The dynamics of 2d and 3d topological glasses

Pierre Collet, Maher Younan

3d Younan’s PhD (almost finished)

Molecular dynamics

Let me begin with the model of molecular dynamics ofE. Aharonov, Bouchbinder, Hentschel, Ilyin, Makedonska,Procaccia, SchupperGas with large blue and small red point particles in theplane IR2 (on the torus) with a soft repulsive potentialbetween them

V(a; b) = (ffa + ffb

2 dist(a; b))12

vanishing for distance dist(a; b) > 2:25(ffa + ffb)and ffblue = 1:4 and ffred = 1

Something like a Lennard-Jones potential

Molecular dynamics

The particles have all the same mass and one lets themevolve with equations of motion, starting at some meanpotential energy " per particle

Each one, or all of them together, are coupled in somereasonable way to a heat bath (a stochastic process whichfixes the mean energy and allows for unbounded (above)fluctuations of the energy)

Molecular dynamics

A typical state (at some low temperature) looks as follows:

Molecular dynamics

Procaccia et al. take the Voronoi decomposition of theconfiguration, that is, construct a polygon V(x) aroundeach particle x such that each point in V(x) is closer tox than to any other particle

Molecular dynamics

Now comes the trick: they color-code it as follows(my variant)White : hexagons (801)Dark colors :5-gons for red (287)

:7-gons for blue (277)Very dark: other values (198)

So: either hexagons, or red (small) in 5-gons and blue(large) in 7-gons

Molecular dynamics

White : hexagons (801)red : 5-gons (287)blue : 7-gons (277)Very dark: other values (198)

Molecular dynamics

We observe that theblue particles ‘‘like’’ to be in 7-gonsred ones in 5-gons

Inventing a topological model

Question

How much of this model can be captured by workingexclusively with these quasi-species, forgetting theparticles?

. . . considering that the original model is a ‘‘glass’’A colored version of Aste-Sherrington

Inventing a topological model

Triangulations

The dual of the Voronoi tessellation is a (Delaunay)triangulation

To each Voronoi diagram there corresponds a triangulation(of the torus) with the nodes (corners) of the triangles atthe center of the particles

Inventing a topological model

I want to ignore position and only consider the topology(connections) in these triangulations

Inventing a topological model

From the quasi-particle picture, we keep the information:Theblue particles ‘‘like’’ to have 7 neighbors and thered ones 5 (Euler: mean degree about 6)

Therefore, I introduce the energy of a triangulation T :

ET ” Xi=blue

(degi ` 7)2 + Xi=red

(degi ` 5)2

Note that this energy is localHowever, the topological constraints of triangulations areglobalAs we will see, such models have glassy states, and whilethey are not ‘‘realistic,’’ their logical analysis is muchsimpler than that of molecular dynamics models

Inventing a dynamics

Topological dynamics

Leaving the energy aside for the moment, one defines adynamics on the set of triangulations given by a flip (this iscalled Gross-Varsted move, Pachner-move, T1-move, flip)Froth, 2-D gravity,. . .

We fix the number n of particlesThe PHASE SPACE G will be the (finite) set of alltriangulations T (of the sphere) with n nodesNB!! It does NOT depend on temperature

Inventing a dynamics

What is a flip?

In particle motion this would have been

Inventing a dynamics

We fix the (large) number n of particles (nodes in thetriangulation), half blue and half redLet G = Gn be the state space of the system, that is, allpossible (colored) triangulations of the sphere, with n nodes:How big is this state space?

THEOREM 1 : The number of elements in G is asymptotically

˛̨G˛̨ı C2n

„25627

«n

n`3

This is a colored variant of a result by Tutte (1962) who showed C1

`25627

´nn`5=2

) The number of states grows only like Cn (not Cn log n)

Inventing a dynamics

THEOREM 2 : The flipping process is irreducible andaperiodic on G

Irreducible: Every state in G can be reached from any other stateAperiodic: No parity of number of flips between the states which can bereached from a given state

Inventing a dynamics

Proof of irreducibility: This is a colored variant of a resultby Wagner (1936) (Collet&E JSP 2005, Negami)

Any configuration can be transformed by flips to a‘‘Christmas tree’’

Inventing a dynamics

It is an induction reducing the number of links at the top

Proof of aperiodicity: There are 3 moves of the Christmastree which change nothing

Inventing a dynamics

The thermal dynamics

Introduce temperature 1=˛ and consider the followingMarkov process on G (Metropolis) depending on the energy:› Choose a link (uniformly)› If the link cannot be flipped - for topological reasons -

try another linkOtherwise: compute Ebefore and Eafter

› If Eafter < Ebefore do the flip and restart› If Eafter – Ebefore do the flip with probability exp(`˛ ´ ‹E),

where ‹E = Eafter ` Ebefore

Lemma : This process satisfies detailed balance, and hasthe unique invariant density on G: Prob(T ) = Z`1exp(`˛ET )

Glassyness

Approach to equilibrium, simulations

Godrèche, Kostov, Yekutieli (PRL 1992), Magnasco(???)

Start with an arbitrary configuration, and repeat theprocess until one reaches an ‘‘equilibrium’’The energy, as a function of the number of flips exhibitsthe typical slowing down of glassy systems

In a typical state, energy ı number of ‘‘defects’’,i.e., how many dblue 6= 7 resp dred 6= 5

ET ” Xi=blue

(degi ` 7)2 + Xi=red

(degi ` 5)2

Glassyness Energy as function of time, T=0.175

367

1096

3283

3283

Defects

Time0.00

100.00

200.00

380.00

10^8 10^9 10^10

GlassynessEnergy as function of time (collapsed)

367

1096

3283

3283

% Defects

Time, rescaled

1

5

10

Glassyness

Properties of the glassy state

I mean by this that the equilibrium state is disordered,with temporal and ‘‘spatial’’ correlations typical of otherglass models

Ben Arous and »Cern «y have studied such questions forso-called ‘‘trap models’’ (random walks on high dimensionalcubes)

Glassyness

Example of how to measure correlations:Define the distance between two triangulations as thenumber of corresponding nodes whose degrees differThen look at d(t0; t0 + fi) for fixed t0 as a function of fi

d(t0; t0 + „t0), should be independent of t0

Glassyness

0.01 0.1 1theta

0

500

1000

1500

2000

2500

3000

unmod

ified n

odes

t0= 2’550’000’000t0= 5’050’000’000t0=15’050’000’000t0=29’100’000’000

Number of unmodified nodes as function of flips

f(„) = D(t ; t + „t )

Glassyness

Spatial Correlation function (like in quantum gravity)

C(r) =

Pij:dist(i;j)=r(di ` dave)(dj ` dave)P

ij:dist(i;j)=r 1

Then take Fourier transform

Glassyness

0 1 2 3 4frequency

0

0.005

0.01

0.015

0.02

amplitud

e

torus, regularsphere, glassy

Power spectra

Glassyness

Explaining why/how the model is glassy

Glassyness

Degeneracy of energy levels

THEOREM 3 : There is a C0 < 1 such that for any n, theminimum energy configuration has energy < C0

For n = 18k, C0 < 54 by construction (C0 = 0 for the torus)

THEOREM 4 : There is a C1 > 1, independent of n such thatthe number of configurations with energy < E grows atleast like (nC1)E

This holds as long as E < n1`"

Glassyness

For the torus, there are configurations with energy 0

Movement of defects

How does the system relax?

The most natural idea is that defects wander around, andwhen 2 defects of opposite charge meet, they annihilateand the energy decreases by 2

Movement of defects

An isolated defect cannot move without increasing theenergy

+

+ +

−

By Metropolis, this means that one needs to wait onaverage "`2 ” e2=T fl 1 time steps before this happens

Movement of Pairs

On the other hand, a pair of +1 ` 1 defects can move withno energy cost:

+

− +

−

Since it moves with no energy cost, the movement of pairsand that of isolated defects occur on 2 different timescales: a pair moves ’’infinitely’’ faster than an isolateddefect

Movement of Pairs

Where does a pair move? A pair performs a 1d random walkalong a 1-dimensional predefined path:

−

+

Pair-Defect Collisions

QuestionHow can defects move at all?

Answer

The only possibility is that an isolated defect moves when a1d random walker (a pair) collides with it

Pair-Defect Collisions

An example of a collision where the pair disappears andthe ‘‘`1’’ defect jumps one site

−−

−

−

+

In a collision, a pair might or might not disappear and thedefect might or might not move

Pair-Defect Collisions

Theorem:

There are 9 topologically distinct collision types (each withdifferent outcomes). The probability that a collision is of agiven type and the relative probability of each outcome areconstants fixed by the topology (independent of T and n).

This means that the average number of defects moved bya pair is constant. The only thing changing is the averagedistance between two collisions

Movement of defects: Gambler’s ruin

Define ‰ fl 1 as the average distance between defects.‰ increases with time: if the energy density is d, then‰ ‰ O

`d`0:5

´

Assume that a pair P is created near a defect A:

+

A

P

+ +

−

Movement of defects: Gambler’s ruin

Since pairs move along lines, we will say that A is atposition 0 and the pair is at position 1. The next defect A0

along the trajectory of the pair is on average at distance ‰

If the pair P returns to the origin 0 (where A is) beforereaching A0, then there is a chance that P disappears andA returns to where it was before the pair was created. Nodefect will have moved. The probability that this happens is1 ` O

`‰`1

´

Movement of defects: Gambler’s ruin

Conclusion 1

Most pairs do nothing: they are created, wander around alittle bit then disappear without moving any defect

Conclusion 2

The diffusion constant of defects is proportional to e2=T

(the creation rate of pairs, constant over time) and ‰`1

(decreasing with time).

With these methods, one can explain the decay rates withhigh precision and also the probabilities of different energychanges (later)

Ultrametricity

Ultrametricity

One can make the state space G (of all triangulations with n

points) into a super-graph whosenodes are the ı 18:8n triangulations, and connecting by alink any two triangulations (nodes) which differ by exactlyone flip

THEOREM 5 : This super-graph has about 18:8n

nodes and diameter at most O (n2)

(Small world. . . )Remarks: 256=27 ı 9:4. For uncolored graphs the diameterof the graph is known to be 12n ` 60 (see Negami)

Ultrametricity

The distance between any two triangulations is thus O (n2).But to connect two triangulations which have about thesame energy, without passing through a much higherenergy, one needs to make big ‘‘detours’’. These facts areresponsible for the ‘‘glassy’’ behavior of the systemThe local landscape in the ‘‘glassy state’’ is in fact quiteuniversal:

Ultrametricity

-2 0 2 4 6 8 10Energy difference

1e-06

0.0001

0.01

1Pr

obab

ility

n=367n=1096n=3283

Local neighborhood of triangulation at T=0.175

The figure is an average over many realizations in equilibrium

This can be explained by the discussion I gave on how defects move

Ultrametricity

This analysis also sheds some new light on the ‘‘ultrametric’’property of glasses, that is, the difficulty to go from onestate to anotherNote that each triangulation with n nodes has 3n ` 6 linkswhich can in principle all be flipped. Thus the graph G has‰ 18:8n nodes and (almost) every node has 3n-6 links. Ofthese, in the glassy state, only 0.05% of all directions areenergy neutral, and only 10`4% are energy improving (whenn = 3283). These numbers are for T = 0:175

The typical state is NOT a minimum, but a SADDLE. Howeverthe index (number of negative directions) is only a smallpercentage of the dimension (3n-6)

Ultrametricity

This observation leads to an explanation of how somethinglike the ultrametric property comes about in this model, andby the simplicity of the argument in basically every modelwith local energyWhenever I flip, the energy goes up by 4, by the precedingargument. And if I flip again, the energy goes - withprobability going to 1 when n ! 1 - again up by 4 (sincethe link will probably touch other nodes)I can repeat this argument n1`" times and find that theminima are in valleys of depths 4n1`". But with much smallerprobability I find lower maxima when flipping into one of thefew horizontal (i.e., energy preserving) links

3d: The Phase Space

Triangulations of S3

4 variables:› n the number of nodes

› e the number of edges

› f the number of faces

› t the number of tetrahedra

2 relations:› n ` e + f ` t = 2

› 2f = 4t

Which leaves 2 variables, say t and n

The Phase Space

But. . .

Problem

No equivalent of Steinitz’s Theorem in 3d. In 2d, thistheorem states that every triangulation (defined in atopological sense) can be geometrically realized in the plane(on the sphere)

Actually it turns out that the majority of triangulations ofS3 are not ‘‘geometrical’’:

› Pfeiffle and Ziegler, Math. Ann. 330 (2004), 829-837

› Goodman and Pollack, Bull. Amer. Math. Soc. 14 (1986),127-129

The Phase Space

Another problem

What is the size of the phase space (number of possibleconfigurations)?

In 2d, Tutte, the number of triangulations of S2 with n

nodes grows as“

44

33

”n

In 3d, the answer is still unknown; Gromov asked whetherthe number of triangulations of S3 with t tetrahedra growas Ct or Ct log t? (interesting for example in 3d quantumgravity)A bound of the form Ct implies that the entropy, defined as the logarithm of

the total number of configurations, is extensive in t. It is known that the total

number of configurations with n nodes grows at least as Cn5=4

The Dynamics

What about the dynamics?

Pachner moves in 3d:

There are 4 such moves, only 2 of which conserve thenumber of particles:

The 2-3 move and its inverse:

The Dynamics

Since we want the number of particles to be conserved, weonly allow these 2 moves

But. . .

Problem

Contrary to the 2d case, the phase space is not irreducibleunder these 2 moves: Dougherty et al. exhibited in DiscreteComput. Geom. 32 (2004), 309-315 an ‘‘unflippable’’triangulation

The Dynamics

How many connected (large) components are therein the phase space?

This is not known. . .

But. . .

One can show that all Delaunay triangulations, the‘‘physical’’ triangulations, are in one large component andany 2 such triangulations can be connected by at mostO (n5) flips.

3d : dynamics

Dynamics in 3d

We do know how a nice ground states looks: It is aface-centered lattice filled with tetras (tetrakiscube)

Perhaps there are configurations with lower energy. A defect is any locally

different configuration

3d : dynamics

˙

3d : dynamics

˙

3d : dynamics

In this configuration the local sphere around any node has6 edges on 4 tetras and 8 edges on 6 tetras. (This alsomeans that there are 24 tetras) It will be useful tointroduce the notationhe3; e4; : : : ; ek; : : : i for a node with ej edges which are on j

tetras

3d : dynamics

Thus the ground state is, locally, of the form e4 = 6 ande6 = 8 with all other ej = 0

A ‘‘good’’ energy function should be 0 on suchconfigurations, and non-negative on others, and thepositive values define ‘‘defects’’

The hope is that at low temperature, there are only fewdefects, which form a gas (as in the 2d case) Our studiesshow that not all possible choices of an energy function dothe job, but some work very well:

3d : dynamics

For example:

H = Xnodes

Hn = Xnodes

jdeg(n) ` 14j2 + h + ‹e6>e4

h = supk:ek6=0

fk; f4 = f6 = 0; f3 = f5 = f7 = 1; others fk = (k ` 6)2

This is again local and neither too soft nor too hard

3d : dynamics

10000 1e+06 1e+08Attempted flips

10000

Ene

rgy

10000 1e+06 1e+08Attempted flips

10000

2D simulation ,16380 nodes

3d : dynamics

1e+06 1e+09Attempted flips

10000

Ene

rgy

3d simulation, 8000 nodes