Marginal stability in infinite dimensional Hard Spheres: the Gardner transition and the fullRSB solution Pierfrancesco Urbani Together with: ! Patrick Charbonneau, Jorge Kurchan, Giorgio Parisi and Francesco Zamponi Nature Communications, 5, 3725 (2014) Cargese, 28/08/2014 1
• The simplest RFOT scheme ! • Microscopic description of marginal stability: statistics of amorphous sphere packings. ! • Infinite dimensional Hard Spheres. First steps. ! • The 1RSB solution, phase diagram and inconsistencies. ! • Stability of the 1RSB solution. The Gardner transition in the replicated theory. ! • The 2RSB solution: a new approximate landscape. ! • The fullRSB solution: analytical prediction for the critical exponents at jamming. ! • The fullRSB marginal stability and the microscopic marginal stability. 2
Idea Think about jammed packings of hard spheres as the infinite pressure limit of metastable glassy states. Apply equilibrium statistical mechanics Mézard-Parisi, tools developed to study glassy systems Parisi-Zamponi 1RSB Phase diagram Mean field phase diagram. Up to now the theory ha developed on the basis o hypothesis for the structur metastables states 3
P ( f ) The distribution of forces between spheres in contact displays a power law behavior at small forces P ( f ) ∼ f θ Analogous to the pseudo-gap in the distribution of frozen fields in the Sherrington-Kirkpatrick model. Wyart, Nagel, Muller The shows a power law divergence at the contact value of g ( r ) Given this and the fact that the packin g ( r ) ∼ ( r − 1) − γ marginally stable , a wonderful scaling theo of mean field critical exponents at jammi can be derived (Wyart) 4
We can hope to do practical calculations in order to give first theoretical estimates for measured physical quantities (compute the exponents!) Starting point of a systematic 1/d expansion The properties of hard spheres at jamming seem to depend slightly on the d (It is not the same for the glass transition where the debate has not been solved yet) Charbonneau, Corwin, Parisi, In the infinite dimensional limit the virial expansion takes a simple f d x ρ ( x ) (1 − log ρ ( x )) + 1 Z Z S [ ρ ( x )] = d x d y ρ ( x ) ρ ( y ) f ( x, y ) f ( x ) = − 2 Mayer functi 5
Z Z − 2 x = { x 1 , . . . , x m } Monasson, Fra The entropy is a function of the molecule density. x a = X + u a Translational and rotational invariance imply ρ ( x ) = ρ ( q ab ) q Write the saddle point equations to obtain the molecule density and them to obtain the entropy. This has been done by Kurchan, Parisi and Zamponi (2012) A wonderful result is that if we take a Gaussian ansatz for the molecule density we can obtain the correct entropy. Gaussian ansatz Big simplification 6
p − 1RSB solution b ϕ 1RSB b b ϕ ϕ d th E 1RS E th packing fraction Parisi Zamponi (05-10) J-line: jammed states 7
1. The jammed packings are expected to be isostatic. The 1RSB ansatz gives hyperstatic packings 2. Scaling of the Debye-Waller factor with the pressure From 1RSB: ∆ EA ∼ Instead it was observed that ∆ EA ∼ p − κ κ ∼ 3. The distribution of forces do show a power law behavior 8
Entropic term Interaction term 2 m m n ! 4 − 1 n + 1 n a where X X X F [ˆ v ] = lim n 1 ! . . . n m ! exp v aa v ab 2 2 n → 0 a =1 n 1 ,...,n m : P a n a = n a,b =1 Take the 1RSB ansatz α 1RSB = b α 1 ( m δ ab − 1) ab Plug it into the expression for the entropy. Find 1RSB saddle point equatio Obtain the solution. Study its stability. Obtain a stability matrix (Simple replica structure) M ab ; cd Study its eigenvalues. In particular: replicon eigenvalue Expand at the third order to obtain the cubic terms and the value of the exponent parameter (MCT dynamical exponents) λ = w 2 (Original motivation for the work) w 1 9
Gardner transi Gardner line 1RSB stable p − 1 1RSB stable 1RSB uns b b 1RSB unstable E ϕ d ϕ The instability affects the Jamming part of the phase di We hope to cure 1RSB inconsistencies going beyond the 1RSB s Kurchan Parisi Urba 10
• Gardner (85) showed that the equilibrium states undergo a fullRSB transition at very low temperature (below the Kauzmann point) ! • Montanari and Ricci-Tersenghi (03) however showed that the Gardner phase affects strongly off equilibrium states. It may be relevant for off- Implications equilibrium dynamics! ! • If in the Gardner phase • Renewal interest from Krzakala and fullRSB solution we ha Zdeborova (13) who were able to modes ! follow adiabatically glassy states • The fullRSB solution is showing their instability. ! stable. Could the fullRSB • Rizzo(13): fullRSB solution. stability reproduce the microscopic marginal s jammed packings? 11
ϕ ∗ b b b ϕ d ϕ ϕ 2RSB ϕ 1RSB b b th th E E 1RS λ = w 2 th > m w 1 Rizzo (13) in the ca Doing a perturbative 2RSB calculation just below the Ising p-spin mo instability line shows that a 2RSB solution (and a fullRSB one) can be found only above a certain packing fraction 12
Behavior at the 2RSB threshold Behavior in the high density limit 13
0 0 1 b b b b b b b b b b b b b ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 0 ∆ 2 ∆ 2 ∆ 2 ∆ 1 ∆ 1 ∆ 1 0 B b b b b b b B b b b b b b b C ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 2 0 ∆ 2 ∆ 2 ∆ 1 ∆ 1 ∆ 1 0 B B C B B C b b b b b b b b b b b b b ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 2 ∆ 2 0 ∆ 2 ∆ 1 ∆ 1 ∆ 1 0 B C B B C B b b b b b b b b b b b b b ∆ 2 ∆ 2 ∆ 2 ∆ 1 ∆ 1 ∆ 1 B ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 C B 0 0 b b ∆ = ∆ = B B C b b b b b b b b b b b b b B B C ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 2 ∆ 2 ∆ 1 ∆ 1 ∆ 1 ∆ 1 0 ∆ 1 ∆ 1 ∆ 1 0 B B C b b b b b b b B b b b b b b B C ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 2 ∆ 2 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 0 ∆ 1 ∆ 1 0 B C B @ @ b b b b b b b A b b b b b b ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 2 ∆ 2 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 0 ∆ 1 0 b b b b b b b b b b b b b b ∆ ab → ∆ ( x ) ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 2 ∆ 2 ∆ 2 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 ∆ 1 0 kRSB fullRSB The mean square displacement profile depends on m in a complicated way! We are interested in the behavior of the Debye-Waller factor ∆ EA = ∆ (1) ∼ m ? Reminder: m 14
" ∂ 2 f ( x, h ) ∂ f ( x, h ) = − 1 " # h 2 ˙ q ( x ) ∂ x ∂ h 2 f (1 , h ) = log Θ p 2 ∆ (1) f (1 , h ) = log(2 cosh( β h )) Once the expression of the fullRSB entropy is produced, one has to optimize it over the mean square displacement profile. Find the variational equations. We must enforce the Parisi equations. Find the variational kRSB equations Find their continuum limit Add Lagrange multipliers to the action and then take the variations with respect to everything. At the end check that the two methods give the same eqns. How to solve the equations Put them on a grid and discretize everything. Solve numerically the kRSB equations for successive values of k. 15
y = x/m [ m, 1] → [1 , 1 /m ] ∆ ( y ) ⇠ y − κ κ ' 1 . 416 16
functions that appear in them. Asymptotic power law behavior of the ∆ ( y ) ∼ y − κ mean square displacement profile j ( y, h ) = − c b 2 y J ( hy b / √ γ ∞ ) Roughly, the solution of the Parisi equation 8 y c p 0 ( hy c ) for h ⇠ � y − c > Analogous to the distribution of < fields in the SK model b y a p 1 ( hy b ) for | h | ⇠ y − b P ( y, h ) ⇠ > : for h � y − b p 2 ( h ) Connected with the g(r) in the gla b = 1 + c 2 Solve for the exponents. W γ = γ = a θ = c − a κ = 1 + c b b − c a 17
Whithin the fullRSB ansatz l = 1 − β 2 (1 − 2 q EA Z 1 N d yP (1 , h ) m 4 (1 .h ) = q EA = 1 1 − 2 q (1) + X m 2 i �1 N Z 1 i =1 d hP (1 , h )[ m 0 (1 , h )] 2 ≤ T 2 = T 2 N r = 1 X m 4 �1 i N i =1 In the Hard Spheres case Z 1 1 = b d h e h b ϕ P ( y, h ) ˜ f 00 ( y, h ) 2 2 �1 Crucial to obtain the values of the exponents (thus, also the microscopic marginal stability condition) Crucial to obtain that the fullRSB jammed packings are isostatic 18
The value extracted fr The value of k seems not to fullRSB solution seems to depend on the dimension well the data. 19
• A Gardner transition is present in the phase diagram and affects the jamming limit of the glass phase ! • fullRSB effects must be included to describe the statistics of packing • The asymptotic fullRSB equations in the jamming limit give the expo jamming ! • The microscopic marginal stability is directly linked to the fullRSB stability ! • The infinite dimensional calculation seems to predict well the beha finite dimensions. • Confirm the numerical analysis at finite pressure. ! • Try to derive the statistics of avalanches using the results on the SK (compute the power law exponent of the avalanches size distributi • Try to detect the Gardner point (insights from the state following calculation) ! • Try to extend the formalism to soft spheres. ! Thank • Try to understand why mean field theory works so well 20
Recommend
More recommend
