Disordered Energy Minimizing Configurations

Transcription

1 . p. /4 Disordered Energy Minimizing Configurations 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 Joint work with Ge Zhang and Frank Stillinger, Princeton University

2 Slow and Rapid Cooling of a Liquid Classical ground states are those classical particle configurations with minimal potential energy per particle.. p. 2/4

3 . p. 2/4 Slow and Rapid Cooling of a Liquid Classical ground states are those classical particle configurations with minimal potential energy per particle. super cooled liquid liquid Volume glass crystal rapid quench very slow cooling glass transition (Tg) Temperature freezing point (Tf) Typically, ground states are periodic with high crystallographic symmetries.

4 . p. 2/4 Slow and Rapid Cooling of a Liquid Classical ground states are those classical particle configurations with minimal potential energy per particle. super cooled liquid liquid Volume glass crystal rapid quench very slow cooling glass transition (Tg) Temperature freezing point (Tf) Typically, ground states are periodic with high crystallographic symmetries. Can classical ground states ever be disordered?

5 Definitions A point process in d-dimensional Euclidean space R d is a distribution of an infinite number of points in R d with configuration r,r 2,... with a well-defined number density ρ (number of points per unit volume). This is statistically described by the n-particle correlation function g n (r,,...,r n ). A lattice Λ in d-dimensional Euclidean space R d is the set of points that are integer linear combinations of d basis (linearly independent) vectors a i, i.e., {n a + n 2 a n d a d n,...,n d Z} The space R d can be geometrically divided into identical regions F called fundamental cells, each of which contains just one point. For example, in R 2 : Every lattice has a dual (or reciprocal) lattice Λ. A periodic point distribution in R d is a fixed but arbitrary configuration of N points (N ) in each fundamental cell of a lattice.. p. 3/4

6 . p. 4/4 Definitions For statistically homogeneous and isotropic point processes in R d at number density ρ, g 2 (r) is a nonnegative radial function that is proportional to the probability density of pair distances r. We call h(r) g 2 (r) the total correlation function. When there is no long-range order in the system, h(r) [or g 2 (r) ] in the large-r limit. We call a point process disordered if h(r) tends to zero sufficiently rapidly such that it is integrable over all space. The nonnegative structure factor S(k) is defined in terms of the Fourier transform of h(r), which we denote by h(k): where k denotes wavenumber. S(k) + ρ h(k), When there is no long-range order in the system, S(k) in the large-k limit, the dual-space analog of the aforementioned direct space condition. In some generalized sense, S(k) can be viewed as a probability density of pair distances of points in reciprocal space.

7 . p. 5/4 Pair Statistics for Spatially Uncorrelated and Ordered Point Processes Poisson Distribution (Ideal Gas) g 2 (r) S(k) r Lattice 2 3 k g 2 (r) S(k) 2 3 r 4π/ 3 8π/ 3 2π/ 3 k

8 . p. 6/4 Hyperuniform Point Processes Torquato and Stillinger, PRE (23) Points can represent molecules of a material, stars in a galaxy, or trees in a forest. Let Ω represent a spherical window of radius R in d-dimensional Euclidean space R d. R Ω R Ω Denote by σ 2 (R) N 2 (R) N(R) 2 the number variance.

9 . p. 6/4 Hyperuniform Point Processes Torquato and Stillinger, PRE (23) Points can represent molecules of a material, stars in a galaxy, or trees in a forest. Let Ω represent a spherical window of radius R in d-dimensional Euclidean space R d. R Ω R Ω Denote by σ 2 (R) N 2 (R) N(R) 2 the number variance. For a Poisson point pattern and many disordered point patterns, σ 2 (R) R d.

10 Hyperuniform Point Processes Torquato and Stillinger, PRE (23) Points can represent molecules of a material, stars in a galaxy, or trees in a forest. Let Ω represent a spherical window of radius R in d-dimensional Euclidean space R d. R Ω R Ω Denote by σ 2 (R) N 2 (R) N(R) 2 the number variance. For a Poisson point pattern and many disordered point patterns, σ 2 (R) R d. We call point patterns whose variance grows more slowly than R d (window volume) hyperuniform (infinite-wavelength density fluctuations vanish). This implies that structure factor S(k) for k. All perfect crystals and quasicrystals are hyperuniform such that σ 2 (R) R d : number variance grows like window surface area.. p. 6/4

11 . p. 7/4 DISORDERED HYPERUNIFORM STATES OF MATTER Disordered hyperuniform many-particle systems can be regarded to be new distinguishable states of disordered matter in that they (i) behave more like perfect 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 no 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 turn out to have technological importance, require new theoretical tools.

12 DISORDERED HYPERUNIFORM STATES OF MATTER Disordered hyperuniform many-particle systems can be regarded to be new distinguishable states of disordered matter in that they (i) behave more like perfect 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 no 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 turn out to have technological importance, require new theoretical tools. The hyperuniformity concept provides a unified means of categorizing and characterizing crystals, quasicrystals and such special disordered systems according to the degree to which they suppress large-scale density fluctuations.. p. 7/4

13 . p. 8/4 Outline Brief Review of Collective-Coordinate Simulations Ensemble Theory of Disordered Ground States

14 . p. 9/4 Creation of Disordered Hyperuniform Ground States Uche, Stillinger & Torquato, Phys. Rev. E 24 Batten, Stillinger & Torquato, Phys. Rev. E 28 Collective-Coordinate Simulations Consider a system of N particles with configuration r N r,r 2,... in a fundamental region Ω (under periodic boundary conditions). The complex collective density variable is defined by ρ(k) = NX exp(ik r j ) j= The real nonnegative structure factor is S(k) = ρ(k) 2 N

15 . p. 9/4 Creation of Disordered Hyperuniform Ground States Uche, Stillinger & Torquato, Phys. Rev. E 24 Batten, Stillinger & Torquato, Phys. Rev. E 28 Collective-Coordinate Simulations Consider a system of N particles with configuration r N r,r 2,... in a fundamental region Ω (under periodic boundary conditions). The complex collective density variable is defined by ρ(k) = NX exp(ik r j ) j= The real nonnegative structure factor is S(k) = ρ(k) 2 N Consider stable bounded pair potentials v(r) in R d with Fourier tranform ṽ(k). The total energy is Φ N (r N ) = X i<j v(r ij ) = N 2 Ω X ṽ(k)s(k) + constant k When ṽ(k) positive k K and zero otherwise, finding configurations in which S(k) is constrained to be zero where ṽ(k) has support is equivalent to globally minimizing Φ(r N ). These hyperuniform ground states are called stealthy and generally highly degenerate.

16 Creation of Disordered Hyperuniform Ground States Uche, Stillinger & Torquato, Phys. Rev. E 24 Batten, Stillinger & Torquato, Phys. Rev. E 28 Collective-Coordinate Simulations Consider a system of N particles with configuration r N r,r 2,... in a fundamental region Ω (under periodic boundary conditions). The complex collective density variable is defined by ρ(k) = NX exp(ik r j ) j= The real nonnegative structure factor is S(k) = ρ(k) 2 N Consider stable bounded pair potentials v(r) in R d with Fourier tranform ṽ(k). The total energy is Φ N (r N ) = X i<j v(r ij ) = N 2 Ω X ṽ(k)s(k) + constant k When ṽ(k) positive k K and zero otherwise, finding configurations in which S(k) is constrained to be zero where ṽ(k) has support is equivalent to globally minimizing Φ(r N ). 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 ).. p. 9/4

17 . p. /4 Creation of Disordered Stealthy Ground States Unconstrained Region K Exclusion Zone S=

18 . p. /4 Creation of Disordered Stealthy Ground States Unconstrained Region K Exclusion Zone S= One class of stealthy potentials involves the following power-law form: ṽ(k) = v ( k/k) n Θ(K k), where n is any whole number. The special case n = is just the simple step function.. v(r) d=3 m= m=2 m= r k In the large-system (thermodynamic) limit with n = and n = 4, we have the following large-r v(k) asymptotic behavior, respectively: v(r) cos(r) r 2 (n = ) ~.5.5 v(r) r 4 (n = 4) m= m=2 m=4

19 Creation of Disordered Stealthy Ground States Unconstrained Region K Exclusion Zone S= One class of stealthy potentials involves the following power-law form: ṽ(k) = v ( k/k) n Θ(K k), where n is any whole number. The special case n = is just the simple step function.. v(r) d=3 m= m=2 m=4 ~ v(k).5.5 m= m=2 m= r k In the large-system (thermodynamic) limit with n = and n = 4, we have the following large-r asymptotic behavior, respectively: v(r) cos(r) r 2 (n = ) v(r) r 4 (n = 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. /4

20 . p. /4 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.

21 . p. /4 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 χ <.5, the stealthy ground states are degenerate, disordered and isotropic..5 S(k) k The success rate to achieve any disordered ground state is %, independent of the dimension. It turns out that χ = /2 is the limit at which there are no longer degrees of freedom that can be constrained independently.

22 . p. /4 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 χ <.5, the stealthy ground states are degenerate, disordered and isotropic..5 S(k) k The success rate to achieve any disordered ground state is %, independent of the dimension. It turns out that χ = /2 is the limit at which there are no longer degrees of freedom that can be constrained independently. These exotic disordered ground states, which possess a hidden order on large length scales, can be viewed as many-particle analogs of spin liquids.

23 . p. 2/4 Creation of Disordered Stealthy Ground States Two Dimensions Three Dimensions (a) χ=.467 (b) χ =.47 2 g 2 (r) r / L x

24 Creation of Disordered Stealthy Ground States Two Dimensions Three Dimensions (a) χ=.467 (b) χ =.47 2 g 2 (r) r / L x Increasing χ induces strong short-range ordering, and eventually the system runs out of degrees of freedom that can be constrained and crystallizes. Maximum χ S(k) 2 3. p. 2/4

25 . p. 3/4 Ensemble Theory of Disordered Ground States Can one formulate an ensemble theory for stealthy degenerate ground states? Which ensemble? A challenging issue is the determination of the entropically favored states at each χ or ρ, i.e., weights associated with ground states. For some ensemble at fixed density ρ, the average energy per particle u for radial potentials in the thermodynamic limit is given by u Φ(rN ) N = ρ 2 where v ṽ(k = ). Z R d v(r)g 2 (r)dr = ρ 2 v 2 v(r = ) + 2(2π) d Z R d ṽ(k)s(k)dk.

26 . p. 3/4 Ensemble Theory of Disordered Ground States Can one formulate an ensemble theory for stealthy degenerate ground states? Which ensemble? A challenging issue is the determination of the entropically favored states at each χ or ρ, i.e., weights associated with ground states. For some ensemble at fixed density ρ, the average energy per particle u for radial potentials in the thermodynamic limit is given by u Φ(rN ) N = ρ 2 where v ṽ(k = ). Z R d v(r)g 2 (r)dr = ρ 2 v 2 v(r = ) + 2(2π) d Z R d ṽ(k)s(k)dk. Consider the same class of stealthy radial potential functions ṽ(k) in R d. Whenever particle configurations in R d exist such that S(k) is constrained to be its minimum value of zero where ṽ(k) has support, the system must be at its ground state or global energy minimum: u = ρ 2 v v(r = ) 2 Remark: Ground-state manifold, which is generally degenerate, is invariant to the specific form of the stealthy potential because the energy is a configuration-independent constant at fixed ρ.

27 Ensemble Theory of Disordered Ground States Can one formulate an ensemble theory for stealthy degenerate ground states? Which ensemble? A challenging issue is the determination of the entropically favored states at each χ or ρ, i.e., weights associated with ground states. For some ensemble at fixed density ρ, the average energy per particle u for radial potentials in the thermodynamic limit is given by u Φ(rN ) N = ρ 2 where v ṽ(k = ). Z R d v(r)g 2 (r)dr = ρ 2 v 2 v(r = ) + 2(2π) d Z R d ṽ(k)s(k)dk. Consider the same class of stealthy radial potential functions ṽ(k) in R d. Whenever particle configurations in R d exist such that S(k) is constrained to be its minimum value of zero where ṽ(k) has support, the system must be at its ground state or global energy minimum: u = ρ 2 v v(r = ) 2 Remark: Ground-state manifold, which is generally degenerate, is invariant to the specific form of the stealthy potential because the energy is a configuration-independent constant at fixed ρ. In the thermodynamic limit, parameter χ is related to the number density ρ in any dimension d via ρ χ = V (K) 2d(2π) d, where V (K) is the volume of a d-dimensional sphere of radius K. Remarks: We see that χ and ρ are inversely proportional to one another. Thus, for fixed K and d, as χ tends to zero, ρ tends to infinity, which corresponds counterintutively to the uncorrelated ideal-gas limit (Poisson distribution). As χ increases from zero, the density ρ decreases.. p. 3/4

28 . p. 4/4 Ensemble Theory of Disordered Ground States Any periodic crystal with a finite basis is a stealthy ground state for all positive χ up to a maximum χ max (ρ min ) determined by its first positive Bragg peak (minimal vector in Fourier space). Lemma: At fixed K, a configuration comprised of the union of m different stealthy ground-state configurations in R d with χ, χ 2,..., χ m, respectively, is itself stealthy with a value χ value given by " m # X χ =, χ i i= which is the harmonic mean of the χ i divided by m. These last two facts can be used to see how complex aperiodic patterns are possible ground states.

29 Ensemble Theory of Disordered Ground States Any periodic crystal with a finite basis is a stealthy ground state for all positive χ up to a maximum χ max (ρ min ) determined by its first positive Bragg peak (minimal vector in Fourier space). Lemma: At fixed K, a configuration comprised of the union of m different stealthy ground-state configurations in R d with χ, χ 2,..., χ m, respectively, is itself stealthy with a value χ value given by " m # X χ =, χ i i= which is the harmonic mean of the χ i divided by m. These last two facts can be used to see how complex aperiodic patterns are possible ground states. Table 2: Periodic stealthy ground states in R with K =. Structure χ max ρ min Integer lattice Periodic with n-particle basis n n 2π 2π = = ( )n Table 2: Periodic stealthy ground states in R 2 with K =. Structure χ max ρ min Kagomé crystal Honeycomb crystal Square lattice Triangular lattice π 3 = π 8π 2 = = π 2 = π 4 = π 2 = π = π 2 = p. 4/4

30 . p. 5/4 Ensemble Theory of Disordered Ground States Table 3: Periodic stealthy ground states in R 3 with K =. Structure χ max ρ min Pyrochlore crystal Diamond crystal Simple hexagonal lattice SC lattice π 4 = π 3 = π 3 2 = = π 3 3 π 9 = π 4 3 π 3 = = π 3 = π 3 3 HCP crystal = = π 3 π FCC lattice = = π 3 MCC lattice BCC lattice 2 2π 9 = π 3 = Table 4: Periodic stealthy ground states in R 4 with K =. Structure χ max ρ min Kag 4 crystal Dia 4 crystal Z 4 lattice D 4 lattice π 2 4 = π 3 = π 2 6 = π 3 = π 2 6 = π 3 = π 2 8 = π 3 =

31 . p. 6/4 Canonical Ensemble Theory of Disordered Ground States We consider the Gibbs canonical ensemble in which the partition function Z is a function of ρ and absolute temperature T. Our main interest is in a theory in the limit T, i.e., the entropically favored ground states in the canonical ensemble.

32 . p. 6/4 Canonical Ensemble Theory of Disordered Ground States We consider the Gibbs canonical ensemble in which the partition function Z is a function of ρ and absolute temperature T. Our main interest is in a theory in the limit T, i.e., the entropically favored ground states in the canonical ensemble. Ground-State Pressure and Isothermal Compressibility Energy Route: The pressure in the thermodynamic limit at T = can be obtained from the energy per particle (taking v = ) via the relation Therefore, for stealthy potentials, p = ρ 2 u ρ The isothermal compressibility κ T ρ ρ p p = ρ2 2. T «κ T = ρ 2. T. of such a system is

33 Canonical Ensemble Theory of Disordered Ground States We consider the Gibbs canonical ensemble in which the partition function Z is a function of ρ and absolute temperature T. Our main interest is in a theory in the limit T, i.e., the entropically favored ground states in the canonical ensemble. Ground-State Pressure and Isothermal Compressibility Energy Route: The pressure in the thermodynamic limit at T = can be obtained from the energy per particle (taking v = ) via the relation Therefore, for stealthy potentials, p = ρ 2 u ρ The isothermal compressibility κ T ρ ρ p p = ρ2 2. T «κ T = ρ 2. T. of such a system is Virial Route: An alternative route to the pressure is through the virial equation, which at T = is given by p = ρ2 2d s ()» = ρ2 2d Z r d dv dr g 2(r)dr Z f(k) h(k)dk R d f(k = ) + (2π) d where f(k) is the Fourier transform of f(r) rdv/dr, when it exists.. p. 6/4

34 . p. 7/4 Canonical Ensemble Theory of Disordered Ground States Consider a radial function u(r) in R d such that its Fourier transform ũ(k) exists with compact support in the interval [, ). Let w(r) = rdu/dr such that its Fourier transform w(k) exists. Proposition : The Fourier transform of the radial function w(r) in R d is given by w(k) = d ũ(k) dũ dk. Corollary: It immediately follows from Proposition that w(k) has the same support as ũ(k) and w(k = ) = d ũ(k = ). Substitution of the general form h(k) = Θ(K k) + P(k) (where P(k) has support for k > K) into virial expression and use of Proposition yields p = 2d» ρ 2 ṽ(k = ) + ρ (2π) d Z R d f(k)dk ρ (2π) d Z f(k) P(k)dk R d = ρ2 2. Remarks: This agrees with pressure obtained via energy route whenever f(k) exists. Since the latter is a stronger condition than the existence of ṽ(k), it is possible to devise ṽ(k) for which f(k) does not exist and hence a virial pressure that either diverges or is nonconvergent.

35 . p. 8/4 Pair Statistics of Disordered Ground States In the ground state of a stealthy potential, the structure factor attains its minimum value S(k) = for < k K, and hence has the general form S(k) = Θ(k K)[ + Q(k)], where Θ(x) = (, x <,, x. Therefore, ρ h(k) = f(k) + P(k), where f(k) = Θ(K k) and P(k) = Θ(k K) Q(k). Remark: The function f(k) is identical to the Mayer-f function for a Gibbs-ensemble hard-sphere system in direct space. It easily shown that the functions v(r) and P(r) are orthogonal to one another: Z R d v(r)p(r)dr =.

36 Pair Statistics of Disordered Ground States Behavior of the Pair Correlation Function Near the Origin Using the general form for S(k) for stealthy ground states and elementary properties of Fourier transforms, it is easy to show pair correlation function at the origin is given by Z g 2 (r = ) = 2dχ + 2d 2 χ K k d Q(k)dk Since g 2 (r) must be nonnegative for all r, we have the following integral condition on Q(k): 2d 2 χ Z Hence, this integral must be positive for χ 2d. K k d Q(k)dk 2dχ Moreover, if g 2 (r) has positive curvature at the origin, Q(k) must obey the additional integral condition: (d + 2) Z K k d+ Q(k)dk. Finally, when g 2 (r = ) =, the results above yield the equality Z K k d Q(k)dk = 2dχ 2d 2 χ and, because the curvature must be positive in this instance, the previous inequality must generally be obeyed.. p. 9/4

37 . p. 2/4 Pseudo-Hard Spheres in Fourier Space From previous considerations, we see that that an important contribution 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 tends to zero, i.e., as the number density ρ tends to infinity..5.5 S(k).5 g 2 (r) k 2 3 r That the structure factor must have the behavior S(k) Θ(k K), χ is perfectly reasonable; it is a perturbation about the ideal-gas limit in which S(k) = for all k.

38 Pseudo-Hard Spheres in Fourier Space From previous considerations, we see that that an important contribution 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 tends to zero, i.e., as the number density ρ tends to infinity..5.5 S(k).5 g 2 (r) k 2 3 r That the structure factor must have the behavior S(k) Θ(k K), χ is perfectly reasonable; it is a perturbation about the ideal-gas limit in which S(k) = for all k. Imagine carrying out a series expansion of S(k) about χ =. We make the ansatz that for sufficiently small χ, S(k) in the canonical ensemble for a stealthy potential can be mapped to g 2 (r) for an effective disordered hard-sphere system for sufficiently small density.. p. 2/4

39 Let us define Pseudo-Hard Spheres in Fourier Space H(k) ρ h(k) = h HS (r = k) There is an Ornstein-Zernike integral equation that defines the appropriate direct 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) (2π) d η C(r). This mapping enables us to exploit the well-developed accurate theories of standard Gibbsian disordered hard spheres in direct space. To get an idea of the large-r asymptotics, consider case χ for any d: ρh(r) r d/2 J d/2(r) (χ ) ρh(r) r cos(r (d + )π/4) (r + ) (d+)/2. p. 2/4

40 . p. 22/4 Pair Statistics: Theory vs. Simulations.5 d=3, χ=.5.5 d=3, χ=. S(k).5 Theoretical Simulation S(k).5 Theoretical Simulation k k.5 χ=.5.5 χ=. g 2 (r).5 d=, Simulation d=2, Simulation d=3, Simulation d=, Theoretical d=2, Theoretical d=3, Theoretical 5 r g 2 (r).5 d=, Simulation d=2, Simulation d=3, Simulation d=, Theoretical d=2, Theoretical d=3, Theoretical 5 r

41 . p. 23/4 Ensemble Theory for Number Variance For an ensemble in d-dimensional Euclidean space R d : h Z i σ 2 (R) = N(R) + ρ h(r)α(r; R)dr R d α(r; R)- scaled intersection volume of 2 windows of radius R separated by r R r α(r;r) Spherical window of radius R.8.6 d=.4 d= r/(2r)

42 . p. 23/4 Ensemble Theory for Number Variance For an ensemble in d-dimensional Euclidean space R d : h Z i σ 2 (R) = N(R) + ρ h(r)α(r; R)dr R d α(r; R)- scaled intersection volume of 2 windows of radius R separated by r R r α(r;r) Spherical window of radius R.8.6 d=.4 d= r/(2r) For large R and under certain conditions, we can show h «R d «R d «R d i σ 2 (R) = 2 d φ A + B + o, D D D where A and B are the volume and Z surface-area coefficients: Z A = S(k = ) = + ρ h(r)dr, B(R) = c(d) h(r)rdr, R d R d D: microscopic length scale, φ: dimensionless density Hyperuniform: A =, B >

43 . p. 23/4 Ensemble Theory for Number Variance For an ensemble in d-dimensional Euclidean space R d : h Z i σ 2 (R) = N(R) + ρ h(r)α(r; R)dr R d α(r; R)- scaled intersection volume of 2 windows of radius R separated by r R r α(r;r) Spherical window of radius R.8.6 d=.4 d= r/(2r) For large R and under certain conditions, we can show h «R d «R d «R d i σ 2 (R) = 2 d φ A + B + o, D D D where A and B are the volume and Z surface-area coefficients: Z A = S(k = ) = + ρ h(r)dr, B(R) = c(d) h(r)rdr, R d R d Definition: D: microscopic length scale, φ: dimensionless density Hyperuniform: A =, B > B = lim L L R L B(R)dR Finding the point process that minimizes B is equivalent to finding the ground state of overlap potential, which is the dual of minimizer of the Epstein zeta function with s = (d + )/2.

44 . p. 24/4 σ 2 (R)/R d Number Variance σ 2 (R): Theory vs. Simulations χ=.5 d=, Simulation d=2, Simulation d=3, Simulation d=, Theoretical d=2, Theoretical d=3, Theoretical σ 2 (R)/R d χ=. d=, Simulation d=2, Simulation d=3, Simulation d=, Theoretical d=2, Theoretical d=3, Theoretical R R σ 2 (R)/R d χ=.2 d=, Simulation d=2, Simulation d=3, Simulation d=, Theoretical d=2, Theoretical d=3, Theoretical R 6 8

45 . p. 25/4 Number Variance σ 2 (R): Theory vs. Simulations Table 5: The coefficent B for various values of χ across dimensions. χ d = d = 2 d = Integer lattice.67 Triangular lattice.58 BCC lattice.245

46 . p. 26/4 Definitions: Nearest-Neighbor Functions H P (r)dr = Probability that a point of the point process lies at a distance between r and r + from another point of the point process. E P (r) = Probability of finding a spherical cavity of radius r centered at an arbitrary point of the point process empty of any other points. Exact series representation: E P (r) = + X ( ) k ρk k! k= H P (r) = E P r. Z R d g k+ (r k+ ) k+ Y j=2 Θ(r r j )dr j. Upper and lower bounds: E P (r) Z(r) E P (r) exp[ Z(r)] where Z(r) = ρs () Z r x d g 2 (x)dx, is the cumulative coordination number: expected number of points within a sphere of radius r.

47 . p. 27/4 Nearest-Neighbor Function: Theory vs. Simulations.5 d=2, χ=.5.5 d=2, χ=. E p (r).5 Simulation Theoretical Lower Bound Theoretical Upper Bound E p (r).5 Simulation Theoretical Lower Bound Theoretical Upper Bound 2 3 r r.5 d=3, χ=.5.5 d=3, χ=. E p (r).5 Simulation Theoretical Lower Bound Theoretical Upper Bound E p (r).5 Simulation Theoretical Lower Bound Theoretical Upper Bound r r

48 T. p. 28/4 Remark About Excited States For a system in equilibrium, the compressibility relation relates the isothermal compressibility κ T to the structure factor at k = : S() = ρk B Tκ T, where k B is Boltzmann s constant. Because κ T is is bounded according to κ T = ρ 2 for finite ρ, S() = because T =, which clearly must be the case for stealthy ground states. Now consider excited states infinitesimally close to the stealthy ground states, i.e., when temperature T is positive and infinitesimally small. Under the assumption that the structure of such excited states will be infintesimally near the ground-state configurations, we can estimate to an excellent approximation how S() varies with T for such excited states: S() c(d) χ T in units k B = v =. Morover, it is expected that this positive value of S() will be the uniform value of S(k) for k K. This behavior of S(k) has indeed been verified by molecular dynamics simulations in the canonical ensemble: S() d=2, χ=. Simulation Theoretical

49 . p. 29/4 CONCLUSIONS 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. Disordered hyperuniform systems are new distinguishable states of matter somewhere between a liquid and a perfect crystal, and appear to be endowed with novel physical properties Disordered ground states are necessarily highly degenerate and their existence for radial pair potentials in low d is surprising. Devising a corresponding ensemble theory is highly nontrivial (many ensembles from which to choose) and until recently had been lacking. We have derived exact general results that apply to any ensemble of stealthy ground states, disordered or not. For the specific case of the canonical ensemble, our theory enables us to obtain predictive analytical results for the structure and thermodynamics of the entropically favored disordered ground states. Proving the existence of disordered stealthy ground states is an outstanding problem. Extensions of these results to compact spaces would be very interesting.

50 . p. 3/4 Hyperuniformity and Number Theory Averaging fluctuating quantity Λ(R) gives coefficient of interest: Λ = lim L L L Λ(R)dR

51 . p. 3/4 Hyperuniformity and Number Theory Averaging fluctuating quantity Λ(R) gives coefficient of interest: Λ = lim L We showed that for a lattice σ 2 (R) = q L L Λ(R)dR ( ) 2πR d [J d/2(qr)] 2, Λ = 2 d π d q Epstein zeta function for a lattice is defined by Z(s) = q q 2s, Re s > d/2. q Summand can be viewed as an inverse power-law potential. For q d+. lattices, minimizer of Z(d + ) 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.

52 . p. 3/4 Quantifying Suppression of Density Fluctuations at Large Scales: D The surface-area coefficient Λ for some crystal, quasicrystal and disordered one-dimensional hyperuniform point patterns. Pattern Λ Integer Lattice / Step+Delta-Function g 2 3/6 =.875 Fibonacci Chain.2 Step-Function g 2 /4 =.25 Randomized Lattice / Zachary & Torquato (29)

53 . p. 32/4 Quantifying Suppression of Density Fluctuations at Large Scales: 2D The surface-area coefficient Λ for some crystal, quasicrystal and disordered two-dimensional hyperuniform point patterns. 2D Pattern Λ/φ /2 Triangular Lattice Square Lattice.564 Honeycomb Lattice Kagomé Lattice Penrose Tiling Step+Delta-Function g 2.62 Step-Function g Zachary & Torquato (29)

54 . p. 33/4 Quantifying Suppression of Density Fluctuations at Large Scales: 3D Contrary 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.

55 . p. 33/4 Quantifying Suppression of Density Fluctuations at Large Scales: 3D Contrary 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/φ 2/3 BCC Lattice FCC Lattice HCP Lattice SC Lattice.2892 Diamond Lattice.4892 Wurtzite Lattice.4284 Damped-Oscillating g Step+Delta-Function g Step-Function g Carried out analogous calculations in high d (Zachary & Torquato, 29), of importance in communications. Disordered point patterns may win in high d (Torquato & Stillinger, 26).

56 Hyperuniform Diffraction Patterns. p. 34/4

57 . p. 35/4 General Scaling Behaviors Hyperuniform particle distributions possess structure factors with a small-wavenumber scaling S(k) k α, α >, including the special case α = + for periodic crystals. Hence, number variance σ 2 (R) increases for large R asymptotically as (Zachary and Torquato, 2) σ 2 (R) R d ln R, α = R d α, α < R d, α > (R + ). Until recently, all known hyperuniform configurations pertained to α.

58 . p. 36/4 Targeted Spectra S k α k /2 S(k) k k 2 k 4 k K Configurations are ground states of an interacting many-particle system with up to four-body interactions.

59 . p. 37/4 Targeted Spectra S k α with α Uche, Stillinger & Torquato (26) Figure : One of them is for S(k) k 6 and other for S(k) k.

60 . p. 38/4 Targeted Spectra S k α with α < Zachary & Torquato (2) (a) (b) Figure 2: Both configurations exhibit strong local clustering of points and possess a highly irregular local structure; however, only one of them is hyperuniform (with S k /2 ).

61 . p. 39/4 Stealthy Hyperuniform Ground and Excited States These soft, long-ranged potentials in two and three dimensions exhibit unusual low-t behavior. Batten, Stillinger and Torquato, Phys. Rev. Lett. (29) Have used disordered stealthy materials to design photonic materials with large complete band gaps. Florescu, Torquato and Steinhardt, PNAS (29)

62 . p. 39/4 Stealthy Hyperuniform Ground and Excited States These soft, long-ranged potentials in two and three dimensions exhibit unusual low-t behavior. Batten, Stillinger and Torquato, Phys. Rev. Lett. (29) Have used disordered stealthy materials to design photonic materials with large complete band gaps. Florescu, Torquato and Steinhardt, PNAS (29) Disordered Hyperuniform Biological Systems

63 . p. 39/4 Stealthy Hyperuniform Ground and Excited States These soft, long-ranged potentials in two and three dimensions exhibit unusual low-t behavior. Batten, Stillinger and Torquato, Phys. Rev. Lett. (29) Have used disordered stealthy materials to design photonic materials with large complete band gaps. Florescu, Torquato and Steinhardt, PNAS (29) Disordered Hyperuniform Biological Systems Which is the Cucumber Image?

64 . p. 4/4 Pair Correlation (Radial Distribution) Function g 2 (r) g 2 (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) = ρs () r xd g 2 (x)dx: Expected number of points contained in a spherical region of radius r.

65 Pair Correlation (Radial Distribution) Function g 2 (r) g 2 (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) = ρs () r xd g 2 (x)dx: Expected number of points contained in a spherical region of radius r. Poisson Point Distribution in R 3 Maximally Random Jammed State in R 3 g 2 (r) g 2 (r) r 2 3 r. p. 4/4

66 Diffraction Pattern for Poisson Point Distribution (Ideal Gas) Ground states are highly degenerate. 2.8 Thermodynamic limit.6.4 S(k) k. p. 4/4