Spherical 2-Designs as Stationary Points of Many-Body Systems
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ORIGINAL PAPER
Spherical 2-Designs as Stationary Points of Many-Body Systems Makoto Miura1
•
Kazushi Ueda2
Received: 20 June 2020 / Revised: 10 November 2020 / Accepted: 13 November 2020 Springer Japan KK, part of Springer Nature 2020
Abstract We show that spherical 2-designs give stationary points of the classical mechanical system of particles in a Euclidean space interacting by a double-well potential. Keywords Spherical design Double-well potential Euclidean design
1 Introduction Spherical designs are introduced in [2] as ‘‘good’’ configurations of points on a sphere; for a positive integer t, a finite subset fx1 ; . . .; xn g of an ðm 1Þ-dimensional sphere Sm1 is a spherical t-design if for any polynomial f(x) in m variables of degree at most t, one has Z n 1 1X f ðnÞdn ¼ f ðxi Þ; ð1:1Þ volðSm1 Þ Sm1 n i¼1 where dn is the standard measure on Sm1 and volðSm1 Þ is the volume of Sm1 . Besides practical application for numerical integration, spherical designs are important objects in algebraic combinatorics, with connections to various branches of mathematics, as explained, e.g., in [1]. In this paper, we give an interpretation of spherical 2-designs as stationary points of many-body systems in classical mechanics.
& Makoto Miura [email protected] Kazushi Ueda [email protected] 1
Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
2
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguroku, Tokyo 153-8914, Japan
123
Graphs and Combinatorics
We consider the system of n particles of the same type in the Euclidean space Rm whose 2-body potential is given by the spherically symmetric double-well potential ð1 d2 Þ2 ; where d is the distance between the particles. We describe the configuration of particles by an m n matrix X ¼ ðxui Þu;i ¼ ðx1 x2 xn Þ 2 Rmn ;
ð1:2Þ
where xi ¼ ðx1i xmi ÞT 2 Rm is the position of the i-th particle, so that the total potential energy is given by X 2 2 VðXÞ ¼ 1 xi xj : ð1:3Þ 1 i\j n
A configuration X of points on a sphere is a spherical 1-design if and only if the center of gravity is the origin; n X
xj ¼ 0:
ð1:4Þ
j¼1
It is a spherical 2-design if and only if, in addition to (1.4), the covariance matrix 1 C :¼ XX T n
ð1:5Þ
is proportional to the identity matrix; C ¼ vIm :
ð1:6Þ
The constant v of proportionality is determined by taking the trace of both sides of (1.6) as v¼
r2 ; m
ð1:7Þ
where r is the radius of the sphere. The main result in this paper is the following: qffiffiffiffiffiffiffiffiffiffiffi m is a stationary point of the Theorem 1.1 A spherical 2-design of radius 2ðmþ1Þ energy (1.3). It is known that, if n [ m, a spherical 2-design exists unless both n and m are odd and m ¼ 1 or m ¼ n 2 [3]. This ensures that Theorem 1.1 is not vacuous. We remark that, for any t and m, there exists a spherical t-design for sufficiently large n, as shown by [6]. The proof of Theorem 1.1 is given in Sect. 2, which is sh
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