American theoretical scientist
Salvatore Torquato is an American theoretical scientist born in Falerna, Italy. His research work has impacted a variety of fields, including physics,
chemistry, applied and pure mathematics, materials science, engineering, 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.
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. Indeed, the impact of his work has extended well beyond the physical sciences, including the biological sciences, discrete geometry and number theory. As of Oct. 2024, his published work has been cited over 55,870 times and his h-index is 122 according to his Google Scholar page.
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, identifying a Kepler-like conjecture for the densest packings of nonspherical particles, 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. He has devised the premier algorithm to reconstruct microstructures of random media. Torquato formulated the first comprehensive cellular automaton model of cancer growth. He has made seminal contributions to the study of random heterogeneous materials, including writing the treatise on this subject called "Random Heterogeneous Materials." He is one of the world's authorities on "materials by design" using optimization techniques, including "inverse" statistical mechanics. In 2003, he introduced a new exotic state of matter called "disordered hyperuniformity", which is intermediate between a crystal and liquid. These states of matter are endowed with novel physical properties.
 A study in 2019 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.
Random heterogeneous media.
The work on the theory of random heterogeneous media 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.
Rigorous theories
In 1990s, 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, 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.
Designer metamaterials via optimization
In 1997, Ole Sigmund and Torquato wrote a seminal paper on the use of the topology optimization method to design metamaterials with negative thermal expansion or those with zero thermal expansion. They also designed 3D anisotropic porous solids with negative Poisson's ratio to optimize the performance of piezoelectric composites. Torquato and coworkers were the first to show that
composites whose interfaces are triply periodic minimal surfaces are optimal for multifunctionality.
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. 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.
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. 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
as well as to problems in number theory.
Liquids and glasses.
Torquato is one of the world leaders in the statistical-mechanical theory of liquid and glassy states of matter. He has made seminal contributions 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 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 of "order metrics and maps" to characterize the degree of order/disorder in many-particle systems. 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, water and disordered ground
states of matter, among other examples.
"g"2-invariant processes
Torquato and Stillinger pioneered the notion of "g"2-invariant processes in which a given nonnegative pair correlation "g"2 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 "g"2.
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, disordered ground states as well as atomic systems with negative Poisson’s ratios over a wide range of temperatures and densities.
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 "T"c 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. This length scale grows
appreciably with decreasing temperature.
Perfect glasses
In a seminal paper published in 2016, Zheng, Stillinger and Torquato introduced the notion of a "perfect glass". 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. This discovery provides singular examples in which entropy and disorder are at odds with one another.
Packing problems.
Torquato is one of the world's foremost authorities 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 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. 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. MRJ packings have come to be viewed as prototypical glasses because they are maximally disordered (according to different order metrics) and infinitely mechanically rigid. Michael Klatt and Torquato characterized various correlation functions as well as transport and electromagnetic properties of MRJ sphere packings.
Dense packings of polyhedra
In a pioneering paper 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. 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. 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.
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. 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. 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 as well as dense packings of ordered and disordered nonspherical particles (including polyhedra) via Monte Carlo methods.
Hyperuniformity.
In a seminal article published in 2003, Torquato and Stillinger introduced the "hyperuniformity" concept to characterize the large-scale density fluctuations of ordered and disordered point configurations. 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