hull & tiling space - peoplepeople.math.gatech.edu/~jeanbel/talkse/knoxville12.pdf · periodic...
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HULL & TILING SPACEof
APERIODIC SOLIDSJean BELLISSARD
Georgia Institute of Technology, AtlantaSchool of Mathematics & School of Physics
e-mail: [email protected]
Sponsoring
CRC 701, Bielefeld
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CollaborationsJ. Pearson, (Atlanta, GA)
J. Savinien, (U. Metz, Metz, France)
A. Julien, (U. Trondheim, Norway)
I. Palmer, (NSA, Washington DC)
R. Parada, (Gatech, Atlanta, GA)
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Some ReferencesC. Janot, Quasicrystals: A Primer, Oxford Univ. Press, New York, (1992).
J. Bellissard, D. Hermmann, M. Zarrouati, in Directions in Mathematical Quasicrystals,CRM Monograph Series, 13, 207-259, M.B. Baake & R.V. Moody Eds., AMS Providence, (2000).
J. Bellissard, C. Radin, S. Shlosman, J. Phys. A: Math. Theor., 43, (2010), 305001.
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Content1. Representing Aperiodic Materials
2. The Tiling Space
3. Thermal Equilibrium
4. Atomic Motion
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I - Representing Aperiodic Solids
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Fundamental Laws1. The Coulomb forces between atomic cores and valence electrons
create chemical bonding and cohesion in solids
2. Electrons are fermions: they resist compression. For free Fermigas (`e−e = average e − e distance, P=pressure)
P ∼ `5e−e
3. In metals, valence electrons are delocalized, approximately free.Atomic cores localize themselves to minimize the Fermi seaenergy (jellium).
4. A pair potential with strong repulsion at short distances, Friedel’soscillations at medium range and exponentially decaying tail,gives a good description of the effective atom-atom interac-tions.
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An example of atom-atom pairpotential in ZnMgSc
icosahedral quasicrystal(de Boissieu et al.,
Nat. Mat., 6, 977-984 (2007))
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A ConjectureJ. Bellissard, D. Hermmann, M. Zarrouati, in Directions in Mathematical Quasicrystals,CRM Monograph Series, 13, 207-259, M.B. Baake & R.V. Moody Eds., AMS Providence, (2000).
J. Bellissard, C. Radin, S. Shlosman, J. Phys. A: Math. Theor., 43, (2010), 305001.
• Representing atoms by a point located at its nucleus position, inthe infinite volume limit, gives a discrete setL ⊂ Rd (d = 1, 2, 3)in space.
• Such a set is uniformly discrete if the minimum distance betweenatom is positive (nonzero).
Conjecture: For an assembly of atoms with an isotropic pair potentialinteraction decaying fast enough at infinity and with a strong enoughrepulsion at short distance, the equilibrium configurations are almostsurely uniformly discrete in the zero temperature limit.
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Delone Sets
• A uniformly discrete set L is Delone is the size of its holes (va-cancies) is bounded from above.
• A Delone sets has finite local complexity (FLC) is for each r > 0it has only a finite number of local patches of radius r modulotranslation.
• A Delone sets is repetitive if for any r > 0, ε > 0 and any localpatch p of radius r, there is R > 0 such that a translated copy ofp occurs in any ball of radius R modulo an error ε.
• A Delone sets is a Meyer if the set of vectors joining any pair ofits points is itself Delone.
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Examples
• Periodic solids or quasicrystals are represented, in the zero tem-perature limit, by repetitive, FLC, Meyer set.
• As soon as the temperature is nonzero, vacancies in a solidhave unbounded size: a Delone set representation can only bean approximation.
•Hence the best that can be expected is to represent a solid by auniformly discrete set. However
Theorem: The groundstate configurations of a system of attractinghard balls are Delone with probability one. (JB, Radin Shlosman, ‘10)
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II - Tiling Space
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Local Patches• A patch of radius r > 0 is a finite set of atoms in a ball of radius
r > 0 centered at one atomic position and translated at theorigin
p = {y − x ; y ∈ L , ‖y − x‖ ≤ r}
• The set of patches of radius r in L is called Q(0)r .
• There is a way to measure the distance between two patches, us-ing the so-called Hausdorffmetric. Let Qr denotes the completionof Q(0)
r .
• A point in Qris a limit of local patches, namely, a configurationof atoms in a ball of radius r with one atom at the origin.
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Local Patches
A local patch
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Compactness
• Theorem: Qr is compact.
• Compactness means
1. Qr is totally bounded: given any ε > 0 there is a finite cover ofQr by balls of radius ε. Hence Qr can be approximate by a finitenumber of points.
2. Qr is closed, meaning that the limit points of any sequence oflocal patch still belongs to Qr.
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Restriction Map
If r′ > r there is a restriction map
πr←r′(q) = p
this map is continuous
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Inverse Limit
• The inverse limit Ξ = lim←(Qr, πr←r′) is the set of increasingcompatible families of patches
1. ξ ∈ Ξ⇒ ξ = (pr)r>02. pr ∈ Qr for all r > 03. πr←r′(p′r) = pr for r′ > r
• Ξ is called the tiling space. Ξ is a compact space.
• A point ξ ∈ Ξ represents a possible configuration of atoms for thesolid
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Known Results
• Result 1: the patch space Qr of an FLC uniformly discrete setis finite.
• Result 2: (J. Kellendonk ‘97) the tiling space of an FLC uniformly dis-crete set is completely discontinuous (each connected subset is apoint). If the set is FLC, repetitive and if no element of the tilingspace is periodic, then the tiling space is a Cantor set.
• Example: the tiling space of a quasicrystal is a Cantor set
•Guess: the patch space Qr of a BMG is invariant by rotation.Modulo rotation it is finite for r small (SRO) (Egami et al ‘84, Miracle ‘04), itcontains continuous paths for r large enough (MRO and LRO)(see e.g. Langer et al. STZ theory).
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The Hull
• Given L uniformly discrete, its Hull is the set of limit points ofthe family {L − a ; a ∈ Rd
} of translate of L.
• The limit is understood in the following sense: a sequence (Ln)n>0 of uniformlydiscrete set converges to L whenever given ε > 0 (small) and R > 0 (large) thereis N integer such that for n > N, each point of the patch of radius R around theorigin in Ln is within a distance at most ε from a point of the corresponding patchfor L.
• Elements of the Hull are atomic configurations.
• The Hull is compact.
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The Hull
The canonical transversal is the set of elements of the Hull with oneatom at the origin.
Theorem: The canonical transversal coincides with the Tiling Space
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III - Thermal Equilibrium
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Gibbs States• For Na-atoms of the species a labelled by (i, a) and located in a
region Λ of volume V = |Λ|with a pair potential interaction Vab,the total potential energy is
UΛ =∑a,b
∑i, j
Vab(|ri,a − r j,b|)
• If µa represents the chemical potential for the species a, thepartition function at temperature T , β = 1/kBT is given by
ZΛ =∑
Na≥0
eβ∑
a Naµa∏a Na!
∏i,a
∫Λ
ddri,a e−βUΛ
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Gibbs States• Average of observables
〈A〉Λ =1
ZΛ
∑Na≥0
eβ∑
a Naµa∏a Na!
∏i,a
∫Λ
ddri,a e−βUΛA(r)
• It defines a probability measure that will be denoted by PΛ, de-pending upon the temperature and chemical potentials.
General results (O. Lanford III, ‘69) show that there are limit proba-bilities for the family {PΛ}Λ as V = |Λ| → ∞.
Any such limits is called a Gibbs state
• Results: the set of translation invariant Gibbs states is notempty
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Gibbs States• Result 1: the set of Gibbs states is compact, convex.
• Result 2: the set of translation invariant Gibbs states is notempty, also compact and convex. Extremal points are ergodicw.r.t. translation.
• Result 3: if P is a limit points of translation invariant Gibbsstates at T → 0 which is ergodic, then the Hull of almost allconfiguration is the same (JB, Hermmann, Zarrouati ‘00).
If the set of uniformly discrete configurations of atoms hasP-probability one, there is a non random r0 > 0 such almost allconfigurations have minimum interatomic distance r0.
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Groundstates
• Points of the Hull are expected to represent the groundstates,namely the pure phases at zero temperature.
• At nonzero but low temperature, various degrees of freedom, likephonons or vacancies, can be described by a statistical mechanicmodel with background lattice provided by configurations ofthe Hull.
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IV - Atomic Motion
A Model and some Speculations
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Atoms Motion in Patches
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Atoms Motion in Patches
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Atoms Motion in Patches
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Atoms Motion in Patches
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Atoms Motion in Patches
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Atoms Motion in Patches
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Atoms Motion in Patches
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Random Jumps
• Atomic motion is due to thermal noise or tunneling.Both mechanisms are random.
• Local motion can be represented by a jumps from local patchesto local patches.
• The jump probability rate P(p→ q) between two patches p and qis proportional to
– The Gibbs factor exp{−β(F(q) − F(p))}, where β = 1/kBT and Fis the free energy of the patch.
– The inverse of the jump time τp→q depending upon the heightof the energy barrier between p and q
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Random JumpsThe free energy of a patch p is given by
F(p) = U(p) −∑a∈A
µa Na + tr{Π Σ(p)
}where
• U is the mechanical energy,
• Na is the number of atoms of species a,
• µa is the corresponding chemical potential,
•Π is the stress tensor
• Σ(p) is the deformation of p
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Random Jumps
• The tunneling time is proportional to
τtunn ∼ exp{
Sp→q
~
}where Sp→q is the tunneling action between the two patches
• The noise transition time is provided by Kramers law given, inthe limit of high viscosity, by
τnoise ∼ exp{
Wp→q
ε2
}where Wp→q is the height of the energy barrier measured fromp and ε is the noise intensity. For thermal noise ε2 = kBT.
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A Markov Process
• The probability Pt(q) at time t of the patch q ∈ Qr is given by
dPt(q)dt
=∑p∈Qr
P(p→ q) Pt(p)
• This equation is not rigorous, unless Qr is finite(namely for tilings with finite local complexity, such as quasicrys-tals)
• Solving this equation gives a calculation of transport coefficients,such as the elasticity tensor or the response of the solid to stress.
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Questions and Problems
• Coarse graining of each Qr allows to replace it by a finite numberof local patches.
How is the Markov process behaving in the continuum limit ?
• Letting the radius r → ∞ is equivalent to look at small scale inthe tiling space Ξ.
How can one control this infinite volume limit ?
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Questions and Problems
• Experimental results at Medium Range, suggest that Qr is fractalfor r ∼ 4-6Å.
• At small stress, the elasticity theory applies and suggests that thelarge scale behavior, namely the small scale in Ξ is universal
Can one approximate this dynamics by a smooth one ?
• Is such a model able to account for discontinuities at large stress ?
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Merci de votre attention !
Thanks for listening !