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Disordered Hyperuniform Point Patterns inPhysics, Mathematics and Biology
Salvatore Torquato
Department of Chemistry,
Department of Physics,
Princeton Institute for the Science and Technology of Materials,
and Program in Applied & Computational Mathematics
Princeton University
http://cherrypit.princeton.edu
. – p. 1/50
States (Phases) of Matter
Source: www.nasa.gov
. – p. 2/50
States (Phases) of Matter
Source: www.nasa.gov
We now know there are a multitude of distinguishable states of matter, e.g.,
quasicrystals and liquid crystals , which break the continuous translational
and rotational symmetries of a liquid differently from a sol id crystal .. – p. 2/50
What Qualifies as a Distinguishable State of Matter?
Traditional Criteria
Homogeneous phase in thermodynamic equilibrium
Interacting entities are microscopic objects, e.g. atoms, molecules or spins
Often, phases are distinguished by symmetry-breaking and/ or some
qualitative change in some bulk property
. – p. 3/50
What Qualifies as a Distinguishable State of Matter?
Traditional Criteria
Homogeneous phase in thermodynamic equilibrium
Interacting entities are microscopic objects, e.g. atoms, molecules or spins
Often, phases are distinguished by symmetry-breaking and/ or some
qualitative change in some bulk property
Broader Criteria
Reproducible quenched/long-lived metastable or nonequilibrium phases, e.g.,
spin glasses and structural glasses
Interacting entities need not be microscopic, but can inclu de building blocks
across a wide range of length scales, e.g., colloids and metamaterials
Endowed with unique properties
. – p. 3/50
What Qualifies as a Distinguishable State of Matter?
Traditional Criteria
Homogeneous phase in thermodynamic equilibrium
Interacting entities are microscopic objects, e.g. atoms, molecules or spins
Often, phases are distinguished by symmetry-breaking and/ or some
qualitative change in some bulk property
Broader Criteria
Reproducible quenched/long-lived metastable or nonequilibrium phases, e.g.,
spin glasses and structural glasses
Interacting entities need not be microscopic, but can inclu de building blocks
across a wide range of length scales, e.g., colloids and metamaterials
Endowed with unique properties
New states of matter become more compelling if they
Give rise to or require new ideas and/or experimental/theor etical tools
Technologically important. – p. 3/50
Pair Statistics in Direct and Fourier SpacesFor particle systems in R
d at number density ρ , g2(r) is a nonnegative radial function that is
proportional to the probability density of pair distances r.
The nonnegative structure factor S(k) ≡ 1 + ρh(k) is obtained from the Fourier transform of
h(r) = g2(r) − 1, which we denote by h(k).
Poisson Distribution (Ideal Gas)
0 1 2 3r
0
0.5
1
1.5
2
g 2(r)
0 1 2 3k
0
0.5
1
1.5
2
S(k)
Liquid
0 1 2 3 4 5r
0
0.5
1
1.5
2
2.5
3
g 2(r)
0 5 10 15 20 25 30k
0
0.5
1
1.5
2
2.5
3
S(k)
Crystal (Angularly Averaged)
0 1 2 3r
g 2(r)
0 4π/√3 8π/√3 12π/√3k
S(k)
. – p. 4/50
CuriositiesDisordered jammed packings
0 4 8 12 16 20 24kD
0
2
4
6
8
S(k
)
40,000−particle random
φ=0.632
jammed packing
S(k) appears to vanish in the limit k → 0: very unusual behavior for a
disordered system .
Harrison-Zeldovich spectrum for density fluctuations in the early Universe:
S(k) ∼ k for sufficiently small k.
. – p. 5/50
HYPERUNIFORMITY
A hyperuniform many-particle system is one in which normalized density
fluctuations are completely suppressed at very large lengths scales .
. – p. 6/50
HYPERUNIFORMITY
A hyperuniform many-particle system is one in which normalized density
fluctuations are completely suppressed at very large lengths scales .
Disordered hyperuniform many-particle systems can be regarded to be new
distinguishable states of disordered matter in that they
(i) behave more like crystals or quasicrystals in the manner in which they
suppress large-scale density fluctuations , and yet are also like liquids and
glasses because they are statistically isotropic structures with n o Bragg
peaks ;
(ii) can exist as both as equilibrium and quenched nonequilibrium phases ;
(iii) and, appear to be endowed with unique bulk physical properties .
Understanding such states of matter, which have technologi cal importance, require
new theoretical tools.
. – p. 6/50
HYPERUNIFORMITY
A hyperuniform many-particle system is one in which normalized density
fluctuations are completely suppressed at very large lengths scales .
Disordered hyperuniform many-particle systems can be regarded to be new
distinguishable states of disordered matter in that they
(i) behave more like crystals or quasicrystals in the manner in which they
suppress large-scale density fluctuations , and yet are also like liquids and
glasses because they are statistically isotropic structures with n o Bragg
peaks ;
(ii) can exist as both as equilibrium and quenched nonequilibrium phases ;
(iii) and, appear to be endowed with unique bulk physical properties .
Understanding such states of matter, which have technologi cal importance, require
new theoretical tools.
All perfect crystals (periodic systems) and quasicrystals are hyperuniform.
. – p. 6/50
HYPERUNIFORMITY
A hyperuniform many-particle system is one in which normalized density
fluctuations are completely suppressed at very large lengths scales .
Disordered hyperuniform many-particle systems can be regarded to be new
distinguishable states of disordered matter in that they
(i) behave more like crystals or quasicrystals in the manner in which they
suppress large-scale density fluctuations , and yet are also like liquids and
glasses because they are statistically isotropic structures with n o Bragg
peaks ;
(ii) can exist as both as equilibrium and quenched nonequilibrium phases ;
(iii) and, appear to be endowed with unique bulk physical properties .
Understanding such states of matter, which have technologi cal importance, require
new theoretical tools.
All perfect crystals (periodic systems) and quasicrystals are hyperuniform.
Thus, hyperuniformity provides a unified means of categorizing and
characterizing crystals, quasicrystals and such special disordered systems.. – p. 6/50
Local Density Fluctuations for General Point Patterns
Torquato and Stillinger, PRE (2003)
Points can represent molecules of a material, stars in a gala xy, or trees in a
forest. Let Ω represent a spherical window of radius R in d-dimensional
Euclidean space Rd.
ΩR ΩR
Denote by σ2(R) ≡ 〈N2(R)〉 − 〈N(R)〉2 the number variance .
. – p. 7/50
Local Density Fluctuations for General Point Patterns
Torquato and Stillinger, PRE (2003)
Points can represent molecules of a material, stars in a gala xy, or trees in a
forest. Let Ω represent a spherical window of radius R in d-dimensional
Euclidean space Rd.
ΩR ΩR
Denote by σ2(R) ≡ 〈N2(R)〉 − 〈N(R)〉2 the number variance .
For a Poisson point pattern and many disordered point patterns, σ2(R) ∼ Rd.
We call point patterns whose variance grows more slowly than Rd (window
volume) hyperuniform . This implies that structure factor S(k) → 0 for k → 0.
All perfect crystals and quasicrystals are hyperuniform such that
σ2(R) ∼ Rd−1: number variance grows like window surface area .
. – p. 7/50
Hyperuniformity and Crystals
-25 -20 -15 -10 -5 0 5 10 15 20 25-20
-15
-10
-5
0
5
10
15
20
Diffraction Pattern for a Square LatticeDots denote Bragg peaks at δ(k-Q) (Lattice spacing = 2π/a )
0 5 10 15 20 25
Wavenumber, k
0.00
0.05
0.10
S(k)
Angular-averaged S(k) for a Square Lattice
2π/a
All perfect crystals are trivially hyperuniform .
But the degree to which they suppress large-scale density
fluctuations varies. Which crystal in Rd minimizes such
fluctuations?
. – p. 8/50
Hidden Order on Large Length Scales
Which is the hyperuniform pattern?
. – p. 9/50
Outline
Hyperuniform Systems
“Inverted” Critical Phenomena
Minimizing Variance: Special Ground State Problem
Variance as an Order Metric at Large Length Scales
Examples of Disordered Hyperuniform Systems:
Natural and Synthetic
. – p. 10/50
ENSEMBLE-AVERAGE FORMULATIONFor an ensemble in d-dimensional Euclidean space R
d:
σ2(R) = 〈N(R)〉h
1 + ρ
Z
Rd
h(r)α(r; R)dri
α(r; R)- scaled intersection volume of 2 windows of radius R separated by r
R
r
0 0.2 0.4 0.6 0.8 1r/(2R)
0
0.2
0.4
0.6
0.8
1
α(r;
R) d=1
d=5
Spherical window of radius R
. – p. 11/50
ENSEMBLE-AVERAGE FORMULATIONFor an ensemble in d-dimensional Euclidean space R
d:
σ2(R) = 〈N(R)〉h
1 + ρ
Z
Rd
h(r)α(r; R)dri
α(r; R)- scaled intersection volume of 2 windows of radius R separated by r
R
r
0 0.2 0.4 0.6 0.8 1r/(2R)
0
0.2
0.4
0.6
0.8
1
α(r;
R) d=1
d=5
Spherical window of radius R
For large R, we can show
σ2(R) = 2dφh
A
„
R
D
«d
+ B
„
R
D
«d−1
+ o
„
R
D
«d−1i
,
where A and B are the “ volume ” and “ surface-area ” coefficients:
A = S(k = 0) = 1 + ρ
Z
Rd
h(r)dr, B = −c(d)
Z
Rd
h(r)rdr,
D: microscopic length scale, φ: dimensionless density
Hyperuniform : A = 0, B > 0
. – p. 11/50
INVERTED CRITICAL PHENOMENAh(r) can be divided into direct correlations , via function c(r), and indirect correlations:
c(k) =h(k)
1 + ρh(k)
. – p. 12/50
INVERTED CRITICAL PHENOMENAh(r) can be divided into direct correlations , via function c(r), and indirect correlations:
c(k) =h(k)
1 + ρh(k)
For any hyperuniform system , h(k = 0) = −1/ρ, and thus c(k = 0) = −∞. Therefore, at the
“critical” reduced density φc, h(r) is short-ranged and c(r) is long-ranged .
This is the inverse of the behavior at liquid-gas (or magnetic) critical points , where h(r) is
long-ranged (compressibility or susceptibility diverges ) and c(r) is short-ranged .
. – p. 12/50
INVERTED CRITICAL PHENOMENAh(r) can be divided into direct correlations , via function c(r), and indirect correlations:
c(k) =h(k)
1 + ρh(k)
For any hyperuniform system , h(k = 0) = −1/ρ, and thus c(k = 0) = −∞. Therefore, at the
“critical” reduced density φc, h(r) is short-ranged and c(r) is long-ranged .
This is the inverse of the behavior at liquid-gas (or magnetic) critical points , where h(r) is
long-ranged (compressibility or susceptibility diverges ) and c(r) is short-ranged .
For sufficiently large d at a disordered hyperuniform state , whether achieved via a nonequilibrium
or an equilibrium route,
c(r) ∼ −1
rd−2+η(r → ∞), c(k) ∼ −
1
k2−η(k → 0),
h(r) ∼ −1
rd+2−η(r → ∞), S(k) ∼ k2−η (k → 0),
where η is a new critical exponent .
One can think of a hyperuniform system as one resulting from an effective pair potential v(r) at
large r that is a generalized Coulombic interaction between like charges . Why? Because
v(r)
kBT∼ −c(r) ∼
1
rd−2+η(r → ∞)
However, long-range interactions are not required to drive a nonequilibrium system to a
disordered hyperuniform state.. – p. 12/50
SINGLE-CONFIGURATION FORMULATION & GROUND STATESWe showed
σ2(R) = 2dφ
„
R
D
«dh
1 − 2dφ
„
R
D
«d
+1
N
NX
i6=j
α(rij ; R)i
where α(r; R) can be viewed as a repulsive pair potential :
0 0.2 0.4 0.6 0.8 1r/(2R)
0
0.2
0.4
0.6
0.8
1
α(r;
R) d=1
d=5
Spherical window of radius R
. – p. 13/50
SINGLE-CONFIGURATION FORMULATION & GROUND STATESWe showed
σ2(R) = 2dφ
„
R
D
«dh
1 − 2dφ
„
R
D
«d
+1
N
NX
i6=j
α(rij ; R)i
where α(r; R) can be viewed as a repulsive pair potential :
0 0.2 0.4 0.6 0.8 1r/(2R)
0
0.2
0.4
0.6
0.8
1
α(r;
R) d=1
d=5
Spherical window of radius R
Finding global minimum of σ2(R) equivalent to finding ground state .
. – p. 13/50
SINGLE-CONFIGURATION FORMULATION & GROUND STATESWe showed
σ2(R) = 2dφ
„
R
D
«dh
1 − 2dφ
„
R
D
«d
+1
N
NX
i6=j
α(rij ; R)i
where α(r; R) can be viewed as a repulsive pair potential :
0 0.2 0.4 0.6 0.8 1r/(2R)
0
0.2
0.4
0.6
0.8
1
α(r;
R) d=1
d=5
Spherical window of radius R
Finding global minimum of σ2(R) equivalent to finding ground state .
For large R, in the special case of hyperuniform systems,
σ2(R) = Λ(R)
„
R
D
«d−1
+ O
„
R
D
«d−3
100 110 120 130R/D
0
0.2
0.4
0.6
0.8
1
Λ(R
)
Triangular Lattice (Average value=0.507826)
. – p. 13/50
Hyperuniformity and Number Theory
Averaging fluctuating quantity Λ(R) gives coefficient of interest:
Λ = limL→∞
1
L
∫ L
0Λ(R)dR
. – p. 14/50
Hyperuniformity and Number Theory
Averaging fluctuating quantity Λ(R) gives coefficient of interest:
Λ = limL→∞
1
L
∫ L
0Λ(R)dR
We showed that for a lattice
σ2(R) =∑
q6=0
(
2πR
q
)d
[Jd/2(qR)]2, Λ = 2dπd−1∑
q6=0
1
|q|d+1.
Epstein zeta function for a lattice is defined by
Z(s) =∑
q6=0
1
|q|2s, Re s > d/2.
Summand can be viewed as an inverse power-law potential . For
lattices , minimizer of Z(d + 1) is the lattice dual to the minimizer of Λ.
Surface-area coefficient Λ provides useful way to rank order crystals,
quasicrystals and special correlated disordered point patterns.
. – p. 14/50
Quantifying Suppression of Density Fluctuations at Large Scales: 1D
The surface-area coefficient Λ for some crystal, quasicrystaland disordered one-dimensional hyperuniform point patterns.
Pattern Λ
Integer Lattice 1/6 ≈ 0.166667
Step+Delta-Function g2 3/16 =0.1875
Fibonacci Chain ∗ 0.2011
Step-Function g2 1/4 = 0.25
Randomized Lattice 1/3 ≈ 0.333333
∗Zachary & Torquato (2009)
. – p. 15/50
Quantifying Suppression of Density Fluctuations at Large Scales: 2D
The surface-area coefficient Λ for some crystal, quasicrystaland disordered two-dimensional hyperuniform point patterns.
2D Pattern Λ/φ1/2
Triangular Lattice 0.508347
Square Lattice 0.516401
Honeycomb Lattice 0.567026
Kagom e Lattice 0.586990
Penrose Tiling ∗ 0.597798
Step+Delta-Function g2 0.600211
Step-Function g2 0.848826
∗Zachary & Torquato (2009)
. – p. 16/50
Quantifying Suppression of Density Fluctuations at Large Scales: 3DContrary to conjecture that lattices associated with the densest
sphere packings have smallest variance regardless of d, we have
shown that for d = 3, BCC has a smaller variance than FCC.
. – p. 17/50
Quantifying Suppression of Density Fluctuations at Large Scales: 3DContrary to conjecture that lattices associated with the densest
sphere packings have smallest variance regardless of d, we have
shown that for d = 3, BCC has a smaller variance than FCC.
Pattern Λ/φ2/3
BCC Lattice 1.24476
FCC Lattice 1.24552
HCP Lattice 1.24569
SC Lattice 1.28920
Diamond Lattice 1.41892
Wurtzite Lattice 1.42184
Damped-Oscillating g2 1.44837
Step+Delta-Function g2 1.52686
Step-Function g2 2.25
Carried out analogous calculations in high d (Zachary &
Torquato, 2009 ), of importance in communications. Disordered
point patterns may win in high d (Torquato & Stillinger, 2006 ).. – p. 17/50
Examples of Homogeneous and Isotropic Hyperuniform Systems
An interesting 1D hyperuniform point pattern is the distribution of the zeros of
the Riemann zeta function (eigenvalues of random Hermitian matrices and bus
arrivals in Cuernavaca) : g2(r) = 1 − sin2(πr)/(πr)2 (Dyson, 1970;
Montgomery, 1973; Krb alek & Seba, 2000).
0 1 2 3 4 5r
0
0.5
1
1.5
g 2(r)
. – p. 18/50
Examples of Homogeneous and Isotropic Hyperuniform Systems
An interesting 1D hyperuniform point pattern is the distribution of the zeros of
the Riemann zeta function (eigenvalues of random Hermitian matrices and bus
arrivals in Cuernavaca) : g2(r) = 1 − sin2(πr)/(πr)2 (Dyson, 1970;
Montgomery, 1973; Krb alek & Seba, 2000).
0 1 2 3 4 5r
0
0.5
1
1.5
g 2(r)
Constructing and/or identifying homogeneous, isotropic h yperuniform
patterns for d ≥ 2 is more challenging .
. – p. 18/50
Examples of Homogeneous and Isotropic Hyperuniform Systems
An interesting 1D hyperuniform point pattern is the distribution of the zeros of
the Riemann zeta function (eigenvalues of random Hermitian matrices and bus
arrivals in Cuernavaca) : g2(r) = 1 − sin2(πr)/(πr)2 (Dyson, 1970;
Montgomery, 1973; Krb alek & Seba, 2000).
0 1 2 3 4 5r
0
0.5
1
1.5
g 2(r)
Constructing and/or identifying homogeneous, isotropic h yperuniform
patterns for d ≥ 2 is more challenging .
In addition to the few examples discussed thus far, we now know of many
more examples.
. – p. 18/50
More Recent Examples of Disordered Hyperuniform Systems
Fermionic point processes : S(k) ∼ k as k → 0 (ground states and/or
positive temperature equilibrium states)
Maximally random jammed (MRJ) particle packings: S(k) ∼ k as k → 0
(nonequilibrium states)
Disordered classical ground states via collective coordinates
Ultracold atoms (nonequilibrium states): Lesanovsky et al. PRE (2014)
Critical absorbing states (nonequilibrium states): Hexner et al. PRL (2015)
Diffusive systems (nonequilibrium states): Jack et al. PRL (2015)
Periodically-driven emulsions (nonequilibrium states): Weijs et. al. PRL (2015)
. – p. 19/50
More Recent Examples of Disordered Hyperuniform Systems
Fermionic point processes : S(k) ∼ k as k → 0 (ground states and/or
positive temperature equilibrium states)
Maximally random jammed (MRJ) particle packings: S(k) ∼ k as k → 0
(nonequilibrium states)
Disordered classical ground states via collective coordinates
Ultracold atoms (nonequilibrium states): Lesanovsky et al. PRE (2014)
Critical absorbing states (nonequilibrium states): Hexner et al. PRL (2015)
Diffusive systems (nonequilibrium states): Jack et al. PRL (2015)
Periodically-driven emulsions (nonequilibrium states): Weijs et. al. PRL (2015)
Natural Disordered Hyperuniform Systems
Avian Photoreceptor Cells (nonequilibrium states)
. – p. 19/50
More Recent Examples of Disordered Hyperuniform Systems
Fermionic point processes : S(k) ∼ k as k → 0 (ground states and/or
positive temperature equilibrium states)
Maximally random jammed (MRJ) particle packings: S(k) ∼ k as k → 0
(nonequilibrium states)
Disordered classical ground states via collective coordinates
Ultracold atoms (nonequilibrium states): Lesanovsky et al. PRE (2014)
Critical absorbing states (nonequilibrium states): Hexner et al. PRL (2015)
Diffusive systems (nonequilibrium states): Jack et al. PRL (2015)
Periodically-driven emulsions (nonequilibrium states): Weijs et. al. PRL (2015)
Natural Disordered Hyperuniform Systems
Avian Photoreceptor Cells (nonequilibrium states)
Nearly Hyperuniform Disordered Systems
Amorphous Silicon (nonequilibrium states)
Supercooled Liquids and Glasses (nonequilibrium states). – p. 19/50
Hyperuniformity and Spin-Polarized Free FermionsOne can map random Hermitian matrices (GUE), fermionic gase s, and zeros of
the Riemann zeta function to a unique hyperuniform point process on R.
. – p. 20/50
Hyperuniformity and Spin-Polarized Free FermionsOne can map random Hermitian matrices (GUE), fermionic gase s, and zeros of
the Riemann zeta function to a unique hyperuniform point process on R.
We provide exact generalizations of such a point process in d-dimensional
Euclidean space Rd and the corresponding n-particle correlation functions ,
which correspond to those of spin-polarized free fermionic systems in Rd.
0 0.5 1 1.5 2r
0
0.2
0.4
0.6
0.8
1
1.2
g 2(r)
d=1d=3
0 2 4 6 8 10k
0
0.2
0.4
0.6
0.8
1
1.2
S(k
)
d=1
d=3
g2(r) = 1 −2Γ(1 + d/2) cos2 (rK − π(d + 1)/4)
K πd/2+1 rd+1(r → ∞)
S(k) =c(d)
2Kk + O(k3) (k → 0) (K : Fermi sphere radius )
Torquato, Zachary & Scardicchio, J. Stat. Mech., 2008
Scardicchio, Zachary & Torquato, PRE, 2009
. – p. 20/50
Hyperuniformity and Jammed PackingsConjecture : All strictly jammed saturated sphere packings are hyperuniform(Torquato & Stillinger, 2003 ).
. – p. 21/50
Hyperuniformity and Jammed PackingsConjecture : All strictly jammed saturated sphere packings are hyperuniform(Torquato & Stillinger, 2003 ).
A 3D maximally random jammed (MRJ) packing is a prototypical glass in that it
is maximally disordered but perfectly rigid (infinite elastic moduli).
Such packings of identical spheres have been shown to be hyperuniform with
quasi-long-range (QLR) pair correlations in which h(r) decays as −1/r4
(Donev, Stillinger & Torquato, PRL, 2005 ).
0 0.5 1 1.5 2kD/2π
0
1
2
3
4
5
S(k)
Data
0 0.2 0.4 0.60
0.02
0.04
Linear fit
This is to be contrasted with the hard-sphere fluid with correlations that decay
exponentially fast .
. – p. 21/50
Hyperuniformity and Jammed PackingsConjecture : All strictly jammed saturated sphere packings are hyperuniform(Torquato & Stillinger, 2003 ).
A 3D maximally random jammed (MRJ) packing is a prototypical glass in that it
is maximally disordered but perfectly rigid (infinite elastic moduli).
Such packings of identical spheres have been shown to be hyperuniform with
quasi-long-range (QLR) pair correlations in which h(r) decays as −1/r4
(Donev, Stillinger & Torquato, PRL, 2005 ).
0 0.5 1 1.5 2kD/2π
0
1
2
3
4
5
S(k)
Data
0 0.2 0.4 0.60
0.02
0.04
Linear fit
This is to be contrasted with the hard-sphere fluid with correlations that decay
exponentially fast .
Apparently, hyperuniform QLR correlations with decay −1/rd+1 are a
universal feature of general MRJ packings in Rd.
Zachary, Jiao and Torquato, PRL (2011) : ellipsoids, superballs, sphere mixtures
Berthier et al., PRL (2011); Kurita and Weeks, PRE (2011) : sphere mixtures
Jiao and Torquato, PRE (2011) : polyhedra. – p. 21/50
Slow and Rapid Cooling of a LiquidClassical ground states are those classical particleconfigurations with minimal potential energy per particle.
Vol
ume
Temperature
quenchglass
liquid
crystal
cooling
. – p. 22/50
Slow and Rapid Cooling of a LiquidClassical ground states are those classical particleconfigurations with minimal potential energy per particle.
Vol
ume
Temperature
rapidquench
glass
super−cooledliquid
liquid
freezingpoint(Tf)
glasstransition(Tg)
crystal
veryslowcooling
Typically, ground states are periodic with high crystallographicsymmetries .
. – p. 22/50
Slow and Rapid Cooling of a LiquidClassical ground states are those classical particleconfigurations with minimal potential energy per particle.
Vol
ume
Temperature
rapidquench
glass
super−cooledliquid
liquid
freezingpoint(Tf)
glasstransition(Tg)
crystal
veryslowcooling
Typically, ground states are periodic with high crystallographicsymmetries .
Can classical ground states ever be disordered ?. – p. 22/50
Creation of Disordered Hyperuniform Ground States
Uche, Stillinger & Torquato, Phys. Rev. E 2004
Batten, Stillinger & Torquato, Phys. Rev. E 2008
Collective-Coordinate Simulations
• Consider a system of N particles with configuration rN ≡ r1, r2, . . . in a fundamental region Ω
(under periodic boundary conditions) at position r. The complex collective density variable is defined by
ρ(k) =
NX
j=1
exp(ik · rj)
. – p. 23/50
Creation of Disordered Hyperuniform Ground States
Uche, Stillinger & Torquato, Phys. Rev. E 2004
Batten, Stillinger & Torquato, Phys. Rev. E 2008
Collective-Coordinate Simulations
• Consider a system of N particles with configuration rN ≡ r1, r2, . . . in a fundamental region Ω
(under periodic boundary conditions) at position r. The complex collective density variable is defined by
ρ(k) =
NX
j=1
exp(ik · rj)
• The real nonnegative structure factor is
S(k) =|ρ(k)|2
N
. – p. 23/50
Creation of Disordered Hyperuniform Ground States
Uche, Stillinger & Torquato, Phys. Rev. E 2004
Batten, Stillinger & Torquato, Phys. Rev. E 2008
Collective-Coordinate Simulations
• Consider a system of N particles with configuration rN ≡ r1, r2, . . . in a fundamental region Ω
(under periodic boundary conditions) at position r. The complex collective density variable is defined by
ρ(k) =
NX
j=1
exp(ik · rj)
• The real nonnegative structure factor is
S(k) =|ρ(k)|2
N
• Consider stable pair potentials v(r) that are bounded with Fourier transform v(k). The total energy is
ΦN (rN ) =X
i<j
v(rij)
=N
2|Ω|
X
k
v(k)S(k) + constant
• For v(k) positive ∀ 0 ≤ |k| ≤ K and zero otherwise, finding configurations in which S(k) is
constrained to be zero where v(k) has support is equivalent to globally minimizing Φ(rN ). These
hyperuniform ground states are called “stealthy” and generally highly degenerate .
• Stealthy patterns can be tuned by varying the parameter χ: number of independently (real) constrained
degrees of freedom to the total number of degrees of freedom d(N − 1).. – p. 23/50
Creation of Disordered Stealthy Ground StatesUnconstrained
Region
Exclusion
Zone S=0
K
. – p. 24/50
Creation of Disordered Stealthy Ground StatesUnconstrained
Region
Exclusion
Zone S=0
K
One class of stealthy potentials involves the following power-law form:
v(k) = v0(1 − k/K)m Θ(K − k),
where n is any whole number. The special case n = 0 is just the simple step function.
10 20r
0
0.01v(r)
m=0m=2m=4
d=3
0 0.5 1 1.5k
0
0.5
1
1.5
v(k)
m=0m=2m=4
~
In the large-system ( thermodynamic ) limit with m = 0 and m = 4, we have the following large-r
asymptotic behavior , respectively:v(r) ∼
cos(r)
r2(m = 0)
v(r) ∼1
r4(m = 4)
While the specific forms of these stealthy potentials lead to the same convergent ground-state
energies, this will not be the case for the pressure and other thermodynamic quantities.. – p. 24/50
Creation of Disordered Stealthy Ground States
In our previous simulations, we began with an initial random distribution of N
points and then found the energy minimizing configurations (with extremely
high precision) using optimization techniques.
. – p. 25/50
Creation of Disordered Stealthy Ground States
In our previous simulations, we began with an initial random distribution of N
points and then found the energy minimizing configurations (with extremely
high precision) using optimization techniques.
For 0 ≤ χ < 0.5, the stealthy ground states are degenerate, disordered and
isotropic .
0 1 2 3k
0
0.5
1
1.5
S(k)
Note the “hard-core” nature of S(k) for stealthy ground states.
The success rate to achieve any disordered ground state is 100%, independent
of the dimension. It turns out that χ = 1/2 is the limit at which there are no
longer degrees of freedom that can be constrained independe ntly.
. – p. 25/50
Creation of Disordered Stealthy Ground States
In our previous simulations, we began with an initial random distribution of N
points and then found the energy minimizing configurations (with extremely
high precision) using optimization techniques.
For 0 ≤ χ < 0.5, the stealthy ground states are degenerate, disordered and
isotropic .
0 1 2 3k
0
0.5
1
1.5
S(k)
Note the “hard-core” nature of S(k) for stealthy ground states.
The success rate to achieve any disordered ground state is 100%, independent
of the dimension. It turns out that χ = 1/2 is the limit at which there are no
longer degrees of freedom that can be constrained independe ntly.
For χ > 1/2, the system undergoes a transition to a crystal phase and the
energy landscape becomes considerably more complex.. – p. 25/50
Creation of Disordered Stealthy Ground StatesTwo Dimensions
(a) χ= 0.04167 (b) χ = 0.41071
Three Dimensions
0 0.1 0.2 0.3 0.4r / L
x
0
1
2
g 2(r)
. – p. 26/50
Creation of Disordered Stealthy Ground StatesTwo Dimensions
(a) χ= 0.04167 (b) χ = 0.41071
Three Dimensions
0 0.1 0.2 0.3 0.4r / L
x
0
1
2
g 2(r)
Increasing χ induces strong short-range ordering , and eventually the system runs out of degrees
of freedom that can be constrained and crystallizes .
0 1 2 3k/K
S(k)
Maximum χ
. – p. 26/50
Ensemble Theory of Disordered Ground StatesTorquato, Zhang & Stillinger, Phys. Rev. X, 2015
Nontrivial : Dimensionality of the configuration space depends on the nu mber density ρ (or χ) and
there is a multitude of ways of sampling the ground-state man ifold, each with its own probability
measure. Which ensemble? How are entropically favored states determined?
Derived general exact relations for thermodynamic propert ies that apply to any ground-state
ensemble as a function of ρ in any d and showed how disordered degenerate ground states arise.
. – p. 27/50
Ensemble Theory of Disordered Ground StatesTorquato, Zhang & Stillinger, Phys. Rev. X, 2015
Nontrivial : Dimensionality of the configuration space depends on the nu mber density ρ (or χ) and
there is a multitude of ways of sampling the ground-state man ifold, each with its own probability
measure. Which ensemble? How are entropically favored states determined?
Derived general exact relations for thermodynamic propert ies that apply to any ground-state
ensemble as a function of ρ in any d and showed how disordered degenerate ground states arise.
From previous considerations, we that an important contrib ution to S(k) is a simple hard-core
step function Θ(k − K), which can be viewed as a disordered hard-sphere system in Fourier
space in the limit that χ ∼ 1/ρ tends to zero, i.e., as the number density ρ tends to infinity.
0 1 2 3k
0
0.5
1
1.5
S(k
)
0 1 2 3r
0
0.5
1
1.5
g 2(r)
That the structure factor must have the behavior
S(k) → Θ(k − K), χ → 0
is perfectly reasonable; it is a perturbation about the ideal-gas limit in which S(k) = 1 for all k.
. – p. 27/50
Ensemble Theory of Disordered Ground StatesTorquato, Zhang & Stillinger, Phys. Rev. X, 2015
Nontrivial : Dimensionality of the configuration space depends on the nu mber density ρ (or χ) and
there is a multitude of ways of sampling the ground-state man ifold, each with its own probability
measure. Which ensemble? How are entropically favored states determined?
Derived general exact relations for thermodynamic propert ies that apply to any ground-state
ensemble as a function of ρ in any d and showed how disordered degenerate ground states arise.
From previous considerations, we that an important contrib ution to S(k) is a simple hard-core
step function Θ(k − K), which can be viewed as a disordered hard-sphere system in Fourier
space in the limit that χ ∼ 1/ρ tends to zero, i.e., as the number density ρ tends to infinity.
0 1 2 3k
0
0.5
1
1.5
S(k
)
0 1 2 3r
0
0.5
1
1.5
g 2(r)
That the structure factor must have the behavior
S(k) → Θ(k − K), χ → 0
is perfectly reasonable; it is a perturbation about the ideal-gas limit in which S(k) = 1 for all k.
We make the ansatz that for sufficiently small χ, S(k) in the canonical ensemble for a stealthy
potential can be mapped to g2(r) for an effective disordered hard-sphere system for sufficiently
small density .. – p. 27/50
Pseudo-Hard Spheres in Fourier SpaceLet us define
H(k) ≡ ρh(k) = hHS(r = k)
There is an Ornstein-Zernike integral eq. that defines FT of appropriate direct correlation function , C(k):
H(k) = C(k) + η H(k) ⊗ C(k),
where η is an effective packing fraction . Therefore,
H(r) =C(r)
1 − (2π)d η C(r).
This mapping enables us to exploit the well-developed accur ate theories of standard Gibbsian
disordered hard spheres in direct space .
0 1 2 3 4k
0
0.5
1
1.5
S(k
)
TheorySimulation
d=3, χ=0.05
0 1 2 3 4k
0
0.5
1
1.5S
(k)
TheorySimulation
d=3, χ=0.1
0 1 2 3 4k
0
0.5
1
1.5
S(k
)
TheorySimulation
d=3, χ=0.143
0 5 10r
0
0.5
1
1.5
g 2(r)
d=1, Simulationd=2, Simulationd=3, Simulationd=1, Theoryd=2, Theoryd=3, Theory
χ=0.05
0 5 10r
0
0.5
1
1.5
g 2(r)
d=1, Simulationd=2, Simulationd=3, Simulationd=1, Theoryd=2, Theoryd=3, Theory
χ=0.1
0 5 10r
0
0.5
1
1.5
g 2(r)
d=1, Simulationd=2, Simulationd=3, Simulationd=1, Theoryd=2, Theoryd=3, Theory
χ=0.143
. – p. 28/50
Stealthy Disordered Ground States and Novel Materials
Until recently, it was believed that Bragg scattering was required to achieve
metamaterials with complete photonic band gaps .
. – p. 29/50
Stealthy Disordered Ground States and Novel Materials
Until recently, it was believed that Bragg scattering was required to achieve
metamaterials with complete photonic band gaps .
Have used disordered, isotropic “stealthy” ground-state configurat ions to
design photonic materials with large complete (both polarizations and all
directions) band gaps .
Florescu, Torquato and Steinhardt, PNAS (2009)
. – p. 29/50
Stealthy Disordered Ground States and Novel Materials
Until recently, it was believed that Bragg scattering was required to achieve
metamaterials with complete photonic band gaps .
Have used disordered, isotropic “stealthy” ground-state configurat ions to
design photonic materials with large complete (both polarizations and all
directions) band gaps .
Florescu, Torquato and Steinhardt, PNAS (2009)
These metamaterial designs have been fabricated for microwave regime.
Man et. al., PNAS (2013)
Because band gaps are isotropic , such photonic materials offer advantages
over photonic crystals (e.g., free-form waveguides ).
. – p. 29/50
Stealthy Disordered Ground States and Novel Materials
Until recently, it was believed that Bragg scattering was required to achieve
metamaterials with complete photonic band gaps .
Have used disordered, isotropic “stealthy” ground-state configurat ions to
design photonic materials with large complete (both polarizations and all
directions) band gaps .
Florescu, Torquato and Steinhardt, PNAS (2009)
These metamaterial designs have been fabricated for microwave regime.
Man et. al., PNAS (2013)
Because band gaps are isotropic , such photonic materials offer advantages
over photonic crystals (e.g., free-form waveguides ).
Other applications include new phononic devices.. – p. 29/50
Which is the Cucumber Image?
. – p. 30/50
Avian Cone PhotoreceptorsOptimal spatial sampling of light requires that photoreceptors be arranged in
the triangular lattice (e.g., insects and some fish).
Birds are highly visual animals, yet their cone photoreceptor patterns are
irregular .
. – p. 31/50
Avian Cone PhotoreceptorsOptimal spatial sampling of light requires that photoreceptors be arranged in
the triangular lattice (e.g., insects and some fish).
Birds are highly visual animals, yet their cone photoreceptor patterns are
irregular .
5 Cone Types
Jiao, Corbo & Torquato, PRE (2014).. – p. 31/50
Avian Cone PhotoreceptorsDisordered mosaics of both total population and individual cone types are
effectively hyperuniform , which has been never observed in any system before
(biological or not). We term this multi-hyperuniformity .
Jiao, Corbo & Torquato, PRE (2014). – p. 32/50
Amorphous Silicon is Nearly HyperuniformHighly sensitive transmission X-ray scattering measureme nts performed at
Argonne on amorphous-silicon (a-Si) samples reveals that t hey are nearly
hyperuniform with S(0) = 0.0075.
Long, Roorda, Hejna, Torquato, and Steinhardt (2013)
This is significantly below the putative lower bound recentl y suggested by de
Graff and Thorpe (2009) but consistent with the recently pro posed nearly
hyperuniform network picture of a-Si (Hejna, Steinhardt and Torquato, 2013).
0 5 10 15 200.0
0.5
1.0
1.5
2.0
k(A )-1
S(k)
a-Si Anneal. [YY] a-Si NHN5 a-Si Impl. [YY] a-Si CRN [XX]
. – p. 33/50
Amorphous Silicon is Nearly HyperuniformHighly sensitive transmission X-ray scattering measureme nts performed at
Argonne on amorphous-silicon (a-Si) samples reveals that t hey are nearly
hyperuniform with S(0) = 0.0075.
Long, Roorda, Hejna, Torquato, and Steinhardt (2013)
This is significantly below the putative lower bound recentl y suggested by de
Graff and Thorpe (2009) but consistent with the recently pro posed nearly
hyperuniform network picture of a-Si (Hejna, Steinhardt and Torquato, 2013).
0 5 10 15 200.0
0.5
1.0
1.5
2.0
k(A )-1
S(k)
a-Si Anneal. [YY] a-Si NHN5 a-Si Impl. [YY] a-Si CRN [XX]
Increasing the degree of hyperuniformity of a-Si appears to be correlated with
a larger electronic band gap (Hejna, Steinhardt and Torquato, 2013).. – p. 33/50
Structural Glasses and Growing Length ScalesImportant question in glass physics: Do growing relaxation times under
supercooling have accompanying growing structural length scales? Lubchenko
& Wolynes (2006); Berthier et al. (2007); Karmakar, Dasgupt a & Sastry (2009); Chandler &
Garrahan (2010); Hocky, Markland & Reichman (2012)
. – p. 34/50
Structural Glasses and Growing Length ScalesImportant question in glass physics: Do growing relaxation times under
supercooling have accompanying growing structural length scales? Lubchenko
& Wolynes (2006); Berthier et al. (2007); Karmakar, Dasgupt a & Sastry (2009); Chandler &
Garrahan (2010); Hocky, Markland & Reichman (2012)
We studied glass-forming liquid models that support an alte rnative view:
existence of growing static length scales (due to increase of the degree of
hyperuniformity ) as the temperature T of the supercooled liquid is decreased
to and below Tg that is intrinsically nonequilibrium in nature.
0 1 2 3 4 5Dimensionless temperature, T / T
g
3
4
5
6
7
8L
engt
h sc
ale,
ξc
. – p. 34/50
Structural Glasses and Growing Length ScalesImportant question in glass physics: Do growing relaxation times under
supercooling have accompanying growing structural length scales? Lubchenko
& Wolynes (2006); Berthier et al. (2007); Karmakar, Dasgupt a & Sastry (2009); Chandler &
Garrahan (2010); Hocky, Markland & Reichman (2012)
We studied glass-forming liquid models that support an alte rnative view:
existence of growing static length scales (due to increase of the degree of
hyperuniformity ) as the temperature T of the supercooled liquid is decreased
to and below Tg that is intrinsically nonequilibrium in nature.
0 1 2 3 4 5Dimensionless temperature, T / T
g
3
4
5
6
7
8L
engt
h sc
ale,
ξc
The degree of deviation from thermal equilibrium is determi ned from a
nonequilibrium indexX =
S(k = 0)
ρkBTκT− 1,
which increases upon supercooling .Marcotte, Stillinger & Torquato (2013)
. – p. 34/50
CONCLUSIONSDisordered hyperuniform many-particle systems can be regarded to be a new
distinguishable state of disordered matter .
Hyperuniformity provides a unified means of categorizing and characterizing
crystals, quasicrystals and special correlated disordere d systems.
The degree of hyperuniformity provides an order metric for the extent to which
large-scale density fluctuations are suppressed in such systems.
Disordered hyperuniform systems appear to be endowed with unusual
physical properties that we are only beginning to discover.
Hyperuniformity has connections to physics (e.g., ground states, quantum
systems, random matrices, novel materials, etc.), mathematics (e.g., geometry
and number theory), and biology .
. – p. 35/50
CONCLUSIONSDisordered hyperuniform many-particle systems can be regarded to be a new
distinguishable state of disordered matter .
Hyperuniformity provides a unified means of categorizing and characterizing
crystals, quasicrystals and special correlated disordere d systems.
The degree of hyperuniformity provides an order metric for the extent to which
large-scale density fluctuations are suppressed in such systems.
Disordered hyperuniform systems appear to be endowed with unusual
physical properties that we are only beginning to discover.
Hyperuniformity has connections to physics (e.g., ground states, quantum
systems, random matrices, novel materials, etc.), mathematics (e.g., geometry
and number theory), and biology .
CollaboratorsRobert Batten (Princeton) Etienne Marcotte (Princeton)
Paul Chaikin (NYU) Weining Man (San Francisco State)
Joseph Corbo (Washington Univ.) Sjoerd Roorda (Montreal)
Marian Florescu (Surrey) Antonello Scardicchio (ICTP)
Miroslav Hejna (Princeton) Paul Steinhardt (Princeton)
Yang Jiao (Princeton/ASU) Frank Stillinger (Princeton)
Gabrielle Long (NIST) Chase Zachary (Princeton)
Ge Zhang (Princeton). – p. 35/50
SCATTERING AND DENSITY FLUCTUATIONS
. – p. 36/50
DENSITY FLUCTUATIONS
Density fluctuations of many-particle systems are of great fundamental
interest, containing important thermodynamic and structu ral information.
For systems in equilibrium at number density ρ,
ρkBTκT =〈N2〉 − 〈N〉2
〈N〉= S(k = 0) = 1 + ρ
∫
Rd
h(r)dr
where h(r) ≡ g2(r) − 1 is the total correlation function, g2(r) is the pair
correlation function, and S(k) = 1 + ρh(k) is the structure factor .
Note that generally ρkTκT 6= S(k = 0).
X =S(k = 0)
ρkBTκT− 1 : Nonequilibrium index
. – p. 37/50
DENSITY FLUCTUATIONS
Density fluctuations of many-particle systems are of great fundamental
interest, containing important thermodynamic and structu ral information.
For systems in equilibrium at number density ρ,
ρkBTκT =〈N2〉 − 〈N〉2
〈N〉= S(k = 0) = 1 + ρ
∫
Rd
h(r)dr
where h(r) ≡ g2(r) − 1 is the total correlation function, g2(r) is the pair
correlation function, and S(k) = 1 + ρh(k) is the structure factor .
Note that generally ρkTκT 6= S(k = 0).
X =S(k = 0)
ρkBTκT− 1 : Nonequilibrium index
Large-scale structure of the universe ( Peebles 1993 )
Probe structure and collective motions in vibrated granula r media ( Warr
and Hansen 1996 )
Coulombic systems ( Martin & Yalcin 1980; Lebowitz 1983 )
Integrable quantum systems ( Bleher, Dyson and Lebowitz 1993 )
. – p. 37/50
Hyperuniform Diffraction Patterns
. – p. 38/50
Hypothesized Disordered Hyperuniform Statesg2-invariant process ≡ one in which a given nonnegative pair correlation g2(r) function remains
invariant as the density varies, for all r, over the range of densities
0 ≤ φ ≤ φ∗
At the terminal density φ∗, the hypothesized system is hyperuniform .
. – p. 39/50
Hypothesized Disordered Hyperuniform Statesg2-invariant process ≡ one in which a given nonnegative pair correlation g2(r) function remains
invariant as the density varies, for all r, over the range of densities
0 ≤ φ ≤ φ∗
At the terminal density φ∗, the hypothesized system is hyperuniform .
Step-function g2
0 1 2 3 4 5r
0
0.5
1
1.5
2
g 2(r)
Asymptotic large- r behavior of c(r) is given by the infinite-space Green’s function for the
d-dimensional Laplace equation:
c(r) =
8
>
>
<
>
>
:
−6r, d = 1,
4 ln(r), d = 2,
−2(d+2)d(d−2)
1rd−2
, d ≥ 3.
. – p. 39/50
Hypothesized Disordered Hyperuniform Statesg2-invariant process ≡ one in which a given nonnegative pair correlation g2(r) function remains
invariant as the density varies, for all r, over the range of densities
0 ≤ φ ≤ φ∗
At the terminal density φ∗, the hypothesized system is hyperuniform .
Step-function g2
0 1 2 3 4 5r
0
0.5
1
1.5
2
g 2(r)
Asymptotic large- r behavior of c(r) is given by the infinite-space Green’s function for the
d-dimensional Laplace equation:
c(r) =
8
>
>
<
>
>
:
−6r, d = 1,
4 ln(r), d = 2,
−2(d+2)d(d−2)
1rd−2
, d ≥ 3.
Step + delta function g2
0 1 2 3 4 5r
0
0.5
1
1.5
2
g 2(r)
. – p. 39/50
Pseudo-Hard Spheres in Fourier SpaceLet us define
H(k) ≡ ρh(k) = hHS(r = k)
There is an Ornstein-Zernike integral equation that defines the appropriatedirect correlation function C(k):
H(k) = C(k) + η H(k) ⊗ C(k),
where η = bχ is an effective packing fraction , where b is a constant to be
determined. Therefore,
H(r) =C(r)
1 − (2π)d η C(r).
This mapping enables us to exploit the well-developed accur ate theories of
standard Gibbsian disordered hard spheres in direct space .
To get an idea of the large- r asymptotics , consider case χ → 0 for any d:
ρh(r) ∼ −1
rd/2Jd/2(r) (χ → 0)
ρh(r) ∼ −1
r(d+1)/2cos(r − (d + 1)π/4) (r → +∞)
. – p. 40/50
Stealthy Disordered Ground State Configurations
Compression Dilation
Direct Space Reciprocal Space
S(k)
. – p. 41/50
General Scaling Behaviors
Hyperuniform particle distributions possess structure factorswith a small-wavenumber scaling
S(k) ∼ kα, α > 0,
including the special case α = +∞ for periodic crystals.
Hence, number variance σ2(R) increases for large R
asymptotically as ( Zachary and Torquato, 2011 )
σ2(R) ∼
Rd−1 ln R, α = 1
Rd−α, α < 1
Rd−1, α > 1
(R → +∞).
Until recently, all known hyperuniform configurations pertainedto α ≥ 1.
. – p. 42/50
Targeted SpectraS ∼ kα
0
k0
S(k)
k4
k2
k
k1/2
K
Configurations are ground states of an interactingmany-particle system with up to four-body interactions .
. – p. 43/50
Targeted SpectraS ∼ kα with α ≥ 1
Uche, Stillinger & Torquato (2006)
Figure 1: One of them is for S(k) ∼ k6 and other for S(k) ∼ k.
. – p. 44/50
Targeted SpectraS ∼ kα with α < 1
Zachary & Torquato (2011)
(a) (b)
Figure 2: Both configurations exhibit strong local clustering of poin ts and possess a highly irreg-
ular local structure; however, only one of them is hyperuniform (with S ∼ k1/2).
. – p. 45/50
DEFINITIONS OF THE CRITICAL EXPONENTS
In the vicinity of or at a hyperuniform state , we have
Exponent Asymptotic behavior
γ S−1(0) ∼ (1 − φφc
)−γ (φ → φ−c )
γ ′ S−1(0) ∼ ( φφc
− 1)−γ′
(φ → φ+c )
ν ξ ∼ (1 − φφc
)−ν (φ → φ−c )
ν ′ ξ ∼ ( φφc
− 1)−ν′
(φ → φ+c )
η c(r) ∼ r2−d−η (φ = φc)
where γ = (2 − η)ν
. – p. 46/50
Definitions
A lattice in d-dimensional Euclidean space Rd is the set of points that are
integer linear combinations of d basis (linearly independent) vectors, i.e., for
basis vectors a1, . . . ,ad,
n1a1 + n2a2 + · · · + ndad | n1, . . . , nd ∈ Z
The space Rd can be geometrically divided into identical regions F called
fundamental cells , each of which contains just one point.
In R2:
A periodic point distribution in Rd is a fixed configuration of N points (where
N ≥ 1) in each fundamental cell of a lattice.
. – p. 47/50
Stealthy Hyperuniform Ground and Excited StatesThese soft, long-ranged potentials in two and three dimensions exhibit
unusual low- T behavior.
Batten, Stillinger and Torquato, Phys. Rev. Lett. (2009)
Have used disordered “stealthy” materials to design photon ic materials with
large complete band gaps .
Florescu, Torquato and Steinhardt, PNAS (2009)
. – p. 48/50
Stealthy Hyperuniform Ground and Excited StatesThese soft, long-ranged potentials in two and three dimensions exhibit
unusual low- T behavior.
Batten, Stillinger and Torquato, Phys. Rev. Lett. (2009)
Have used disordered “stealthy” materials to design photon ic materials with
large complete band gaps .
Florescu, Torquato and Steinhardt, PNAS (2009)
Disordered Hyperuniform Biological Systems
. – p. 48/50
Stealthy Hyperuniform Ground and Excited StatesThese soft, long-ranged potentials in two and three dimensions exhibit
unusual low- T behavior.
Batten, Stillinger and Torquato, Phys. Rev. Lett. (2009)
Have used disordered “stealthy” materials to design photon ic materials with
large complete band gaps .
Florescu, Torquato and Steinhardt, PNAS (2009)
Disordered Hyperuniform Biological Systems
Which is the Cucumber Image?
. – p. 48/50
Pair Correlation (Radial Distribution) Function g2(r)
g2(r) ≡ Probability density function associated with finding
a point at a radial distance r from a given point.
Expected cumulative coordination number at number density ρ
Z(r) = ρs1(1)∫ r
0xd−1g2(x)dx: Expected number
of points contained in a spherical region of radius r.
. – p. 49/50
Pair Correlation (Radial Distribution) Function g2(r)
g2(r) ≡ Probability density function associated with finding
a point at a radial distance r from a given point.
Expected cumulative coordination number at number density ρ
Z(r) = ρs1(1)∫ r
0xd−1g2(x)dx: Expected number
of points contained in a spherical region of radius r.
Poisson Point Distribution in R3 Maximally Random Jammed State in R
3
0 1 2 3
r
0
1
2
3
4
5
g 2(r)
0 1 2 3r
0
1
2
3
4
5
g 2(r)
. – p. 49/50
Diffraction Pattern for Poisson Point Distribution (Ideal Gas)Ground states are highly degenerate.
0 2 4 6 8 10
k
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S(k)
Thermodynamic limit
. – p. 50/50