Related papers: On the three-dimensional shape of a crystal
In the case of one extra dimension, well known Newton's potential $\phi (r_3)=-G_N m/r_3$ is generalized to compact and elegant formula $\phi(r_3,\xi)=-(G_N m/r_3)\sinh(2\pi r_3/a)[\cosh(2\pi r_3/a)-\cos(2\pi\xi/a)]^{-1}$ if…
Let $K$ be a convex polyhedron and $\mathscr F$ its Wulff energy, and let $\mathscr C(K)$ denote the set of convex polyhedra close to $K$ whose faces are parallel to those of $K$. We show that, for sufficiently small $\epsilon$, all…
This work formulates and gives grounds for general principles and theorems that question the energy function doctrine and its quantum version as a genuine law of nature without borders of adequacy. The emphasis is on the domain where the…
We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher…
We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a…
In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also…
In this study, a variational method for the inverse problem of self-assembly, i.e., a reconstruction of the interparticle interaction potential of a given structure, is applied to three-dimensional crystals. According to the method, the…
The local character of self-gravity along with the number of spatial dimensions are critical issues when computing the potential and forces inside massive systems like stars and disks. This appears from the discretisation scale where each…
We consider the possibility that classical dynamical systems display motion in their lowest energy state, forming a time analogue of crystalline spatial order. Challenges facing that idea are identified and overcome. We display arbitrary…
Most physical systems, whether classical or quantum mechanical, exhibit spherical symmetry. Angular momentum, denoted as $\ell$, is a conserved quantity that appears in the centrifugal potential when a particle moves under the influence of…
A theory of flexibility and rigidity is developed for general infinite bar-joint frameworks (G,p). Determinations of nondeformability through vanishing flexibility are obtained as well as sufficient conditions for deformability. Forms of…
We outline a regular way for solving Maxwell's equations. We take, as the starting point, the notion of vector potentials. The rationale for introducing this notion in electrodynamics is that the set of Maxwell's equations is seemingly…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
A combination of classical density-functional theory and thermodynamic perturbation theory is applied to a survey of finite-temperature trends in the relative stabilities of one-component crystals and quasicrystals interacting via effective…
In complex crystals close to melting or at finite temperatures, different types of defects are ubiquitous and their role becomes relevant in the mechanical response of these solids. Conventional elasticity theory fails to provide a…
We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle…
Wigner crystallization of free electrons at room temperature is explored for a new class of metallic ultrathin (transdimensional) materials whose properties can be controlled by their thickness. Our calculations of the critical electron…
Freezing of charge-stabilized colloidal suspensions and relative stabilities of crystals and quasicrystals are studied using thermodynamic perturbation theory. Macroion interactions are modelled by effective pair potentials combining…
We demonstrate that the physics which resolves naturalness problems need not take the form of new particles and can sometimes manifest itself as higher dimensional operators. As a proof of principle, we present a simple model where the…
We measure the local contact forces at both the top and bottom boundaries of three-dimensional face-centered-cubic and hexagonal-close-packed granular crystals in response to an external force applied to a small area at the top surface.…