: what s the density of RCP in high dimensions? Conjecture: are disordered packings more optimal than ordered ones?

23 Conjecture: P > (c) becomes valid in the high-dimensional limit (I) Theoretical conjecture of g 2 in high d (neglect correlations) Torquato and Stillinger, Exp. Math., d Large d (II) Factorization of P > (c)

24 Comparison with other theories Edwards theory Isostatic packings (z = 2d) with unique volume fraction Jin, Charbonneau, Meyer, Song, Zamponi, PRE (2010) Agree with Minkowski lower bound Glass transition RT Parisi and Zamponi, Rev. Mod. Phys. (2010) Isostatic packings (z = 2d) ranging volume fraction increases with dimensions No unified conclusion at the mean-field level (infinite d). Neither dynamics nor jamming. Does RCP in large d have higher-order correlations missed by theory?: Test of replica th. Edwards solution seems to corresponds to th. Higher entropy state.

25 Generalizing the theory of monodisperse sphere packings Polydisperse spheres Non-spherical objects (dimers, triangles, tetrahedra, spherocylinders, ellipsoids ) Distribution of radius P(r) Distribution of angles P( ) Extra degree of freedom treated as in Onsager 1949

26 Op$mizing random packings in the Kepler&conjecture:"" Random&sphere"packings:" 0.74 space of object shapes Simula:on"results"on" packings"of"ellipsoids:& RCP"! Ellipsoids"pack"denser"than" spheres"! Peak"at"aspect"ra:o"! Spheres"appear"as"a" singular&limit& " 1.4 h the experimental result for the regular candies (cross). Inset: Donev"et"al,"Science"2004" 22

27 Edwards prediction Non- spherical objects: d L = L d z<2d f

28 Edwards prediction Non- spherical objects: d L = L d Sta$s$cal theory of Voronoi volume z<2d f

29 Edwards prediction Non- spherical objects: d L = L d Sta$s$cal theory of Voronoi volume Evalua$ng the probability of degenerate configura$ons: ellipsoids are hypoconstrained z<2d f

30 Voronoi for non-spherical shapes

31 General non-spherical shapes

32 Object shape Decomposition Effective Voronoi interaction Sphere One sphere Single points Dimer Two spheres Pairs of points Trimer Three spheres Triplets Spherocylinder N spheres Lines Ellipsoid Two spheres Pairs of points and anti-points Tetrahedron Four spheres Quartets of points and anti-points Irregular polyhedron: N faces, M vertices N unequal spheres M points and N anti-points 26

33 Spherocylinders. Separa&on)lines:) ) ) ) ) ) Four)different)interac&ons:) Line..Line. Line..Point. Point..Line. Point..Point.. Exact.equa8on.for.each.case...analy8c.expressions...for.VB. 27

34 Calcula&on)of)coordina&on)number:) Degenerate)configura&ons) Mechanical)equilibrium:) 3)force)equa&ons) 2)torque)equa&ons) (torque)along)symmetry)axis) vanishes)) Linearly)independent?) Z c =2d f = 10 Effec&ve)number)of)degrees)of)freedom) can)be)reduced!) 28

35 Degenerate(configura-ons( Mechanical(equilibrium:( 3(force(equa-ons( 2(torque(equa-ons( (torque(along(symmetry(axis( vanishes)( Linearly)independent?) Effec-ve(number(of(degrees(of(freedom( can(be(reduced!( 29

36 Degenerate(configura-ons( Mechanical(equilibrium:( 3(force(equa-ons( 2(torque(equa-ons( (torque(along(symmetry(axis( vanishes)( Linearly)independent?) Maximal)degenerate)configura6on:(Condi-on( of(force(balance(automa-cally(implies(torque( balance!( Z( ) =2h d f ( )i 30

37 Theoretical predictions 10 9 a 0.74 Theory b b FCC z( ) 8 ) Lens-shaped 0.64 particles theory Ellipsoids theory Lens-shaped particles theory RCP 0.62 Dimers theory Dimers theory Lens-shaped particles theory Spherocylinders theory Spherocylinders theory Dimers theory Theory Spherocylinders 2.0theory Oblate ellipsoids Prolate ellipsoids Spherocylinders/ Dimers Wednesday, 0.60 July , ) Simulations spherocylinder M&M candy spherocylinder spherocylinder spherocylinder dimer dimer spherocylinder spherocylinder oblate ellipsoid spherocylinder prolate ellipsoid spherocylinder Abreu et al Donev et al Lu et al Jia et al Williams et al Schreck et al Faure et al Kyrylyuk et al Bargiel et al Donev et al Wouterse et al Donev et al Zhao et al

38 Results'for'packing'frac2on:'dimers' d 0.70 φ(α) Dimers theory Schreck & O Hern, α 32

39 Results'for'packing'frac2on:' spherocylinders' c FCC RCP Spherocylinders theory Zhao et al, 2012 Kyrylyuk et al, 2011 Lu et al, 2010 φ(α) α 33

40 Edwards phase diagram for many α shapes α 12 c III. Aspherical FCC Tetrahedra (Haji-Akbari et al 2009) Tetrahedra (Jaoshvili et al 2010) 10 Aspherical ellipsoids (Donev et al 2004) II. Rotationally symmetric Prolate ellipsoids (Donev et al 2004) z(φ) 8 Oblate ellipsoids (Donev et al 2004) Spherocylinders (Zhao et al 2012) Spherocylinders (Wouterse et al 2009) Spherocylinders (Williams & Philipse 2003) Spherical ordered branch Dimers (Schreck & O Hern 2011) 6 I. Spherical Spherical random branch RCP Dimers theory Spherocylinders theory Analytic continuation, Eq.(3) φ 34 RCP is not singular: analytical continuation of spheres

41 Summary Ode to Edwards! 1. Edwards ensemble to predict RCP for spheres. 2. Edwards ensemble for non-spherical particles. 3. Edwards ensemble for packings in large dimensions to compare with replica theory of hard sphere glasses. 4. Edwards replica trick or cavity method for proper average over quenched disorder for force distribution for any system: spheres, non-spheres, friction and frictionless, any dimension. 5. Extending Maxwell argument: Cavity method at RS level for solution-no solution transition to calculate Zc from frictionless isostatic grains to frictional grains. 6. Edwards CAVEAT: 1-5 done at expense of drastic (yet controlled) approximations.

42 Cavity Method for Force Transmission Z Z Z(X, T) = exp[ W(~x )/X]D~x exp[ H( ~ f)/t ]D ~ f Edwards volume ensemble predicts: (Z) Cavity method predicts Z: and Force Distribution: Z( ) P (f)

43 Force transmission problem: back to Edwards (simplest model) Edwards model = q-model = annealed disorder average Fix Z =4 f 1 Find P (f) f 3 f 2 with constraint f ~f = ( ~ f 1 + ~ f 2 + ~ f 3 )

44 Boltzmann equation for P(f) f 3 f 1 f 2 2f 2 : component quenched disorder P (f) = Z f Boltzmann equation: assuming uncorrelated forces (MF) P (f 1, 1)P (f 2, 2) ( 1, 2) (f 1f 1 2 f 2 )d 1 d 2 df 1 df 2 annealed disorder Edwards: Tiresomely complicated function well modelled by integrating between 0 and 1 Fourier transform: P (f) = f p e f p

45 Annealed versus quenched disorder Experimentally: first find the distribution for a fixed (quenched) packing, then average over the ensemble of packings Average must be carried over a physical observable: free energy, not the partition function. quenched disorder annealed disorder F = kt ln Z F = kt ln Z Replica trick(edwards-anderson) ln Z =lim(z n 1)/n n!0 Cavity Method Granular matter: Performed average over forces then over contact network

46 Building the factor graph of contacts from a packing Interac/on(par/cle)* site(contact*force) Q i (f n i,f t i ) a({f n,f t, ˆn a, ˆt a ) a({f n,f t, ˆn a, ˆt a )= X i2@a ~f a i! X i2@a ~r a i ~ f a i! Y i2@a (fi n ) (µfi n f t i ) Constraint: force balance + torque balance + repulsive + Coulomb 40

47 Compute marginal belief for a fix contact network Q i (f n i,f t i ) Q b!i (f n i,f t i ) Q a!i (f n i,f t i ) b Q i (fi n,fi t ) i a a particle contact force j Q c!j (f n j,f t j ) c Cavity field: Q a!i (f n i,f t i )= 1 Z a!i Z dˆt i Y j2@a c=@j no average over nj dfj n dfj t dˆt j Q c!j (fj n,fj t ) a ({f n,f t, ˆn, ) i a Belief propagation Q i (f n i,f t i )= 1 Z i Qa!i (f n i,f t i )Q b!i (f n i,f t i ), {a, b} 41

48 Force probability over an ensemble of random graphs P (f n ) Degree distribution Joint distribution of contacts positions on one particle z 1 Solved with Population Dynamics Q(Q! )= 1 X Z Y z R(z) ({ˆn j }) dˆn j DQ!j Q(Q!j ) hq i! F! Q!j Z z j=1 P(f n,f t )=hq i (f n,f t )i = 1 applez 2 DQ a!i Q(Q a!i )Q a!i (f n,f t ) Z Cavity equation (A) Prediction: 10 0 P (f) f =0 2D frictionless ( (B) 10 0 z c =4) signature of jamming 3D frictionless ( z c =6) P(f/<f>) = exp(-2f/<f>) P(f/<f>) = exp(-2f/<f>) f/<f> f/<f> 42

49 Cavity Method: General Formalism Algorithm The Population Dynamics Algorithm" G(V, E), Initialize all cavity fields { i!a (f i )}. Draw an integer z with (edge-perspective) degree distribution P(z). Then pick at random z-1fields j!b (f j ) from the population of N fields. Generate a set of relative contact directions with uniform distribution ; Particles do not overlap. n i(1),...,n i(z 1) update the new cavity field i!a (f i ) by using the incoming fields according to cavity equation. Update all cavity fields to generate a new population. Rescale <f>=const. Run until convergence, or until the number of iterations exceeds T max. 43 Lin Bo (CCNY) Cavity Method for Jammed Disordered Packings of Hard Particles

50 Force probability over an ensemble of random graphs P µ (f n,f t ) (C) D (µ= ) 10 0 (D) D (µ= ) P (f n ) 10 0 P (f n ) P (f) f n P (f) f n 10-2 exp(-1.4f n /<f n >) exp(-3.5f t /<f t >) 10-2 exp(-3.8f t /<f t >) exp(-1.6f n /<f n >) f f 44

51 Comparison with simulations Cavity method P. Wang et al. Physica A (2010) 45

52 ! (f) 3 Solution-no solution transition at Zc Q(Q! )= 1 Z Initial fields X Z z R(z) z Q (f n ) ({ˆn j }) 2D frictionless ( z=3.9 < z c min ) zy 1 j=1 dˆn j DQ!j Q(Q!j ) Peaks= no solution f n z<z c (µ) (A) 10 0 < W n > hq! F! Q!j i order parameter: WIDTH 2D frictionless f Q (f n ) Broad= solution 2D frictionless ( z=4.5 > min z c ) 10-4 W n 10-3 z = 3.9 z = 4.0 z = 4.1 data discretization Time Steps f n z>z c (µ) 46

53 Comparison with simulations Cavity method Silbert et al. PRE (2002) z c (µ) 2D z c (µ) µ µ Consistent with interpretation of z c (µ) as a lower bound 47

54 z =4 Z =3 Definition of jammed state: isostatic condition on Z z = geometrical coordination number. Determined by the geometry of the packing. Z z 2d =6 Z =mechanical coordination number. Determined by force/torque balance. 4=d +1 Z 2d =6 µ =0 µ =

55 Generalizing the theory of monodisperse sphere packings Theory of monodisperse spheres Polydisperse (binary) spheres Non-spherical objects (dimers, triangles, tetrahedrons, spherocylinders, ellipses, ellipsoids ) Distribution of radius P(r) Distribution of angles P( ) Extra degree of freedom Onsager 1949

56 Binary packings Result of binary packings RCP (Z = 6) Danisch, Jin, Makse, PRE (2010)

57 The partition function for hard spheres Volume Ensemble + Force Ensemble 1. The Volume Function: W (geometry) 2. Definition of jammed state: force and torque balance Solution under different degrees of approximations

58 Jammed packings in infinite dimensions Signal Sloane Most efficient design of signals (Information theory) Sampling theorem Optimal packing (Sphere packing problem) High-dimensional point Rigorous bounds Minkowsky lower bound: Kabatiansky-Levenshtein upper bound: Question: what s the density of RCP in high dimensions? Conjecture: are disordered packings more optimal than ordered ones?

59 Sphere packings in high dimensions Signal Sloane Most efficient design of signals (Information theory) Sampling theorem High-dimensional point Optimal packing (Sphere packing problem) Rigorous bounds Minkowsky lower bound: Kabatiansky-Levenshtein upper bound: Question: what s the density of RCP in high dimensions?

60 Q (f n ) Determination of a lower bound on Q(Q! )= 1 Z X Z z R(z) z zy 1 ({ˆn j }) dˆn j DQ!j Q(Q!j ) j=1 z min hq i! F! Q!j average coordination number 2D frictionless ( z=3.9 < z c min ) Q (f n ) D frictionless ( z=4.5 > z c min ) W n c (µ) (A) < W n > z = 3.9 z = 4.0 z = 4.1 data discretization f n Time Steps 2D frictionless zc min (µ) (B) zc min (0)- zc min (µ) f n 10 0 =2/3 = µ µ zc min (µ)- zc min ( ) 54