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In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.
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We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $Asubset mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $sin [p-2, p-1)$, we prove that the order of separation (as $Nto infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we obtain the same separation results for `greedy $s$-energy points. We deduce our results from an upper regularity property of the $s$-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz $s$-energy problem), and we show that this property holds under a local smoothness assumption on the set $A$.
In a microcanonical ensemble (constant $NVE$, hard reflecting walls) and in a molecular dynamics ensemble (constant $NVEmathbf{PG}$, periodic boundary conditions) with a number $N$ of smooth elastic hard spheres in a $d$-dimensional volume $V$ having a total energy $E$, a total momentum $mathbf{P}$, and an overall center of mass position $mathbf{G}$, the individual velocity components, velocity moduli, and energies have transformed beta distributions with different arguments and shape parameters depending on $d$, $N$, $E$, the boundary conditions, and possible symmetries in the initial conditions. This can be shown marginalizing the joint distribution of individual energies, which is a symmetric Dirichlet distribution. In the thermodynamic limit the beta distributions converge to gamma distributions with different arguments and shape or scale parameters, corresponding respectively to the Gaussian, i.e., Maxwell-Boltzmann, Maxwell, and Boltzmann or Boltzmann-Gibbs distribution. These analytical results agree with molecular dynamics and Monte Carlo simulations with different numbers of hard disks or spheres and hard reflecting walls or periodic boundary conditions. The agreement is perfect with our Monte Carlo algorithm, which acts only on velocities independently of positions with the collision versor sampled uniformly on a unit half sphere in $d$ dimensions, while slight deviations appear with our molecular dynamics simulations for the smallest values of $N$.
Positive definite functions on spheres have received an increasing interest in many branches of mathematics and statistics. In particular, the Schoenberg sequences in the spectral representation of positive definite functions have been studied by several mathematicians in the last years. This paper provides a set of relations between Schoenberg sequences defined over real as well as complex spheres of different dimensions. We illustrate our findings describing an application to strict positive definiteness.
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised to a positive power $p$. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the $600$-cell for several ranges of $p$ in different dimensions. Our methods apply to a much broader class of potential functions, those which are absolutely monotonic up to a particular order as functions of the cosine of the geodesic distance. In addition, a preliminary numerical study is presented which suggests optimality of several other highly symmetric configurations and weighted designs in low dimensions. In one case we improve the best known lower bounds on a minimal sized weighted design in $mathbb{CP}^4$. All these results point to the discreteness of minimizing measures for the $p$-frame energy with $p$ not an even integer.
We study distance spheres: the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is too dense and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be straightened into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.
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