The crystallization conjecture: a review

We modify the "floating crystal" trial state for the classical Homogeneous Electron Gas (also known as Jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground state energy of Jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in Density Functional Theory. Finally, in the third system each electron interacts with a periodic image of itself, that is, periodic boundary conditions are imposed on the interaction potential.

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“…where ζ BCC (s) is the Epstein Zeta function of the (density one) BCC lattice, see [38,39] and [2, p. 43].…”

Section: Three Definitions Of the Ground State Energymentioning

confidence: 99%

Floating Wigner crystal with no boundary charge fluctuations

2019

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“…It is widely believed that as the value of ℓ is increased with all other parameters fixed, the global energy minimizer of E ε should be either constant or spatially periodic, with period approaching a constant independent of ℓ as ℓ → ∞. Proving such a crystallization result would be one of the main challenges in the theory of energy-driven pattern formation and is currently out of reach (for a recent review, see [1]), except for the case d = 1,ū ε ∈ (−1, 1) fixed and ε > 0 sufficiently small [2,22,26] (for a very recent result in that direction in higher dimensions, see [7]). On the other hand, it is known that forū ε ∈ (−1, 1) fixed, global energy minimizers are not constant as soon as ε ≪ 1 and ℓ 1 [3,24].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…where χ ε is as in (2.18) with u replaced with u ε . The term E (1) ε is continuous with respect to the weak convergence of measures, hence whereχ ε (x) := χ ε (xℓ/ℓ ε ) and…”

Section: )mentioning

confidence: 99%

Emergence of nontrivial minimizers for the three-dimensional Ohta–Kawasaki energy

Knüpfer

Muratov

Novaga

2020

Pure Appl. Analysis

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This paper is concerned with the diffuse interface Ohta-Kawasaki energy in three space dimensions, in a periodic setting, in the parameter regime corresponding to the onset of non-trivial minimizers. We identify the scaling in which a sharp transition from asymptotically trivial to non-trivial minimizers takes place as the small parameter characterizing the width of the interfaces between the two phases goes to zero, while the volume fraction of the minority phases vanishes at an appropriate rate. The value of the threshold is shown to be related to the optimal binding energy solution of Gamow's liquid drop model of the atomic nucleus. Beyond the threshold the average volume fraction of the minority phase is demonstrated to grow linearly with the distance to the threshold. In addition to these results, we establish a number of properties of the minimizers of the sharp interface screened Ohta-Kawasaki energy in the considered parameter regime. We also establish rather tight upper and lower bounds on the value of the transition threshold.

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“…Crystallization problems have received constant attention in the last decades. The reader is referred to the recent survey by Blanc & Lewin [4] for a comprehensive account on the literature. To the best of our knowledge, crystallization results in periodic landscapes are still currently unavailable.…”

Section: Introductionmentioning

confidence: 99%

Crystallization in a One-Dimensional Periodic Landscape

Friedrich

Stefanelli

2020

J Stat Phys

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We consider the crystallization problem for a finite one-dimensional collection of identical hard spheres in a periodic energy landscape. This issue arises in connection with the investigation of crystalline states of ionic dimers, as well as in epitaxial growth on a crystalline substrate in presence of lattice mismatch. Depending on the commensurability of the radius of the sphere and the period of the landscape, we discuss the possible emergence of crystallized states. In particular, we prove that crystallization in arbitrarily long chains is generically not to be expected.2010 Mathematics Subject Classification. 82D25.

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The crystallization conjecture: a review

Cited by 133 publications

References 209 publications

Floating Wigner crystal with no boundary charge fluctuations

Floating Wigner crystal with no boundary charge fluctuations

Emergence of nontrivial minimizers for the three-dimensional Ohta–Kawasaki energy

Crystallization in a One-Dimensional Periodic Landscape

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