Ground states and formal duality relations in the Gaussian core model.
@article{Cohn2009GroundSA, title={Ground states and formal duality relations in the Gaussian core model.}, author={Henry Cohn and Abhinav Kumar and Achill Sch{\"u}rmann}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2009}, volume={80 6 Pt 1}, pages={ 061116 } }
We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core…
32 Citations
Duality relations for the classical ground states of soft-matter systems
- Physics
- 2010
Bounded interactions are particularly important in soft-matter systems, such as colloids, microemulsions, and polymers. In this paper, we extend the results of a recent letter [S. Torquato and F. H.…
Ground state at high density
- Mathematics
- 2010
Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global…
The Gaussian core model in high dimensions
- Computer Science, MathematicsDuke Mathematical Journal
- 2018
Lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential are proved, which are sharp to within a constant factor as $n \to \infty$ with $s$ fixed.
Ground State at High Density
- Physics
- 2011
Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global…
Inverse Statistical Mechanics, Lattice Packings, and Glasses
- Mathematics
- 2013
Computer simulation methods enable the investigation of systems and properties that are intractable by purely analytical or experimental approaches. Each chapter of this dissertation contains an…
Formal duality and generalizations of the Poisson summation formula
- MathematicsDiscrete Geometry and Algebraic Combinatorics
- 2013
Using the Poisson summation formula, the notion of formal duality is reformulated as a combinatorial phenomenon in finite abelian groups and new examples related to Gauss sums are given.
High-dimensional generalizations of the kagomé and diamond crystals and the decorrelation principle for periodic sphere packings
- Materials Science
- 2011
In this paper, we introduce constructions of the high-dimensional generalizations of the kagome and diamond crystals. The two-dimensional kagome crystal and its three-dimensional counterpart, the…
Order and disorder in energy minimization
- Mathematics
- 2010
How can we understand the origins of highly symmetrical objects? One way is to characterize them as the solutions of natural optimization problems from discrete geometry or physics. In this paper, we…
Classification of Formal Duality with an Example in Sphere Packing UROP + Final Paper
- Mathematics
- 2016
We study the notion of formal duality, which was introduced and developed by Cohn, Kumar, Reiher, and Schürmann through their study of ground state configurations of particles in Euclidean space.…
References
SHOWING 1-10 OF 81 REFERENCES
New duality relations for classical ground states.
- PhysicsPhysical review letters
- 2008
We derive new duality relations that link the energy of configurations associated with a class of soft pair potentials to the corresponding energy of the dual (Fourier-transformed) potential. We…
Counterintuitive ground states in soft-core models.
- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2008
This paper disproves a conjecture of Torquato and Stillinger, who predicted that dilute ground states of the Gaussian core model in dimensions 2 through 8 would be Bravais lattices, and shows that in dimensions 5 and 7, there are in fact lower-energy non-Bravais latticing.
Gaussian core model phase diagram and pair correlations in high Euclidean dimensions.
- MathematicsThe Journal of chemical physics
- 2008
The goals of this paper are to characterize the behavior of the pair correlation function g(2) in various density regimes and to understand the phase properties of the Gaussian core model (GCM) as parametrized by dimension d.
Phase transitions in the Gaussian core system
- Materials Science
- 1976
Some aspects of phase transition behavior have been studied for a classical system of particles which interact in pairs via repelling Gaussian potentials. In a specific low‐temperature, low‐density…
Phase diagram of the Gaussian-core model.
- Physics, Materials SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2005
TheGaussian-core model, a classical system of point particles interacting via a Gaussian-shaped, purely repulsive potential, is traced with high numerical accuracy and it is found that the fluid-bcc-fcc triple-point temperature is about one third of the maximum freezing temperature.
Mathematical physics: Going to ground
- Physics, EducationNature
- 2006
A proof showing that, for certain interactions, periodic ‘ground states’ exist provides a new perspective on this, one of the oldest questions in physics.
Fluid and solid phases of the Gaussian core model
- Physics
- 2000
We study the structural and thermodynamic properties of a model of point particles interacting by means of a Gaussian pair potential first introduced by Stillinger (Stillinger F H 1976 J. Chem. Phys.…
Crystalline ground states for classical particles.
- MathematicsPhysical review letters
- 2005
Pair interactions whose Fourier transform is non-negative and vanishes above a wave number K(0) are shown to give rise to periodic and aperiodic infinite volume ground state configurations (GSCs) in…
Erratum: Study of melting and freezing in the Gaussian core model by molecular dynamics simulation
- Materials Science
- 1978
Molecular dynamics calculations have been carried out to establish quantitative properties of the Gaussian core model near its crystal–fluid transition. Two densities have been considered, for both…
New Conjectural Lower Bounds on the Optimal Density of Sphere Packings
- MathematicsExp. Math.
- 2006
An optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies to obtain upper bounds on the density, and hence has implications for linear programming bounds, and proves that the density estimate can never exceed the Cohn– Elkies upper bound.