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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
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)
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
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
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
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`"
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
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
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