Salvatore Torquato

Last updated
Salvatore Torquato
S Torquato.JPG
NationalityAmerican
Alma mater
Known for
Awards
Scientific career
Fields statistical mechanics
condensed matter physics
materials science
applied mathematics
biophysics
Institutions
Doctoral advisor George Stell
Website torquato.princeton.edu

Salvatore Torquato is an American theoretical scientist born in Falerna, Italy. His research work has impacted a variety of fields, including physics, [6] chemistry, [7] applied and pure mathematics, [8] materials science, [9] engineering, [10] and biological physics. He is the Lewis Bernard Professor of Natural Sciences in the department of chemistry and Princeton Institute for the Science and Technology of Materials at Princeton University. He has been a senior faculty fellow in the Princeton Center for Theoretical Science, an enterprise dedicated to exploring frontiers across the theoretical natural sciences. He is also an associated faculty member in three departments or programs at Princeton University: physics, applied and computational mathematics, and mechanical and aerospace engineering. On multiple occasions, he was a member of the schools of mathematics and natural sciences at the Institute for Advanced Study, Princeton, New Jersey. [11]

Contents

Research accomplishments

Torquato's research work is centered in statistical mechanics and soft condensed matter theory. A common theme of Torquato’s research work is the search for unifying and rigorous principles to elucidate a broad range of physical phenomena. Often his work has challenged or overturned conventional wisdom, which led to resurgence of various fields or new research directions.[ citation needed ] Indeed, the impact of his work has extended well beyond the physical sciences, including the biological sciences, discrete geometry and number theory.[ peacock prose ] Currently,[ when? ] his published work has been cited over 46,650 times and his h-index is 113 according to his Google Scholar page. [12]

Torquato has made fundamental contributions to our understanding of the randomness of condensed phases of matter through the identification of sensitive order metrics. He is one of the world's experts on packing problems, including pioneering the notion of the "maximally random jammed" state of particle packings, [13] [14] identifying a Kepler-like conjecture for the densest packings of nonspherical particles, [15] and providing strong theoretical evidence that the densest sphere packings in high dimensions (a problem of importance in digital communications) are counterintuitively disordered, not ordered as in our three-dimensional world. [16] He has devised the premier algorithm to reconstruct microstructures of random media. [17] Torquato formulated the first comprehensive cellular automaton model of cancer growth. [18] He has made seminal contributions to the study of random heterogeneous materials, including writing the highly acclaimed[ according to whom? ] treatise on this subject called "Random Heterogeneous Materials." [19] He is one of the world's authorities on "materials by design" using optimization techniques, [20] [21] including "inverse" statistical mechanics. More recently[ when? ] he introduced a new exotic state of matter called "disordered hyperuniformity", [22] which is intermediate between a crystal and liquid. These states of matter are endowed with novel physical properties. [23] [24] [25] [26] A recent[ when? ] study has uncovered that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which are called effectively limit-periodic. [27]

Random heterogeneous media

Torquato is a world authority on the theory of random heterogeneous media.[ according to whom? ] [28] This area dates back to the work of James Clerk Maxwell, Lord Rayleigh and Einstein, and has important ramifications in the physical and biological sciences. Random media abound in nature and synthetic situations, and include composites, thin films, colloids, packed beds, foams, microemulsions, blood, bone, animal and plant tissue, sintered materials, and sandstones. The effective transport, mechanical and electromagnetic properties are determined by the ensemble-averaged fields that satisfy the governing partial differential equations. Thus, they depend, in a complex manner, upon the random microstructure of the material via correlation functions, including those that characterize clustering and percolation.

A 3D realization of binary Pb-Sn alloy obtained using the Yeong-Torquato reconstruction algorithm. PbSnalloy.jpg
A 3D realization of binary Pb-Sn alloy obtained using the Yeong-Torquato reconstruction algorithm.

Rigorous theories

Over two decades ago,[ as of? ] rigorous progress in predicting the effective properties had been hampered because of the difficulty involved in characterizing the random microstructures. Torquato broke this impasse by providing a unified rigorous means of characterizing the microstructures and macroscopic properties of widely diverse random heterogeneous media. His contributions revolutionized the field, which culminated in his treatise, [29] written almost two decades ago, has been cited over 5,300 times and continues to greatly influence the field. In an article published in Physical Review X in 2021, Torquato and Jaeuk Kim formulated the first “nonlocal” exact formula for the effective dynamic dielectric constant tensor for general composite microstructures that accounts for multiple scattering of electromagnetic waves to all orders. [30] [31]

Designer metamaterials via optimization

In 1997, Ole Sigmund and Torquato wrote a seminal paper[ peacock prose ] on the use of the topology optimization method to design metamaterials with negative thermal expansion or those with zero thermal expansion. [32] They also designed 3D anisotropic porous solids with negative Poisson's ratio to optimize the performance of piezoelectric composites. [33] Torquato and coworkers were the first to show that composites whose interfaces are triply periodic minimal surfaces are optimal for multifunctionality. [34]

Degeneracy of pair statistics and structure reconstructions

Torquato and colleagues pioneered a novel and powerful inverse optimization procedure to reconstruct or construct realizations of disordered many-particle or two-phase systems from lower-order correlation functions. [35] [36] [37] An outcome is the quantitative and definitive demonstration that pair information of a disordered many-particle system is insufficient to uniquely determine a representative configuration and identified more sensitive structural descriptors beyond the standard three-, four-body distribution functions, which is of enormous significance in the study of liquid and glassy states of matter. [38]

Canonical n-point correlation function

In 1986, Torquato formulated a unified theoretical approach to represent exactly a general n-point "canonical" correlation function Hn from which one can obtain and compute any of the various types of correlation functions that determine the bulk properties of liquids, glasses and random media, as well as the generalizations of these correlation functions. [39] The wealth of structural information contained in Hn is far from understood. More recently, Torquato and colleagues are discovering connections of special cases of the Hn to the covering and quantizer problems of discrete geometry [40] as well as to problems in number theory. [41]

Liquids and glasses

Torquato is one of the world leaders[ according to whom? ] in the statistical-mechanical theory of liquid and glassy states of matter. He has made seminal contributions[ peacock prose ] to the understanding of the venerable hard-sphere model, which has been invoked to study local molecular order, transport phenomena, glass formation, and freezing behavior in liquids. Other notable research advances concern the theory of water, simple liquids, and general statistical-mechanical theory of condensed states of matter. He has been at the forefront of identifying and applying sensitive correlation functions and descriptors to characterize liquid and glassy structures beyond standard pair statistics. He also is known[ according to whom? ] for extending the machinery of liquid-state theory to characterize the structure of random media.

Toward the quantification of randomness

Torquato and colleagues pioneered the powerful notion[ peacock prose ] of "order metrics and maps" to characterize the degree of order/disorder in many-particle systems. [42] Such descriptors were initially applied to suggest an alternative to the ill-defined random close packed state of sphere packings. Order metrics have been employed by many investigators to characterize the degree of disorder in simple liquids, water and structural glasses. Torquato along with his co-workers have used order metrics to provide novel insights into the structural, thermodynamical, and dynamical nature of molecular systems, such as Lennard-Jones liquids and glasses, [43] water [44] and disordered ground states of matter, [45] among other examples.

g2-invariant processes

Torquato and Stillinger pioneered the notion of g2-invariant processes in which a given nonnegative pair correlation g2 function remains invariant over the range of densities 0≤ø≤ø*, where ø* is the maximum achievable density subject to satisfaction of certain necessary conditions on g2. [46] [47]

Inverse statistical mechanics: ground and excited states

During the first decade of the present millennium, Torquato and his collaborators pioneered inverse statistical-mechanical methodologies to find optimized interaction potentials that lead spontaneously and robustly to a target many-particle configuration, including nanoscale structures, at zero temperature (ground states) and positive temperatures (excited states). Novel target structures include low coordinated 2D and 3D crystal ground states, [48] [49] disordered ground states as well as atomic systems with negative Poisson’s ratios over a wide range of temperatures and densities. [50]

Growing length dcales upon supercooling a liquid

In 2013, Marcotte, Stillinger and Torquato demonstrated that a sensitive signature of the glass transition of atomic liquid models is apparent well before the transition temperature Tc is reached upon supercooling as measured by a length scale determined from the volume integral of the direct correlation function c(r), as defined by the Ornstein-Zernike equation. [51] This length scale grows appreciably with decreasing temperature.

Perfect glasses

In a seminal paper[ peacock prose ] published in 2016, Zheng, Stillinger and Torquato introduced the notion of a "perfect glass". [52] Such amorphous solids involve many-body interactions that remarkably eliminate the possibilities of crystalline and quasicrystalline phases for any state variables, while creating mechanically stable amorphous glasses that are hyperuniform down to absolute zero temperature. Subsequently, it was shown computationally that perfect glasses possess unique disordered classical ground states up to trivial symmetries and hence have vanishing entropy: a highly counterintuitive situation. [53] This discovery provides singular examples in which entropy and disorder are at odds with one another.[ citation needed ]

Packing problems

A realization of the maximally random jammed packing in 3D Euclidean space with periodic boundary conditions Rcp64.jpg
A realization of the maximally random jammed packing in 3D Euclidean space with periodic boundary conditions

Torquato is one of the world's foremost authorities[ according to whom? ] on packing problems, such as how densely or randomly nonoverlapping particles can fill a volume. They are among the most ancient and persistent problems in mathematics and science. Packing problems are intimately related to condensed phases of matter, including classical ground states, liquids, crystals and glasses. While the preponderance of work before 2000 considered sphere packings, Torquato and his colleagues spearheaded the study of the densest and disordered jammed packings of nonspherical particles (e.g., ellipsoids, polyhedra, superballs, among other shapes) since then, which has resulted in an explosion of papers on this topic.

Maximally random jammed packings

In a seminal[ peacock prose ] physical review letters in 2000, Torquato together with Thomas Truskett and Pablo Debenedetti demonstrated that the venerable notion of random close packing in sphere packings is mathematically ill-defined and replaced it with a new concept called the maximally random jammed state. [54] This was made possible by pioneering the idea of scalar metrics of order (or disorder), which opened new avenues of research in condensed-matter physics, and by introducing mathematically precise jamming categories. [55] MRJ packings have come to be viewed as prototypical glasses because they are maximally disordered (according to different order metrics) and infinitely mechanically rigid. [56] Michael Klatt and Torquato characterized various correlation functions as well as transport and electromagnetic properties of MRJ sphere packings. [57]

Dense packings of polyhedra

In a pioneering paper[ peacock prose ] published in the PNAS in 2006, John Conway and Torquato analytically constructed packings of tetrahedra that doubled the density of the best known packings at that time. In another seminal paper published in Nature in 2009, Torquato and Jiao determined the densest known packings of the non-tiling Platonic solids (tetrahedra, octahedron, icosahedron and dodecahedron) as well as the thirteen Archimedean solids. [58] The Torquato-Jiao conjecture states that the densest packings of the Platonic and Archimedean solids with central symmetry (which constitute the majority of them) are given by their corresponding densest Bravais lattice packings. [59] They also conjectured that the optimal packing of any convex, identical polyhedron without central symmetry generally is not a Bravais lattice packing. To date, there are no counterexamples to these conjectures, which are based on certain theoretical considerations. Torquato’s work on polyhedra spurred a flurry of activity in the physics and mathematics communities to determine the densest possible packings of such solids, including dramatic improvements on the density of regular tetrahedra. [60] [61] [62]

Disordered sphere packings may win in high dimensions

Torquato and Stillinger derived a conjectural lower bound on the maximal density of sphere packings in arbitrary Euclidean space dimension d whose large-d asymptotic behavior is controlled by 2-(0:77865...)d. This work may remarkably provide the putative exponential improvement on Minkowski’s 100-year-old bound for Bravais lattices, the dominant asymptotic term of which is 1/2d. [63] These results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic, implying the existence of disordered classical ground states for some continuous potentials – a counterintuitive and profound result.

Packing algorithms

Donev, Stillinger and Torquato formulated a collision-driven molecular dynamics algorithm to create dense packings of smoothy shaped non-spherical particles, within a parallelepiped simulation domain, under both periodic or hard-wall boundary conditions. [64] Torquato and Jiao devised the so-called adaptative-shrinking-cell optimization scheme to generate dense packings of ordered and disordered spheres across dimensions using linear programming [65] as well as dense packings of ordered and disordered nonspherical particles (including polyhedra) via Monte Carlo methods. [66]

Hyperuniformity

A realization of stealthy hyperuniform point configuration Stealthy.jpg
A realization of stealthy hyperuniform point configuration

In a seminal article[ peacock prose ] published in 2003, Torquato and Stillinger introduced the "hyperuniformity" concept to characterize the large-scale density fluctuations of ordered and disordered point configurations. [67] A hyperuniform many-particle system in d-dimensional Euclidean space Rd is characterized by an anomalous suppression of large-scale density fluctuations relative to those in typical disordered systems, such as liquids and amorphous solids. As such, the hyperuniformity concept generalizes the traditional notion of long-range order to include not only all perfect crystals and quasicrystals, but also exotic disordered states of matter, which have the character of crystals on large length scales but are isotropic like liquids.

Disordered hyperuniform systems and their manifestations were largely unknown in the scientific community about two decades ago. Now there is a realization that these systems play a vital role in a number of problems across the physical, materials, mathematical, and biological sciences. Torquato and co-workers have contributed to these developments [68] by showing that these exotic states of matter can be obtained via both equilibrium and nonequilibrium routes and come in both quantum mechanical and classical varieties. The study of hyperuniform states of matter is an emerging multidisciplinary field, influencing and linking developments across the physical sciences, mathematics and biology. In particular, the hybrid crystal-liquid attribute of disordered hyperuniform materials endows them with unique or nearly optimal, direction-independent physical properties and robustness against defects, which makes them an intense subject of research.

Generalizations of hyperuniformity to two-phase media, scalar fields, vector fields and spin systems

Torquato generalized the hyperuniformity concept to heterogeneous media. [69] [70] More recently,[ when? ] Torquato extended hyperuniformity to encompass scalar random fields (e.g., concentration and temperature fields, spinodal decomposition), vector fields (e.g., turbulent velocity fields) and statistically anisotropic many-particle systems. [71] This study led to the idea of "directional hyperuniformity" in reciprocal space. Torquato, Robert Distasio, Roberto Car and colleagues have generalized the hyperuniformity idea to spin systems. [72] Recently, Duyu Chen and Torquato formulated a Fourier space-based optimization approach to construct, at will, two-phase hyperuniform media with prescribed spectral densities. [73] To more completely characterize density fluctuations of point configurations, Torquato, Kim and Klatt carried out an extensive theoretical and computational study of the higher-order moments or cumulants, including the skewness, excess kurtosis, and the corresponding probability distribution function of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order, and determined when a central limit theorem was achieved. [74]

Hyperuniformity in quantum systems

Torquato, together Antonello Scardicchio, has rigorously shown the certain ground states of fermionic systems in any space dimension d are disordered and hyperuniform. [75] Daniel Abreu, Torquato and colleagues proved that Weyl–Heisenberg ensembles are hyperuniform. Such ensembles include as a special case a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels, which is responsible for the quantum Hall effect. [76] More recently, it was shown that there are interesting quantum phase transitions in long-range interacting hyperuniform spin chains in a transverse field. [77]

Hyperuniform patterns of avian photoreceptors in chicken retina Cones-mosaic.jpg
Hyperuniform patterns of avian photoreceptors in chicken retina

Hyperuniformity in biology

Jiao, Torquato, Joseph Corbo and colleagues presented the first example of disordered hyperuniformity found in biology, namely, photoreceptor cells in avian retina. [78] Birds are highly visual animals with five different cone photoreceptor subtypes, yet their photoreceptor patterns are irregular, which less than ideal to sample light. By analyzing chicken cone photoreceptors, consisting of five different cell types, it was found that the disordered patterns are hyperuniform, but with a twist - both the total population and the individual cell types are simultaneously hyperuniform. This multihyperuniformity property is crucial for the acute color vision possessed by birds. Elsewhere, Lomba, Torquato and co-workers presented the first statistical-mechanical model that rigorously achieves disordered multihyperuniformity in ternary mixtures to sample the three primary colors: red, blue and green. [79]

Stealthy and hyperuniform disordered ground states

Torquato, Stillinger and colleagues pioneered the collective-coordinate numerical optimization approach to generate systems of particles interacting with isotropic "stealthy" bounded long-ranged pair potentials (similar to Friedel oscillations) whose classical ground states are counterintuitively disordered, hyperuniform, and highly degenerate across space dimensions. [80] [81] [82] "Stealthy" means that there is zero scattering for a range of wavevectors around the origin. A singular feature of such systems is that dimensionality of the configuration space depends on the fraction of such constrained wave vectors compared to the number of degrees of freedom. Nonetheless, a statistical-mechanical theory for stealthy ground-state thermodynamics and structure has been formulated. [83]

Novel disordered photonic materials

About a decade ago,[ as of? ] it was believed that photonic crystals (dielectric networks with crystal symmetries) were required to achieve large complete (both polarizations and all directions) photonic band gaps. Such materials can be thought[ according to whom? ] of a omnidirectional mirrors but for a finite range of frequencies. By mapping the aforementioned "stealthy" disordered ground-state particle configurations to corresponding dielectric networks, Marain Florescu, Paul Steinhardt and Torquato discovered the first disordered network solids with complete photonic band gaps comparable in size to photonic crystals but with the added advantage that the band gaps are completely isotropic. [84] It was shown both theoretically and experimentally that the latter property enables one to design free-form waveguides not possible with crystals. [85] [86]

Disordered hyperuniform materials with optimal transport and elastic properties

Zhang, Stillinger and Torquato showed that stealthy disordered two-phase systems can attain nearly maximal effective diffusion coefficients over a broad range of volume fractions while also maintaining isotropy. [87] Torquato and Chen discovered that the effective thermal (or electrical) conductivities and elastic moduli of 2D disordered hyperuniform low-density cellular networks are optimal under the constraint of statistical isotropy. Elsewhere, Torquato found that hyperuniform porous media possess singular fluid flow characteristics. [88]

Disordered hyperuniform materials with novel wave characteristics

Kim and Torquato demonstrated that stealthy disordered two-phase systems can be made to be perfectly transparent to both elastic and electromagnetic waves for a wide range of incident frequencies. [89] [90]

Creation of large disordered hyperuniform systems via computational and experimental methods

Recently,[ when? ] Torquato and co-workers have formulated protocols to create and synthesize large hyperuniform samples that are effectively hyperuniform down to the nanoscale, which had been a stumbling block. Kim and Torquato formulated a new tessellation-based computational procedure to design extremely large perfectly hyperuniform disordered dispersions (more than 108 particles) for materials discovery via 3D printing techniques. [91] Self-assembly techniques offer a path to fabricate large samples at much smaller length scales. More recently, Ma, Lomba and Torquato a feasible experimental protocol to create very large hyperuniform systems was proposed using binary paramagnetic colloidal particles. [92] The strong and long-ranged dipolar interaction induced by a tunable magnetic field is free from screening effects that attenuate long-ranged electrostatic interactions in charged colloidal systems.

Characterization of the hyperuniformity of quasicrystals

Zachary and Torquato computed the hyperuniformity order metric, derived from the asymptotic number variance, for first time for quasicrystals: 1D Fibonacci chain and 2D Penrose tiling. The characterization of the hyperuniformity of quasicrystals via the structure factor S(k) is considerably more subtle than that for crystals because the former are characterized by a dense set of Bragg peaks. To do so, Erdal Oguz, Joshua Socolar, Steinhardt and Torquato employed the integrated structure factor to ascertain the hyperuniformity of quasicrystals. [93] The same authors demonstrated elsewhere that certain one-dimensional substitution tilings can either be hyperuniform or anti-hyperuniform. [94] Cheney Lin, Steinhardt and Torquato determined how the hyperuniformity metric in quasicrystals depends on the local isomorphism class. [95]

Hyperuniformity in the distribution of the prime numbers

Torquato, together with Matthew De Courcy-Ireland and Zhang, discovered that the prime numbers in a distinguished limit are hyperuniform with dense Bragg peaks (like a quasicrystal) but positioned at certain rational wavenumbers, like a limit-periodic point pattern, but with an “erratic” pattern of occupied and unoccupied sites. [96] The discovery of this hidden multiscale order in the primes is in contradistinction to their traditional treatment as pseudo-random numbers.

Honors and awards

Torquato is a fellow of the American Physical Society, [97] fellow of the Society for Industrial and Applied Mathematics (SIAM) [98] and Fellow of the American Society of Mechanical Engineers. [99] He is the recipient of the 2017 ASC Joel Henry Hildebrand Award, [100] the 2009 APS David Adler Lectureship Award in Material Physics, [101] SIAM Ralph E. Kleinman Prize, [102] Society of Engineering Science William Prager Medal [103] and ASME Richards Memorial Award. [104] He was a Guggenheim Fellow. [105] He has been a Member of the Institute for Advanced Study on four separate occasions. He recently received a Simons Foundation Fellowship in Theoretical Physics. [106]

Selected publications

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References

  1. ↑ "2017 National Award Recipients - American Chemical Society". American Chemical Society. Retrieved 20 November 2018.
  2. ↑ "Simons Fellows Awardees: Theoretical Physics | Simons Foundation". Archived from the original on 2013-01-15. Retrieved 2013-02-16.
  3. ↑ "2009 David Adler Lectureship Award in the Field of Materials Physics Recipient". American Physical Society. Retrieved 2015-02-26.
  4. ↑ "Ralph E. Kleinman Prize". SIAM. 1970-01-01. Retrieved 2018-11-20.
  5. ↑ "Princeton University - Torquato selected to receive Prager Medal". Archived from the original on 2014-05-03. Retrieved 2014-05-02.
  6. ↑ "Simons Fellows Awardees: Theoretical Physics | Simons Foundation". Archived from the original on 2013-01-15. Retrieved 2013-02-16.
  7. ↑ "2017 National Award Recipients - American Chemical Society". American Chemical Society. Retrieved 20 November 2018.
  8. ↑ "Ralph E. Kleinman Prize". SIAM. 1970-01-01. Retrieved 2018-11-20.
  9. ↑ "2009 David Adler Lectureship Award in the Field of Materials Physics Recipient". American Physical Society. Retrieved 2015-02-26.
  10. ↑ "Princeton University - Torquato selected to receive Prager Medal". Archived from the original on 2014-05-03. Retrieved 2014-05-02.
  11. ↑ "Salvatore Torquato - Salvatore Torquato - Department of Chemistry". Archived from the original on 2014-05-15. Retrieved 2016-03-31.
  12. ↑ "Salvatore Torquato - Google Scholar Citations". scholar.google.com. Retrieved 20 November 2018.
  13. ↑ Torquato, S.; Truskett, T. M.; Debenedetti, P. G. (2000). "Is Random Close Packing of Spheres Well Defined?". Physical Review Letters . 84 (10): 2064–2067. arXiv: cond-mat/0003416 . Bibcode:2000PhRvL..84.2064T. doi:10.1103/physrevlett.84.2064. PMID   11017210. S2CID   13149645.
  14. ↑ Donev, A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly, R.; Torquato, S.; Chaikin, P. M. (2004). "Improving the Density of Jammed Disordered Packings using Ellipsoids". Science . 303 (5660): 990–993. Bibcode:2004Sci...303..990D. CiteSeerX   10.1.1.220.1156 . doi:10.1126/science.1093010. PMID   14963324. S2CID   33409855.
  15. ↑ Torquato, S.; Jiao, Y. (2009). "Dense Packings of the Platonic and Archimedean Solids". Nature . 460 (7257): 876–9. arXiv: 0908.4107 . Bibcode:2009Natur.460..876T. doi:10.1038/nature08239. PMID   19675649. S2CID   52819935.
  16. ↑ Torquato, S.; Stillinger, F. H. (2006). "New Conjectural Lower Bounds on the Optimal Density of Sphere Packings". Experimental Mathematics . 15 (3): 307. arXiv: math/0508381 . doi:10.1080/10586458.2006.10128964. S2CID   9921359.
  17. ↑ Yeong, C. L. Y.; Torquato, S. (1998). "Reconstructing Random Media". Physical Review E . 57 (1): 495. Bibcode:1998PhRvE..57..495Y. doi:10.1103/physreve.57.495.
  18. ↑ Kansal, A. R.; Torquato, S.; Harsh, G. R.; Chiocca, E. A.; Deisboeck, T. S. (2000). "Simulated Brain Tumor Growth using a Three-Dimensional Cellular Automaton". Journal of Theoretical Biology . 203 (4): 367–82. CiteSeerX   10.1.1.305.2356 . doi:10.1006/jtbi.2000.2000. PMID   10736214.
  19. ↑ Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties. New-York: Springer-Verlag.
  20. ↑ Sigmund, O.; Torquato, S. (1997). "Design of Materials with Extreme Thermal Expansion using a Three-Phase Topology Optimization Method". Journal of the Mechanics and Physics of Solids . 45 (6): 1037. Bibcode:1997JMPSo..45.1037S. CiteSeerX   10.1.1.152.2950 . doi:10.1016/S0022-5096(96)00114-7.
  21. ↑ Torquato, S. (2009). "Inverse Optimization Techniques for Targeted Self-Assembly". Soft Matter . 5 (6): 1157. arXiv: 0811.0040 . Bibcode:2009SMat....5.1157T. doi:10.1039/b814211b. S2CID   9709789.
  22. ↑ Torquato, S.; Stillinger, F. H. (2003). "Local Density Fluctualtions, Hyperuniform Systems, and Order Metrics". Physical Review E . 68 (6): 041113. Bibcode:2003PhRvE..68f9901T. doi: 10.1103/physreve.68.069901 .
  23. ↑ Florescu, M.; Torquato, S.; Steinhardt, P. J. (2009). "Designer Disordered Materials with Large, Complete Photonic Band Gaps". Proceedings of the National Academy of Sciences . 106 (49): 20658–63. arXiv: 1007.3554 . Bibcode:2009PNAS..10620658F. doi: 10.1073/pnas.0907744106 . PMC   2777962 . PMID   19918087.
  24. ↑ Jiao, Y.; Lau, T.; Haztzikirou, H.; Meyer-Hermann, M.; Corbo, J. C.; Torquato, S. (2014). "Avian Photoreceptor Patterns Represent a Disordered Hyperuniform Solution to a Multiscale Packing Problem". Physical Review E . 89 (2): 022721. arXiv: 1402.6058 . Bibcode:2014PhRvE..89b2721J. doi:10.1103/physreve.89.022721. PMC   5836809 . PMID   25353522.
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  33. ↑ Sigmund, O.; Torquato, S.; Aksay, I. A. (1998). "On the design of 1-3 piezocomposites using topology optimization". J. Mater. Res. 13 (4): 1038–1048. Bibcode:1998JMatR..13.1038S. doi:10.1557/JMR.1998.0145. S2CID   16788281.
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  35. ↑ Rintoul, M. D.; Torquato, S. (1997). "Reconstruction of the structure of dispersions". J. Colloid Interface Sci. 186 (2): 467–476. Bibcode:1997JCIS..186..467R. doi:10.1006/jcis.1996.4675. PMID   9056377.
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