arXiv subjects
Publications and source records attributed to Laurent Bétermin.
We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices $Λ\subset \mathbb{R}^2$ with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be seen as the inhomogeneous counterpart to Montgomery's 1988 result, which proved that the hexagonal lattice minimizes the maximum.
We investigate two-dimensional crystallization phenomena, i.e. minimality of a lattice's patch for interaction energies, with pair potentials of type $(x,y)\mapsto V(\|x-y\|)$ where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^2$ and $V:\mathbb{R}_+^*\to\mathbb{R}$ is a function. For the Heitmann-Radin sticky disk potential $V=V_{\text{HR}}$, we prove, using Brass' key result from [\textit{Computational Geometry}, 6:195--214, 1996], that crystallization occurs for any fixed norm, with a classification of minimizers and minimal energies according to the kissing number associated to $\|\cdot\|$. The minimizer is proved to be, up to affine transform, a patch of the triangular or the square lattice, which shows how to easily get anisotropy in a crystallization phenomenon. We apply this result to the $p$-norms $\|\cdot\|_p$, $p\geq 1$, which allows us to construct an explicit family of norms for which crystallization holds on any given lattice. We also solve part of a crystallization problem studied in [\textit{Arch. Ration. Mech. Anal.}, 240:987--1053] where points are constrained to be on $\mathbb{Z}^2$. Moreover, we numerically investigate the minimization problem for the energy per point among lattices for the Lennard-Jones potential $V=V_{\text{LJ}}:r\mapsto r^{-12}-2r^{-6}$ as well as the Epstein zeta function associated to a $p$-norm $\|\cdot\|_p$, i.e. when $V=V_s:r\mapsto r^{-s}$, $s>2$. Our simulations show a new and unexpected phase transition for the minimizers with respect to $p$.
We consider two-dimensional systems of point particles located on rectangular lattices and interacting via pairwise potentials. The goal of this paper is to investigate the phase transitions (and their nature) at fixed density for the minimal energy of such systems. The 2D rectangle lattices we consider have an elementary cell of sides $a$ and $b$, the aspect ratio is defined as $Δ=b/a$ and the inverse particle density $A = a b$; therefore, the ``symmetric'' state with $Δ=1$ corresponds to the square lattice and the ``non-symmetric'' state to the rectangular lattice with $Δ\ne 1$. For certain types of the interaction potential, by changing continuously the particle density, such lattice systems undertake at a specific value of the (inverse) particle density $A^*$ a structural transition from the symmetric to the non-symmetric state. The structural transition can be either of first order ($Δ$ unstick from its symmetric value $Δ=1$ discontinuously) or of second order ($Δ$ unstick from $Δ=1$ continuously); the first and second-order phase transitions are separated by the so-called tricritical point. We develop a general theory on how to determine the exact values of the transition densities and the location of the tricritical point. The general theory is applied to the double Yukawa and Yukawa-Coulomb potentials.
We study the ground-state properties of one-dimensional fluids of classical (i.e., non-quantum) particles interacting pairwisely via a potential, at the fixed particle density $ρ$. Restricting ourselves to periodic configurations of particles, two possibilities are considered: an equidistant chain of particles with the uniform spacing $A=1/ρ$ and its simplest non-Bravais modulation, namely a bipartite lattice composed of two equidistant chains, shifted with respect to one another. Assuming the long range of the interaction potential, the equidistant chain dominates if $A$ is small enough, $0<A<A_c$. At a critical value of $A=A_c$, the system undergoes a continuous second-order phase transition from the equidistant chain to a bipartite lattice. The energy and the order parameter are singular functions of the deviation from the critical point $A-A_c$ with universal (i.e., independent of the model's parameters) mean-field values of critical exponents. The tricritical point at which the curve of continuous second-order transitions meets with the one of discontinuous first-order transitions is determined. The general theory is applied to the Lennard-Jones model with the $(n,m)$ Mie potential for which the phase diagram is constructed. The inclusion of a hard-core around each particle reveals a non-universal critical phenomenon with an $m$-dependent critical exponent.
The Riemann zeta function $ζ(s):= \sum_{n=1}^{\infty} 1/n^s$ can be interpreted as the energy per point of the lattice $\mathbb{Z}$, interacting pairwisely via the Riesz potential $1/r^s$. Given a parameter $Δ\in (0,1]$, this physical model is generalized by considering the energy per point $E(s,Δ)$ of a periodic one-dimensional lattice alternating the distances between the nearest-neighbour particles as $2/(1+Δ)$ and $2Δ/(1+Δ)$, keeping the lattice density equal to one independently of $Δ$. This energy trivially satisfies $E(s,1)=ζ(s)$ at $Δ=1$, it can be easily expressed as a combination of the Riemann and Hurwitz zeta functions, and extended analytically to the punctured $s$-plane $\mathbb{C} \setminus \{ 1\}$. In this paper, we perform numerical investigations of the zeros of the energy $\{ ρ=ρ_x+{\rm i}ρ_y\}$, which are defined by $E(ρ,Δ)=0$. The numerical results reveal that in the Riemann limit $Δ\to 1^-$ theses zeros include the anticipated critical zeros of the Riemann zeta function with $\Re(ρ_x)=\frac{1}{2}$ as well as an unexpected -- comparing to the Riemann Hypothesis -- infinite series of off-critical zeros. The analytic treatment of these off-critical zeros shows that their imaginary components are equidistant and their real components diverge logarithmically to $-\infty$ as $Δ\to 1^-$, i.e., they become invisible at the Riemann's $Δ=1$.
It is well-known that any Lennard-Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among $2$-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical $(12,6)$ Lennard-Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in [L. Bétermin and P. Zhang, \textit{Commun. Contemp. Math.}, 17 (2015), 1450049] and [L. Bétermin, \textit{SIAM J. Math. Anal.}, 48 (2016), 3236--3269]. The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated.
We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H.~Montgomery. Minimal theta functions. \textit{Glasgow Mathematical Journal}, 30(1):75--85, 1988]. The studied theta functions are generalizations of the Jacobi theta-2 and theta-4 functions. Contrary to Montgomery's result, we show that, among lattices, the hexagonal lattice is the unique maximizer of both families of theta functions. As an immediate consequence, we obtain a new universal optimality result for the hexagonal lattice among two-dimensional alternating charged lattices and lattices shifted by the center of their unit cell.
The two-dimensional Epstein zeta function formulated on a rectangular lattice with spacings $a_x=1$ and $a_y=Δ$, $ζ^{(2)}(s,Δ) = \frac{1}{2} \sum_{j,k} (j^2+Δ^2 k^2)^{-s}$ $(\Re(s)>1)$ where the sum goes over all integers except of the origin $(j,k)=(0,0)$, is studied. It can be analytically continued to the whole complex $s$-plane except for the point $s=1$. The nontrivial zeros $\{ ρ=ρ_x+{\rm i}ρ_y \}$ of the Epstein zeta function, defined by $ζ^{(2)} (ρ,Δ)=0$, split into ``critical'' zeros (on the critical line $ρ_x=\frac{1}{2}$) and ``off-critical'' zeros ($ρ_x\ne\frac{1}{2}$). According to the present numerical calculation, the critical zeros form open or closed curves $ρ_y(Δ)$ in the plane $(Δ,ρ_y)$. Two nearest critical zeros merge at special points, referred to as left/right edge zeros, which are defined by a divergent tangent ${\rm d}ρ_y/{\rm d}Δ\vert_{Δ^*}$. Each of these edge zeros gives rise to a continuous curve of off-critical zeros which can thus be generated systematically. As a rule, each curve of off-critical zeros joins a pair of left and right edge zeros. It is shown that in the regions of small/large values of the anisotropy parameter $Δ$ the Epstein zeta function can be approximated adequately by a function which reveals an equidistant distribution of critical zeros along the imaginary axis in the limits $Δ\to 0$ and $Δ\to\infty$. It is also found that for each $Δ\in (0,Δ_c^*]\cup [1/Δ_c^*,\infty)$ with $Δ_c^*\approx 0.141733$ there exists a pair of \emph{real} off-critical zeros, their $ρ_x$ components go to the borders $0$ and $1$ of the critical region in the limits $Δ\to 0,\infty$. As a rule, each curve of off-critical zeros joins a pair of left and right edge zeros.
The Riesz potential $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones type potentials $f_{n,m}^{\rm{LJ}}(r):=a r^{-n}-b r^{-m}$, $n>m$ that are widely used in Molecular Simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the Body-Centred-Cubic lattice (BCC), Face-Centred-Cubic lattice (FCC), Simple Hexagonal lattices (SH) and Hexagonal Close-Packing structure (HCP), globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for $s<0$ and the HCP lattice is computed to have higher energy than the FCC (for $s>3/2$) and BCC (for $s<3/2$) lattices. In the Lennard-Jones case with exponents $3<m<n$, the ground state among lattices is confirmed to be a FCC lattice whereas a HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type ``FCC-SH" and ``FCC-HCP-SH" (when the HCP lattice is added) as the inverse density $V$ increases are observed for a large spectrum of exponents $(n,m)$. In the SH phase, the variation of the ratio $Δ$ between the inter-layer distance $d$ and the lattice parameter $a$ is studied as $V$ increases. In the critical region of exponents $0<m<n<3$, the SH phase with an extreme value of the anisotropy parameter $Δ$ dominates. If one limits oneself to rigid lattices, the BCC-FCC-HCP phase diagram is found. For $-2<m<n<0$, the BCC lattice is the only energy minimizer. Choosing $-4<m<n<-2$, the FCC and SH latices become minimizers.
Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two- and three-body contributions. Our main focus is on characterizing those configurations which cannot be deformed without changing distances between first neighbors or angles formed by pairs of first neighbors. Such configurations are called {\it angle-rigid}. We tackle this question in the class of finite configurations in $\mathbb{Z}^2$, seen as planar three-dimensional point sets. A sufficient condition preventing angle-rigidity is presented. This condition is also proved to be necessary when restricted to specific subclasses of configurations.
We consider pairwise interaction energies and we investigate their minimizers among lattices with prescribed minimal vectors (length and coordination number), i.e. the one corresponding to the crystal's bonds. In particular, we show the universal minimality -- i.e. the optimality for all completely monotone interaction potentials -- of strongly eutactic lattices among these structures. This gives new optimality results for the square, triangular, simple cubic (SC), face-centred-cubic (FCC) and body-centred-cubic (BCC) lattices in dimensions 2 and 3 when points are interacting through completely monotone potentials. We also show the universal maximality of the triangular and FCC lattices among all lattices with prescribed bonds. Furthermore, we apply our results to Lennard-Jones type potentials, showing the minimality of any universal minimizer (resp. maximizer) for small (resp. large) bond lengths, where the ranges of optimality are easily computable. Finally, a numerical investigation is presented where a phase transition of type "square-rhombic-triangular" (resp. "SC-rhombic-BCC-rhombic-FCC") in dimension $d=2$ (resp. $d=3$) among lattices with more than 4 (resp. 6) bonds is observed.
Certain types of neurons, called "grid cells", have been shown to fire on a triangular grid when an animal is navigating on a two-dimensional environment, whereas recent studies suggest that the face-centred-cubic (FCC) lattice is the good candidate for the same phenomenon in three dimensions. The goal of this paper is to give new evidences of these phenomena by considering a infinite set of independent neurons (a module) with Poisson statistics and periodic spread out Gaussian tuning curves. This question of the existence of an optimal grid is transformed into a maximization problem among all possible unit density lattices for a Fisher Information which measures the accuracy of grid-cells representations in $\mathbb{R}^d$. This Fisher Information has translated lattice theta functions as building blocks. We first derive asymptotic and numerical results showing the (non-)maximality of the triangular lattice with respect to the Gaussian parameter and the size of the firing field. In a particular case where the size of the firing fields and the lattice spacing match with experiments, we have numerically checked that it is possible to find a value for the Gaussian parameter above which the triangular lattice is always optimal. In the case of a radially symmetric distribution of firing locations, we also characterize all the lattices that are critical points for the Fisher Information at fixed scales belonging to an open interval (we call these lattices "volume stationary"). It allows us to compare the Fisher Information of a finite number of lattices in dimension 2 and 3 and to give another evidences of the optimality of the triangular and FCC lattices.
The condition number for eigenvalue computations is a well--studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with exact first order asymptotic.
The Embedded-Atom Model (EAM) provides a phenomenological description of atomic arrangements in metallic systems. It consists of a configurational energy depending on atomic positions and featuring the interplay of two-body atomic interactions and nonlocal effects due to the corresponding electronic clouds. The purpose of this paper is to mathematically investigate the minimization of the EAM energy among lattices in two and three dimensions. We present a suite of analytical and numerical results under different reference choices for the underlying interaction potentials. In particular, Gaussian, inverse-power, and Lennard-Jones-type interactions are addressed.
The goal of this work is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume $V$ with an alternation of charges $\pm 1$ at lattice points, among periodic distribution of charges and lattice structures. We assume that the charges are interacting through two types of radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the $d$-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials. Numerically, we investigate several particular cases in dimensions $2$, $3$ and $8$. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For $d=2$, we observe a phase transition of the type 'triangular-rhombic-square-rectangular' for the minimizer's shape as the density decreases.
We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density -- also called crystallization -- is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.
We consider a deformation $E_{L,Λ}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,Λ)$ and two parameters $(m,t)\in (0,\infty)$, initially proposed by Terry Gannon. We show that the minimizers of the lattice theta function are the maximizers of $E_{L,Λ}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalization of the logarithm, called lattice-logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterized by a variational problem over a class of one-dimensional lattice-logarithm.
We study the minimality properties of a new type of "soft" theta functions. For a lattice $L\subset \mathbb{R}^d$, a $L$-periodic distribution of mass $μ_L$ and an other mass $ν_z$ centred at $z\in \mathbb{R}^d$, we define, for all scaling parameter $α>0$, the translated lattice theta function $θ_{μ_L+ν_z}(α)$ as the Gaussian interaction energy between $ν_z$ and $μ_L$. We show that any strict local or global minimality result that is true in the point case $μ=ν=δ_0$ also holds for $L\mapsto θ_{μ_L+ν_0}(α)$ and $z\mapsto θ_{μ_L+ν_z}(α)$ when the measures are radially symmetric with respect to the points of $L\cup \{z\}$ and sufficiently rescaled around them (i.e. at a low scale). The minimality at all scales is also proved when the radially symmetric measures are generated by a completely monotone kernel. The method is based on a generalized Jacobi transformation formula, some standard integral representations for lattice energies and an approximation argument. Furthermore, for the honeycomb lattice $\mathsf{H}$, the center of any primitive honeycomb is shown to minimize $z\mapsto θ_{μ_{\mathsf{H}}+ν_z}(α)$ and many applications are stated for other particular physically relevant lattices including the triangular, square, cubic, orthorhombic, body-centred-cubic and face-centred-cubic lattices.