Statistics of energy partitions for many-particle systems in arbitrary dimension

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Funct. Anal. Other Math. (210) manuscript No.(will be inserted by the editor)

Statistics of energy partitions for many-particle systemsinarbitrary dimension

Vincenzo Aquilanti · Andrea Lombardi ·Mikhail B. Sevryuk

Received: 27 November 2012 / Accepted: 4 December 2012

Abstract In some previous articles, we defined several partitions of the total kinetic energyT of a system ofN classical particles inRd into components corresponding to various modesof motion. In the present paper, we propose formulas for the mean values of these compo-nents in the normalizationT = 1 (for anyd andN) under the assumption that the masses ofall the particles are equal. These formulas are proven at the“physical level” of rigor and nu-merically confirmed for planar systems (d = 2) at 36N6 100. The case where the massesof the particles are chosen at random is also considered. Thepaper complements our articleof 2008 [Russian J Phys Chem B 2(6):947–963] where similar numerical experiments werecarried out for spatial systems (d = 3) at 36 N6 100.

Keywords Multidimensional systems of classical particles· Instantaneous phase-spaceinvariants· Kinetic energy partitions· Formulas for the mean values· Hyperangularmomenta

Mathematics Subject Classification (2010)53A17 · 93C25· 70G10· 70B99

1 Introduction

The integral characteristic of motion in a system of classical particles is the total kineticenergyT. However, for quite different types of motion, the value ofT can obviously bethe same. For instance, consider a system of two particles with the fixed center-of-massand letr = r(t) be the vector connecting the particles. Depending on the forces and initialconditions, this two-particle system canvibrate (if the direction ofr is not changing) or

Vincenzo AquilantiDipartimento di Chimica, Universita degli Studi di Perugia, Perugia (Italy)E-mail: [email protected]

Andrea LombardiDipartimento di Chimica, Universita degli Studi di Perugia, Perugia (Italy)E-mail: [email protected]

Mikhail B. Sevryuk (B)V.L. Tal′roze Institute of Energy Problems of Chemical Physics of theRussia Academy of Sciences, Moscow(Russia)E-mail: [email protected]

http://arxiv.org/abs/1301.3040v1

2 V. Aquilanti, A. Lombardi, M.B. Sevryuk

rotate (if the length ofr is not changing), and the kinetic energy in both the cases mayattain any positive value. However, these two kinds of motion are in fact extremes: a typicalmotion is a mixture of vibrations and rotations. Of course, in the case of two particles, thetotal kinetic energyT can be straightforwardly and naturally decomposed as the sum of twoterms corresponding to vibrations and rotations. Indeed, let r = dr/dt = r‖+ r⊥ wherer‖is the component parallel tor andr⊥ is the component orthogonal tor . Then

T =m1m2

2(m1+m2)|r |2 = Tvib +Trot,

whereTvib =

m1m2

2(m1+m2)

∣∣r‖∣∣2 = T cos2 θ

is the vibrational energy and

Trot =m1m2

2(m1+m2)

∣∣r⊥∣∣2 = T sin2 θ

is the rotational energy (herem1, m2 are the masses of the particles andθ is the anglebetweenr andr ).

In the case of three or more particles, there are much more kinds of motion: one hasto distinguish rotations of the system in question (the “cluster” or “ aggregate”) as a whole,changes in the principal moments of inertia, various rearrangements of particles in the clus-ter, etc. In this situation, it becomes a rather non-trivialand ambiguous task to define thecomponents ofT corresponding to variousmodesof the motion. This problem has beendiscussed in the literature for decades and several approaches to kinetic energy partitioninghave been proposed; see e.g. the well known papers [1,2,3,4], the recent studies by Marsdenand coworkers [5,6], and references therein.

The importance of exploring the contributions of various motion modes to the total ki-netic energyT stems, to a large extent, from the fact that such energy components may mostprobably be used as effectiveglobal indicatorsof dynamical features and critical phenomena(e.g. phase transitions) in classical clusters. If these energy terms are defined in a sufficiently“symmetric” and “invariant” manner and can be computed automatically and fast from thecoordinates and velocities of the particles, then one expects to be able to straightforwardlydetect structural metamorphoses in the cluster by observing abrupt changes in the way thetotal kinetic energy is distributed among the modes. In the case of large clusters, this is cru-cial for applications because it is much easier to trace a fewindicators than to analyze a hugecollection of data pertaining to all the particles.

Starting in 2002, we have published a series of papers [7,8,9,10,11] where we pro-posed and preliminarily tested a number of such global indicators on the basis of the so-calledhyperspherical approachto cluster dynamics. In particular, several partitions of thetotal kinetic energyT with various amazing features were defined in [9] for particles in theconventional three-dimensional space and in [10] for particles in the Euclidean space of anarbitrary dimensiond. Moreover, our long paper [10] contains alsorigorous mathematicalproofsof many properties (including the invariance under certaingroup actions) of the termsof these partitions. A refinement of one of the partitions wasperformed in [11] and four newterms were introduced. Some perspectives of the partitionsin question are discussed fromthe general viewpoint of the methods of molecular dynamics (with the particles being atomsor ions) in the short reviews [12,13,14]. Applications include studies of such phenomenaand processes in chemical physics as phase transitions in small neutral argon clusters Arn

with n= 3,13,38,55 [8,15,16], dynamics and thermodynamics of small ionic argon clusters

Statistics of energy partitions in arbitrary dimension 3

Ar+n with n = 3,6,9 [17], the prototypical exchange reaction F+H2 → H+HF of atomicfluorine and molecular hydrogen [15], and ultrafast relaxation dynamics of krypton atomicmatrices doped with a nitrogen monoxide molecule [18,19] (some of these applications aresurveyed in [14]). These studies have confirmed the usefulness of the kinetic energy parti-tions for examining dynamics of classical nanoaggregates.

However, neither the theory developed in the articles [8,9,10] nor the applications con-sidered in [8,15,16,17,18,19] allow one to conclude how theenergy partitions defined in[9,10] look like for “typical” systems (roughly speaking, for a random choice of the coordi-nates and velocities of the particles) or how the statisticsof the terms depends on the numberN of the particles and on their masses. The statistics of the kinetic energy components forsystems in the conventional spaceR3 was studied numerically in our paper [11] in the range36N6 100 for two extreme situations: in the case where all the particles have equal massesand in the case where the masses are chosen at random. The mainobservation of [11] is thatin the situation of equal masses, the mean values of almost all the terms of the partitions inthe normalizationT = 1 arevery simple(in fact, linear fractional forN> 4) functions ofN.For some of the terms, the paper [11] proposed also generalizations of the formulas for themean values to an arbitrary dimensiond of the space.

The aim of the present article is fourfold. First, we rigorously prove the properties of thenew energy termsEout

1 , Eout2 , Ein

1 , Ein2 announced in our previous paper [11]. Second, in the

situation of equal masses, we suggest formulas for the mean values ofall the energy com-ponents (except for the so-called unbounded ones) forN particles inRd with anarbitrary d:the mean values turn out to be simple rational functions ofd, N, and min(d,N−1). Third,we prove these formulas at the “physical level” of rigor. Fourth, we examine numerically thestatistics of all the energy terms, both in the situation of equal masses and in the situation ofrandom masses, forN particleson the planeR2 in the same range 36 N 6 100 as in [11].In the situation of equal masses, our numerical experimentsdo confirm the formulas for themean values of the energy terms.

It is worthwhile to note that two-dimensional physics is a well developed and rapidlyprogressing field of science, see e.g. the works [20,21,22,23,24,25,26,27,28,29] and ref-erences therein (by the way, the preprint [24] contains 680 references). Of course, here thewords “two-dimensional” can be used either in the mathematical sense or in the sense ofmonolayer films. In particular, the Nobel Prize in Physics of2010 was awarded jointly toAndre K. Geim and Konstantin S. Novoselov “for groundbreaking experiments regardingthe two-dimensional material graphene” [26,27,30]. Moreover, a paradigm was recentlyproposed in which theactual space-time is a fundamentally(1+1)-dimensional universebut it is “wrapped up” in such a way that it appears higher dimensional (say, 2+1 and 3+1)at larger distances [31,32,33].

The paper is organized as follows. Some fundamental theoretical concepts related to ourapproach to kinetic energy partitions are recalled in Sections 2 and 3 while the partitionsthemselves are defined in Section 4. Almost all the material of Sections 2–4 is in fact con-tained in our articles [9,10] and is reproduced here for the reader’s convenience. The newenergy termsEout

1 , Eout2 , Ein

1 , Ein2 of [11] are defined and explored in Section 5. The formulas

for the mean values of the bounded energy components forN particles of equal masses inR

d are proposed in Section 6 and substantiated in Section 7. Thenumerical experiments ford = 2 and their results are described in Section 8. Concluding remarks follow in Section 9.


2 Instantaneous phase-space invariants

Consider a system ofN classical particles in the Euclidean spaceRd (d,N ∈N= {1; 2; . . .})with massesm1,m2, . . . ,mN. Let r1 = r1(t), r2 = r2(t), . . . , rN = rN(t) be the radii vectorsof these particles with respect to the origin. Introduce thenotation

M =N

∑α=1

mα , qα = (mα/M)1/2rα

(16 α 6 N). In terms of the total massM of the system and themass-scaledradii vectorsqα , the total kinetic energyT and the total angular momentumJ of this system are expressedin an especially simple way:

T =12

N

∑α=1

mα |rα |2 =

M2

N

∑α=1

|qα |2, (1)

J2 = ∑16i< j6d

J2i j , Ji j =

N

∑α=1

mα(r i;α r j;α − r j;α r i;α) = MN

∑α=1

(qi;α q j;α −q j;α qi;α) (2)

(16 i < j 6 d). Herer i;α andqi;α denote respectively theith components of the vectorsrα and qα and, as usual, the dot over a letter means the time derivative. For d = 3, theformulas (1) and (2) give the conventional total kinetic energy and total angular momentumof a system ofN particles. Thus, the mass-scaled radii vectorsqα(t) provide a more adequatedescription of the current state of a system of classical particles than the radii vectorsrα(t)themselves.

Definition 1 The position matrixof a system ofN classical particles inRd is the d×Nmatrix Z whose columns are the mass-scaled radii vectorsq1,q2, . . . ,qN.

Of course, the position matrix at any given time instantt depends on the Cartesian co-ordinate frame chosen. The choice of another coordinate frame (with the same origin) isdescribed by a transformationZ(t) RZ(t) with an orthogonald× d matrix R∈ O(d).On the other hand, according to the general duality concept for thephysical spaceand theabstract “kinematic space” [9,10,11] (which are respectivelyRd andRN in our case), onecan also consider transformations of the formZ(t) Z(t)Q∗ with orthogonalN×N matri-cesQ ∈ O(N) where the asterisk designates transposing. Such transformations correspondto changes in the “type” of the coordinate frame. For instance, the passage from Cartesiancoordinates to Jacobi or Radau–Smith coordinate frames (see e.g. the papers [4,10,34,35]and references therein) is equivalent to a multiplication of the position matrix from the rightby a suitable fixed orthogonal matrix [10].

Example 1 Suppose that theN > 2 particles in question move without external forces andtheir center-of-mass coincides with the origin. Then in anyCartesian coordinate frame

N

∑α=1

m1/2α qα = M−1/2

N

∑α=1

mα rα ≡ 0.

Choose an arbitrary matrixQ∈ O(N) with the entries in the last row equal to

QNα = (mα/M)1/2, 16 α 6 N.

Then thelast columnof the matrixZ(t)Q∗ is zero at any time momentt (and one maytherefore regard the “kinematic space” to beR

N−1).


We are interested in characterizing systems of classical particles inRd by various quan-tities that are determined, at any time instantt0, by the positionsZ(t0) and velocitiesZ(t0)of the particles1 at this time momentt0 only2 (rather than by the whole trajectory

{Z(t)

∣∣

tbegin6 t 6 tend}

) and are invariant under orthogonal coordinate transformations3 both in thephysical space and in the “kinematic space”. Being inspiredby the discussion above and, inparticular, by Example 1, one arrives at the following mathematical model.

Let d,n ∈ N. Consider the space(Rd×n)2 of the pairs of real d×n matrices. We willwrite down these pairs as(Z, Z). On the space(Rd×n)2, there acts the group O(d)×O(n):

(R,Q)(Z, Z) = (RZQ∗,RZQ∗), R∈ O(d), Q∈ O(n). (3)

Definition 2 An instantaneous phase-space invariantof systems of classical particles inRd

is a collection of functions

fd,n : Md,n×Rd×n → R, Md,n ⊂ R

d×n, n> n0

possessing the following two properties.First, the definition domainsMd,n ×R

d×n of the functionsfd,n and the functionsfd,nthemselves are invariant under the action (3) of the group O(d)×O(n):

Z ∈Md,n =⇒ RZQ∗ ∈Md,n & fd,n(RZQ∗,RZQ∗) = fd,n(Z, Z)

for anyZ ∈ Rd×n (n> n0), R∈ O(d), andQ∈ O(n).

Second, the value of the functionfd,n remains the same if one augments both the matri-cesZ andZ by the(n+1)th column equal to zero:

Z ∈Md,n =⇒ (Z 0) ∈Md,n+1 & fd,n+1((Z 0),(Z 0)

)= fd,n(Z, Z)

for anyZ ∈ Rd×n (n> n0).

Most of the instantaneous phase-space invariants we will deal with will depend on theparameterM (the total mass of the system of particles).

Example 2 Let the center-of-mass of a system ofN > 2 particles inRd coincide with theorigin. As was explained in Example 1, one can choose a (non-Cartesian) coordinate frameFin which the last column of the position matrixZ(t) of such a system is identically zero. Thefirst N−1 columns ofZ(t) constitute thed× (N−1) matrix Zreduced(t). Any instantaneousphase-space invariantf of the system in question can be calculated either using thed×Nposition matrices in the initial Cartesian coordinate frame (n= N) or using thed× (N−1)reducedposition matricesZreducedin the coordinate frameF (n= N−1).

In the sequel, it will be convenient to introduce the notation

m= min(d,n).

As is very well known (see e.g. the manuals [36,37,38,39]), any d×n matrixZ ∈Rd×n can

be decomposed as the product of three matrices

Z = Dϒ X∗, D ∈ O(d), X ∈ O(n), (4)

1 Whence the words “phase-space” in Definition 2 below.2 Whence the word “instantaneous” in Definition 2 below.3 Whence the word “invariant” in Definition 2 below.


where all the entries of thed× n matrix ϒ are zeroes with the possible exception of thediagonal entries:

ϒ11 = ξ1, ϒ22 = ξ2, . . . , ϒmm = ξm, ξ1 > ξ2 > · · ·> ξm > 0.

The representation (4) is called thesingular value decomposition(SVD) of the matrixZ.The numbersξ1,ξ2, . . . ,ξm are called thesingular valuesof the matrixZ and are determineduniquely byZ although the orthogonal factorsD andX in the equality (4) arenot. If n> dthen thed singular values ofZ are the square roots of the eigenvalues of the symmetricd×dmatrix ZZ∗. If n6 d then then singular values ofZ are the square roots of the eigenvaluesof the symmetricn×n matrix Z∗Z.

It is clear that each singular value ofZ is an instantaneous phase-space invariant; to bemore precise, for each 16 i 6 d, the ith singular valueξi : Rd×n → R is an instantaneousphase-space invariant forn> i. More generally, any function of the singular values ofZis an instantaneous phase-space invariant, and conversely, any instantaneous phase-spaceinvariant independent ofZ is a function of the singular values ofZ.

Along with the SVD of a matrixZ, it is often expedient to consider asigned singu-lar value decomposition(signed SVD) [10,40]. A signed SVD of ad× n matrix Z is arepresentation (4) where again all the entries of thed×n matrixϒ are zeroes with the pos-sible exception of the diagonal entriesϒ11,ϒ22, . . . ,ϒmm, but it is no longer assumed thatϒ11>ϒ22> · · · >ϒmm > 0. Of course, the numbers|ϒ11|, |ϒ22|, . . . , |ϒmm| constitute in thiscase an unordered collection of the singular values ofZ. The differentiability properties of asigned SVD are usuallybetterthan those of the SVD (see the papers [10,40] and referencestherein).

Recall that the standardFrobenius inner producton the spaceRd×n of reald×n matricesis defined by the formula

〈Za,Zb〉 = Tr[Za(Zb)∗

]=

n

∑α=1

d

∑i=1

Zaiα Zb

iα , (5)

where Tr denotes the trace of a square matrix. The corresponding matrix norm‖·‖ given by

‖Z‖2 = 〈Z,Z〉= Tr(ZZ∗) =n

∑α=1

d

∑i=1

Z2iα =

m

∑σ=1

ξ 2σ

is called theFrobenius norm[36,37,39], theEuclidean norm[37,38], thel2-norm [37], theSchur norm[37], theHilbert–Schmidt norm[37], or thespherical norm[38].

In the context of the pairs(Z, Z) ∈ (Rd×n)2, the instantaneous phase-space invariant

ρ = ‖Z‖

is called thehyperradius(of the system of particles whose position matrix isZ).

Remark 1 In the sequel, we will use the following notation. Given a pair (Z, Z) ∈ (Rd×n)2

and a certain object (number, vector, etc.)X = X(Z) dependent onZ, we will defineX =X(Z, Z) as

X=ddt

X(Z(t)

)∣∣∣∣t=0

,

whereZ : (−ε ,ε)→ Rd×n (0< ε ≪ 1) is an arbitrary matrix-valued function such that

Z(0) = Z,dZ(t)

dt

∣∣∣∣t=0

= Z.


In all the cases below,X will be well defined for all the pairs(Z, Z) with the possible ex-ception of matricesZ lying in a set of positive codimension. For suchZ, one will always beable to define the resulting quantities we will be interestedin by continuity.

3 Hyperangular momenta

Instantaneous phase-space invariants dependent onZ are exemplified by the so-calledhyper-angular momentaof a system of particles (with respect to the origin), namely, thephysicalangular momentum J, the kinematic angular momentum Kdual to J, the grand angularmomentumΛ , and thesingular angular momentum L[8,9,10,11,13]. These non-negativequantities are defined by the formulas

J2 = ∑16i< j6d

J2i j , Ji j = M

n

∑α=1

(Ziα Z jα −Z jα Ziα), (6)

K2 = ∑16α<β6n

K2αβ , Kαβ = M

d

∑i=1

(Ziα Ziβ −Ziβ Ziα), (7)

Λ2 = M2 ∑16i, j6d

16α ,β6ni< j or i= j & α<β

(Ziα Z jβ −Z jβ Ziα)2, (8)

L2 = ∑16σ<τ6m

L2στ , Lστ = M(ξσ ξτ −ξτ ξσ ) (9)

(ξ1,ξ2, . . . ,ξm being the singular values ofZ and their derivativesξσ being defined in Re-mark 1). Of course, the formula (6) coincides with (2) ifZ is the position matrix in a Carte-sian coordinate frame andn= N. In our previous papers [8,9,10,13], we denoted the singu-lar angular momentum byLξ .

The correct definition (7) of the kinematic angular momentumK was first given in ourarticle [9] in the particular cased = 3 although the words “kinematic angular momentum”themselves were used earlier [7,8]. The grand angular momentumΛ was first introduced bySmith [2], also in the particular dimensiond= 3 (and mainly for systems of three particles).The singular angular momentumL was first defined in the paper [8], again in the particu-lar cased = 3 only. For an arbitrary dimensiond of the physical space, the hyperangularmomentaJ, K, Λ , andL were introduced in our article [10].

Theorem 1 All the four hyperangular momenta J, K,Λ , and L are instantaneous phase-space invariants.

Theorem 2 The hyperangular momenta J, K, andΛ can be alternatively computed as

J2 = M2n

∑α ,β=1

[

Γ (1)αβ Γ (3)

αβ −Γ (2)αβ Γ (2)

βα

]

, (10)

Γ (1)αβ =

d

∑i=1

Ziα Ziβ , Γ (2)αβ =

d

∑i=1


d

∑i=1

Ziα Ziβ ,

K2 = M2d

∑i, j=1

[

∆ (1)i j ∆ (3)

i j −∆ (2)i j ∆ (2)

ji

]

, (11)


∆ (1)i j =

n

∑α=1

ZiαZ jα , ∆ (2)i j =

n

∑α=1

Ziα Z jα , ∆ (3)i j =

n

∑α=1

Ziα Z jα ,

Λ2 = M2(

‖Z‖2‖Z‖2−〈Z, Z〉2)

. (12)

There hold the inequalities J2+L2 6Λ2 and K2+L2 6Λ2.

Remark 2 One of the important consequences of Theorem 2 is as follows.The formulas(6)–(9) imply that the numbers of components of the hyperangular momentaJ, K, Λ , andLare equal respectively tod(d−1)/2, n(n−1)/2, dn(dn−1)/2, andm(m−1)/2. However,the alternative formulas (11) and (12) show thatK andΛ can be calculated (likeJ) using nogreater thanCdn operations where the constantCd depends on the dimensiond only.

4 Energy partitions

The instantaneous phase-space invariant

T =M2‖Z‖2 (13)

is called thetotal kinetic energy(of the system of particles described by the position matrixZ and its time derivativeZ). Of course, the formula (13) coincides with (1) ifZ is the timederivative of the position matrix in a Cartesian coordinateframe andn= N.

The main subject of this article is statistical properties of the terms of various partitionsof the total kinetic energyT (almost all of these partitions were introduced in our paper[9]for d = 3 and in the subsequent paper [10] for an arbitrary dimensiond of the physicalspace). In order to define the partitions in question, one hasto make oneself more acquaintedwith the action (3) of the group O(d)×O(n) [10].

Given ad×n matrix Z, consider the manifolds of all the matrices of the form

{RZ∣∣R∈ SO(d)

},

{ZQ∣∣Q∈ SO(n)

},

{RZQ

∣∣ R∈ SO(d), Q∈ SO(n)

}.

Denote respectively the tangent spaces to these manifolds at point Z by

Πleft(Z) ={

Z+RZ∣∣ R ∈ so(d)

},

Πright(Z) ={

Z+ZQ∣∣ Q ∈ so(n)

},

Π (Z) ={

Z+RZ+ZQ∣∣R ∈ so(d), Q ∈ so(n)

}

[recall thatso(d) andso(n) are the spaces of skew-symmetricd×d andn×n matrices, re-spectively]. The multidimensional planesΠleft(Z), Πright(Z), andΠ (Z) are affine subspacesof the spaceRd×n of d×n matrices.

Theorem 3 The intersection of the spacesΠleft(Z) andΠright(Z) consists of the only matrixZ:

Πleft(Z)∩Πright(Z) = {Z} ⇐⇒ dimΠleft(Z)+dimΠright(Z) = dimΠ (Z)

⇐⇒ Π (Z)−Z =[Πleft(Z)−Z

]⊕[Πright(Z)−Z

]

if and only if all thepositivesingular values of the matrix Z are pairwise distinct.


Given now ad×n matrix Z, denote byZleft, Zright, andZrot theorthogonal projectionsofZ onto the spacesΠleft(Z), Πright(Z), andΠ (Z), respectively, in the sense of the Frobeniusinner product (5). Let alsoZI = Z− Zrot denote the component ofZ orthogonal toΠ (Z).

Introduce the following non-negative quantities:

TΛ =Λ2

2Mρ2 called thegrand angular energy,

Tρ =Mρ2

2called thehyperradial energy,

Trot =M2‖Zrot‖2 called therotational energy,

T I =M2‖ZI‖2 called theinertial energy,

Tξ =L2

2Mρ2 called theshape energy,

TΛ , Tρ , andTξ being defined forρ > 0 only (concerningρ , see Remark 1).

Theorem 4 There hold the identities and inequalities

T = TΛ +Tρ , (14)

T = Trot+T I , (15)

Trot 6 TΛ 6 T, Tρ 6 T I 6 T, TΛ −Trot = T I −Tρ = Tξ 6min(TΛ ,TI ),

Tρ =M

2ρ2

(m

∑σ=1

ξσ ξσ

)2

, TI =M2

m

∑σ=1

ξ 2σ .

Definition 3 The equalityT = TΛ +Tρ is called theSmith decompositionof the total kineticenergyT. The equalityT = Trot +TI is called theorthogonal decompositionof the totalkinetic energyT.

The Smith decomposition (14) was deduced for the first time bySmith [2] in the partic-ular caseN = d = 3.

The identity (15) is in fact just the Pythagorean theorem: itfollows immediately fromthe relationsZ = Zrot+ ZI and〈Zrot, ZI 〉= 0. The identity (14) is an immediate consequenceof the equality (12). Indeed, sinceρ = ‖Z‖, one hasρρ = 〈Z, Z〉. Now

2TM

= ‖Z‖2 =‖Z‖2‖Z‖2−〈Z, Z〉2

ρ2 +〈Z, Z〉2

ρ2 =Λ2

M2ρ2 + ρ2 =2TΛM

+2Tρ

M.

Introduce now the non-negative quantities

Text =M2‖Zleft‖2 called theexternal energy,

T int =M2‖Zright‖2 called theinternal energy,

TJ =J2

2Mρ2 called theouter angular energy,

TK =K2

2Mρ2 called theinner angular energy


as well as the quantities

Tres= Trot−Text−T int called theresidual energy,

Tac= Trot−TJ −TK called theangular coupling energy,

TJ, TK , andTac being defined forρ > 0 only. The energiesTres andTac can be negative.

Definition 4 The equalityTrot = Text+T int +Tres (16)

is called theprojective partitionof the rotational energyTrot. The equality

Trot = TJ +TK +Tac (17)

is called thehyperspherical partitionof the rotational energyTrot.

Theorem 5 There hold the inequalities

TJ+Tξ 6 TΛ , TK +Tξ 6 TΛ ,

TJ 6 Text6 Trot, TK 6 T int 6 Trot, Tres6 Tac.

If d 6 2 or n= 1 then TJ = Text. Recall that the case n= 1 corresponds to the situationswhere either there is only one particle(N= 1) or the system consists of two particles(N= 2)whose center-of-mass is fixed and coincides with the origin.If d = 1 or n6 2 then TK = T int.Recall that the case n= 2 corresponds to the situations where either there are two particles(N = 2) or the system consists of three particles(N = 3) whose center-of-mass is fixed andcoincides with the origin.

Finally, suppose that all thepositivesingular values of the matrixZ are pairwise distinct(besides these positive singular values, the matrixZ is allowed to possess zero singular valueof arbitrary multiplicity). Then, according to Theorem 3, the componentZrot ∈

[Π (Z)−Z

]

of Z can be uniquely decomposed as

Zrot = Zout+ Zin, Zout ∈[Πleft(Z)−Z

], Zin ∈

[Πright(Z)−Z

].

Introduce now the non-negative quantities

Eout =M2‖Zout‖2 called thetangent(or singular) external energy,

Ein =M2‖Zin‖2 called thetangent(or singular) internal energy

as well as the quantity

Ecoupl= Trot−Eout−Ein called thetangent(or singular, or Coriolis) coupling energy

which can be negative. In our previous papers [9,10], the quantities Eout, Ein, andEcoupl

were called respectively theouter term, theinner term, and thecoupling. The energyEcoupl

was called the Coriolis coupling term in [5] (see also [6]).

Definition 5 The equalityTrot = Eout+Ein +Ecoupl (18)

is called thesingular value expansion(or justsingular expansion) of the rotational energyTrot.


The singular value expansion (18) is defined only if all the positive singular values ofthe matrixZ are pairwise distinct. However, this condition is met for generic matricesZ ∈Rd×n [10]. Some partitions of the total kinetic energyT essentially equivalent to the equalityT = Eout+Ein +Ecoupl+T I have been known long ago (see e.g. the papers [1,3]).

Theorem 6 Let Z= Dϒ X∗ be asignedSVD(4) of a d×n matrix Z. Suppose that all thethree derivativesD, ϒ , andX are well defined(see Remark 1). Then

D ∈ so(d), X ∈ so(n), Z = DϒX∗+Dϒ X∗+Dϒ X∗,

Zrot = Dϒ X∗+Dϒ X∗, ZI = Dϒ X∗,

(Z+ DϒX∗) ∈ Πleft(Z), (Z+Dϒ X∗) ∈ Πright(Z).

In particular, if all the positive singular values of the matrix Z are pairwise distinct then

Zout = Dϒ X∗, Zin = Dϒ X∗.

Theorem 7 All the 14 energy quantities TΛ , Tρ , Trot, TI , Tξ , Text, Tint, TJ, TK , Tres, Tac,Eout, Ein, and Ecoupl introduced in this section are instantaneous phase-space invariants.

For a fixed total kinetic energyT, the energiesTΛ , Tρ , Trot, T I , Tξ , Text, T int, TJ, TK ,Tres, andTac [the terms of the Smith decomposition (14), the orthogonal decomposition (15),the projective partition (16), and the hyperspherical partition (17)] areboundedaccording toTheorems 4 and 5:

06 TΛ 6 T, 06 Tρ 6 T, 06 Trot 6 T, 06 T I 6 T, 06 Tξ 6 T,

06 Text6 Trot, 06 T int 6 Trot, |Tres|6 Trot,

06 TJ 6 Trot, 06 TK 6 Trot, |Tac|6 Trot.

On the other hand, the termsEout > 0, Ein > 0, andEcoupl6 Trot of the singular valueexpansion (18) arenot: Eout, Ein, and−Ecoupl can be arbitrarily large for any fixed valueT > 0 [10,11]. That is why we call the equality (18) anexpansionrather than apartition anduse the letterE for its terms rather than the letterT.

Remark 3 In the physically most important case whereZ is the position matrix of a sys-tem of N particles (acluster) whose center-of-mass is fixed at the origin, the meaning ofall the termsTΛ , Tρ , Trot, T I , Tξ , Text, T int, TJ, TK , Tres, Tac, Eout, Ein, andEcoupl is dis-cussed in detail in our previous papers [9,10,13] (where also the names of these terms andthe words “orthogonal decomposition”, “projective partition”, “hyperspherical partition”,“singular value expansion” are justified). To be brief, the hyperradial energyTρ correspondsto the contribution (to the total kinetic energyT) of changes in the size of the cluster as awhole and the shape energyTξ , to the contribution of changes in the shape of the cluster.The inertial energyTI = Tρ +Tξ describes the contribution of all the changes in the singularvaluesξ1,ξ2, . . . ,ξω whereω = min(d,N− 1). The external energyText is the contribu-tion (to the total kinetic energyT) of rotations of the cluster inRd (about the origin) as awhole. These rotations leave the symmetricN×N matrixZ∗Z invariant. The internal energyT int is the contribution of so-called “kinematic rotations” [9,10,11,35] of the cluster in the“kinematic space”RN−1 as a whole. “Kinematic rotations” are the cluster rearrangementsthat leave the symmetricd×d matrixZZ∗ invariant. The rotational energyTrot correspondsto the contribution of rotations of both types, physical ones and “kinematic” ones, and theresidual energyTres, to the coupling between these two types of rotations. Finally, the grand


angular energyTΛ = Trot+Tξ describes the joint contribution of conventional rotations ofthe cluster, “kinematic rotations”, and “rotations” of thevector(ξ1,ξ2, . . . ,ξω) of the singu-lar values inRω . The physical meaning of the terms of the hyperspherical partition (17) andthe singular value expansion (18) is much less clear.

5 Refinement of the singular value expansion

The singular value expansion (18) of the rotational energyTrot can be partitioned further asfollows [11]. Suppose that among the singular valuesξ1,ξ2, . . . ,ξm of the matrixZ, thereare k 6 m positive numbers, and these numbers are pairwise distinct (which ensures theexistence of the singular value expansion):

ξ1 > ξ2 > · · ·> ξk > ξk+1 = · · ·= ξm = 0.

Then the symmetricd×d matrix ZZ∗ and the symmetricn×n matrix Z∗Z havek positivepairwise distinct eigenvaluesξ 2

1 ,ξ 22 , . . . ,ξ 2

k , whereas the remaining eigenvalues (if any) ofeach of these matrices are equal to zero. We will denote the standard inner product inRd

andRn by x · y and the Euclidean norm(x · x)1/2 of a vectorx by |x| (in fact, we alreadyused the notation|x| in Sections 1 and 2).

Denote byu1,u2, . . . ,uk the unit eigenvectors of the matrixZZ∗ corresponding to theeigenvaluesξ 2

1 ,ξ 22 , . . . ,ξ 2

k , respectively. These vectors are determined unambiguously up tomultiplication by−1. LetU0 be the zero subspace (of dimensiond−k) of the matrixZZ∗,i.e.,U0 = {u ∈ R

d | ZZ∗u = 0}. Denote byai the orthogonal projection of the derivativeui

of the vectorui (see Remark 1) ontoU0 in the sense of the standard inner productx ·y in Rd

(16 i 6 k). Introduce the non-negative quantities

Eout1 =

M2

k

∑i=1

ξ 2i

k

∑j=1

(u j · ui)2 called theunbounded component of the outer term,

Eout2 =

M2

k

∑i=1

ξ 2i |ai |

2 called thebounded component of the outer term.

Similarly, denote byv1,v2, . . . ,vk the unit eigenvectors of the matrixZ∗Z corresponding tothe eigenvaluesξ 2

1 ,ξ 22 , . . . ,ξ 2

k , respectively. These vectors are again determined unambigu-ously up to multiplication by−1. LetV0 be the zero subspace (of dimensionn− k) of thematrix Z∗Z, i.e.,V0 = {v ∈ R

n | Z∗Zv = 0}. Denote bybα the orthogonal projection of thederivativevα of the vectorvα ontoV0 in the sense of the standard inner productx · y in Rn

(16 α 6 k). Introduce the non-negative quantities

Ein1 =

M2

k

∑α=1

ξ 2α

k

∑β=1

(vβ · vα)2 called theunbounded component of the inner term,

Ein2 =

M2

k

∑α=1

ξ 2α |bα |

2 called thebounded component of the inner term.

It is clear that the energiesEout1 andEout

2 do not change if one multiplies some of the vectorsu1,u2, . . . ,uk by −1 and the energiesEin

1 andEin2 do not change if one multiplies some of

the vectorsv1,v2, . . . ,vk by−1.


Remark 4 It is obvious thatui · u j = −u j · ui andui · ui = 0 for any 16 i, j 6 k. Conse-quently,

k

∑i=1

ξ 2i

k

∑j=1

(u j · ui)2 =

12

k

∑i, j=1

(ξ 2i +ξ 2

j )(ui · u j)2 = ∑

16i< j6k

(ξ 2i +ξ 2

j )(ui · u j)2,

so that

Eout1 =

M2 ∑

16i< j6k

(ξ 2i +ξ 2

j )(ui · u j)2, (19)

and analogously

Ein1 =

M2 ∑

16α<β6k

(ξ 2α +ξ 2

β )(vα · vβ )2. (20)

For an arbitrary dimensiond of the physical space, the termsEout1 [in the form (19)],Ein

1[in the form (20)],Eout

2 , andEin2 were introduced in our previous paper [11]. However, in the

particular cased = 3, the energiesEin1 andEin

2 were defined in the literature earlier (usingother names). For instance, the formula (33) of the article [5] expresses the total kineticenergyT of a cluster ofN > 5 particles as the sum of five components. In our notation,those components areEout, Ecoupl, T I , Ein

1 , andEin2 .

Remark 5 Let Z = Dϒ X∗ be a (signed) SVD (4) of the matrixZ with

ϒ11 = ξ1, ϒ22 = ξ2, . . . , ϒkk = ξk, ϒk+1,k+1 = · · ·=ϒmm = 0.

Thenui can be chosen to be theith columnDi of the matrixD for each 16 i 6 k. Indeed,ZZ∗ = Dϒϒ ∗D∗. If ei denotes theith unit coordinate vector ofRd, then

ZZ∗Di = Dϒϒ ∗D∗Di = Dϒϒ ∗ei = ξ 2i Dei = ξ 2

i Di .

Similarly, vα can be chosen to be theα th columnXα of the matrixX for each 16 α 6 k.Indeed,Z∗Z = Xϒ ∗ϒ X∗. If eα denotes theα th unit coordinate vector ofRn, then

Z∗ZXα = Xϒ ∗ϒ X∗Xα = Xϒ ∗ϒ eα = ξ 2α Xeα = ξ 2

αXα .

Theorem 8 All the four additional energy quantities Eout1 , Eout

2 , Ein1 , and Ein

2 are instanta-neous phase-space invariants, and there hold the decompositions

Eout = Eout1 +Eout

2 , Ein = Ein1 +Ein

2 . (21)

Moreover, the inequalitiesEout

2 6 Trot, Ein2 6 Trot

are valid.

In our previous paper [11], this theorem was just announced,so we will prove it here.

Proof Examine the unbounded and bounded componentsEin1 and Ein

2 of the inner term.Consider a transformationZ′ =RZQ∗, Z′ =RZQ∗ with R∈ O(d), Q∈ O(n). Then(Z′)∗Z′ =QZ∗ZQ∗ and

V ′0 = QV0,

v′α = Qvα , v′α = Qvα , b′α = Qbα (16 α 6 k).


Consequently, the quantitiesEin1 andEin

2 remain unchanged after such a transformation. Ifone augments both the matricesZ andZ by the(n+1)th column equal to zero:Z′ = (Z 0),Z′ = (Z 0), then

R(n+1)×(n+1) ∋ (Z′)∗Z′ =

(Z∗Z 0

0 0

)

.

Thus,

V ′0 =

{(vw) ∈ R

n+1∣∣ v ∈V0, w∈ R

},

v′α = (vα 0), v′α = (vα 0), b′α = (bα 0) (16 α 6 k),

and the quantitiesEin1 andEin

2 again remain unchanged. Thus, these quantities are instanta-neous phase-space invariants.

The relationsEin1 +Ein

2 = Ein andEin2 6 Trot will be verified only in the case where the

pair (Z, Z) admits a signed SVDZ = Dϒ X∗ of the matrixZ with ϒ as in Remark 5 andwith thewell definedderivativesD, ϒ , andX (see Remark 1). The opposite case is highlydegenerate and of no practical importance; in fact, the equality Ein

1 + Ein2 = Ein and the

inequalityEin2 6 Trot will follow for the exceptional pairs(Z, Z) by continuity.

According to Remark 5, one can setvα = Xα and thereforevα = Xα (16 α 6 k), whereXα denotes theα th column of the matrixX andXα denotes theα th column of the matrixX.According to Theorem 6,Zin = Dϒ X∗ andZrot = Dϒ X∗+Dϒ X∗. SinceEin

1 , Ein2 , Ein, and

Trot are instantaneous phase-space invariants, we can assume without loss of generality thatD is the identityd×d matrix andX is the identityn×n matrix. Then

Zin =ϒ X∗, Zrot = Dϒ +ϒ X∗,

vα = eα is theα th unit coordinate vector ofRn,

Z∗Z = diag(ξ 2

1 ,ξ 22 , . . . ,ξ 2

k , 0,0, . . . ,0︸︷︷︸

n−k

),

andV0 is the subspace of all the vectorsv of the form

v =(

0,0, . . . ,0︸︷︷︸

k

,w1,w2, . . . ,wn−k).

Thus,vβ · vα = Xβα , bα =

(0,0, . . . ,0︸︷︷︸

k

, Xk+1,α , Xk+2,α , . . . , Xnα)

(16 α ,β 6 k). Consequently,

Ein1 =

M2

k

∑α=1

ξ 2α

k

∑β=1

X2βα , Ein

2 =M2

k

∑α=1

ξ 2α

n

∑β=k+1

X2βα , (22)

and

Ein1 +Ein

2 =M2

k

∑α=1

ξ 2α

n

∑β=1

X2βα =

M2‖ϒ X∗‖2 =

M2‖Zin‖2 = Ein.

Moreover, the lastn−k columns of thed×n matrix Dϒ vanish. Therefore,

Trot =M2‖Zrot‖2 =

M2‖Dϒ +ϒ X∗‖2


is no less thanM/2 multiplied by the sum of the squares of all the entries of thelastn−kcolumns of thed×n matrixϒ X∗, i.e., is no less thanEin

2 [see (22)].The unbounded and bounded componentsEout

1 andEout2 of the outer term can be treated

the same way. Note only that if one augments both the matricesZ and Z by the(n+1)thcolumn equal to zero:Z′ = (Z 0), Z′ = (Z 0), thenZ′(Z′)∗ = ZZ∗, U ′

0 = U0, andu′i = ui ,

u′i = ui , a′i = ai for each 16 i 6 k. �

However, it is rather difficult to reveal the physical meaning of the quantitiesEout1 , Eout

2 ,Ein

1 , andEin2 . Of course,Eout

1 andEin1 can be arbitrarily large for any fixed valueT > 0 (like

Eout andEin).

6 Equal masses and random masses

To study the statistical properties of various components (defined in Sections 4 and 5) ofthe total kinetic energyT of systems of classical particles inRd, one has to describe pre-cisely the sampling procedure for the coordinates and velocities of the particles. We willconsider systems ofN> 2 particles with the center-of-mass at the origin. As in our previouspaper [11], two situations will be dealt with:particles with equal massesandparticles withrandom masses.

Let wα andwα (16 α 6 N) be 2N independent random vectors, each being uniformlydistributed in the unit ball inRd centered at the origin. Now set

wcm =1N

N

∑α=1

wα , wcm =1N

N

∑α=1

wα

(the subscript “cm” is for “center-of-mass”). The massesmα of the particles in the situationof equalmasses are equal to 2/N:

m1 = m2 = · · ·= mN =2N, M =

N

∑α=1

mα = 2,

and one computes

gα = wα −wcm, gα = wα − wcm, 16 α 6 N.

In the situation ofrandommasses, one first chooses the massesmα of the particles accordingto the formula

mα = 2ηα

(N

∑β=1

ηβ

)−1

, 16 α 6 N, M =N

∑α=1

mα = 2,

whereη1,η2, . . . ,ηN are independent random variables uniformly distributed between 0 and1. The vectorsgα andgα are then computed as

gα = m−1/2α (wα −wcm), gα = m−1/2

α (wα − wcm), 16 α 6 N.

In both the situationsN

∑α=1

m1/2α gα =

N

∑α=1

m1/2α gα = 0. (23)


Finally, the mass-scaled radii vectorsqα and their time derivativesqα are calculated in boththe situations as

qα = C1gα , qα = C2gα , 16 α 6 N, (24)

where the positive factorsC1 andC2 are determined from the condition

N

∑α=1

|qα |2 =

N

∑α=1

|qα |2 = 1. (25)

In both the situationsN

∑α=1

m1/2α qα =

N

∑α=1

m1/2α qα = 0 (26)

according to (23), so that thed×N position matrixZ with columnsq1,q2, . . . ,qN and itstime derivativeZ with columnsq1, q2, . . . , qN do correspond to a system ofN particles withthe massesm1,m2, . . . ,mN and with the center-of-mass at the origin (see Example 1). More-over, the normalization (25) is equivalent to that‖Z‖ = ‖Z‖ = 1. Together withM = 2 thisgivesρ = T = 1.

Remark 6 A random pointw uniformly distributed in the unit ball inRd centered at theorigin can be generated asw = κ1/ds whereκ is a random variable uniformly distributedbetween 0 and 1 ands is a random point uniformly distributed on the unit sphereS

d−1 ⊂Rd

centered at the origin (κ andsbeing independent). There are many methods to chooses ford> 2, see e.g. the papers [41,42,43,44,45] and references therein. The most “elegant” (andprobably the best known but not the fastest) algorithm is theso-called Muller or Brown–Muller procedure [42] which runs as follows. Letχ1,χ2, . . . ,χd be independent randomvariablesnormallydistributed with zero mean and the same standard deviationς > 0. Thenthe point

s=(χ1

d,

χ2

d, . . . ,

χd

d

)

, d2 =d

∑i=1

χ2i (d> 0),

is uniformly distributed onSd−1. Indeed, the joint probability density function

1

ς d(2π)d/2exp

(

−d2

2ς 2

)

of χ1,χ2, . . . ,χd depends ond only.Of course, ford = 2 one can just set

s= (cosϕ , sinϕ),

whereϕ is uniformly distributed between 0 and 2π. Ford = 3 the standard spherical coor-dinates inR3 lead to the choice [11]

s=((1−h2)1/2 cosϕ , (1−h2)1/2 sinϕ , h

),

whereh andϕ are independent random variables uniformly distributed inthe intervals−16h6 1 and 06 ϕ 6 2π.

The following conjecture is the main statement of this paper.


Conjecture 1 Introduce the notation

ν = N−1, ω = min(d,ν).

For any d> 1 and N> 2, the mathematical expectationsE of theboundedterms TΛ , Tρ ,Trot, TI , Tξ , Text, Tint, Tres, TJ, TK , Tac, Eout

2 , and Ein2 of the Smith decomposition(14),

the orthogonal decomposition(15), the projective partition(16), the hyperspherical parti-tion (17), and the singular value expansion(18), (21) in the situation ofequalmasses aregiven by the formulas

ETΛ = 1−1

dν, ETρ =

1dν

, ETrot = 1−ωdν

, ET I =ωdν

, (27)

ETξ =ω −1

dν, (28)

EText =ω(2d−ω −1)

2dν, ET int =

ω(2ν −ω −1)2dν

, ETres= 0, (29)

ETJ =d−1dν

, ETK =ν −1dν

, ETac= 1−d+ν +ω −2

dν, (30)

EEout2 = 1−

ωd, EEin

2 = 1−ων. (31)

Since we use the normalizationT = 1, the formulas (27)–(31) give in fact the mathemat-ical expectations of the ratiosTΛ/T, Tρ/T, Trot/T, T I/T, Tξ/T, Text/T , T int/T, Tres/T,TJ/T, TK/T, Tac/T, Eout

2 /T, andEin2 /T . These formulas reflect vividly the duality of the

physical spaceRd and the “kinematic space”Rν (see Example 1). The expression for eachof the quantitiesETΛ , ETρ , ETrot, ET I , ETξ , ETres, andETac is symmetric with respect tod andν . If one interchangesd andν , the expressions forEText, ETJ, andEEout

2 turn intothose forET int, ETK, andEEin

2 , respectively, and vice versa.Note that all the four formulas (27), (28), and (30) for the mean valuesETΛ , ETξ , ETJ,

andETK of the energy termsH corresponding to the momentaΛ , L, J, andK have the form

EH =γ −1dν

,

where, roughly speaking,γ is the dimension of the space where the “rotation” responsiblefor the momentum in question takes place. This space is the matrix spaceRd×ν for Λ , thespace

{(ξ1,ξ2, . . . ,ξω)

}of the collections of singular values forL, the physical spaceRd

for J, and the “kinematic space”Rν for K.

Example 3 Consider a system of two particles (N = 2) of arbitrary massesm1 andm2 withthe center-of-mass at the origin ofRd. ForN = 2 it is not hard to obtain [10,11] that

T =m1m2

2M|r |2,

Tξ = T int = Tres= TK = Tac= Ein = Ein1 = Ein

2 = Eout1 = Ecoupl≡ 0,

Tρ = T I = T cos2 θ ,

TΛ = Trot = Text = TJ = Eout = Eout2 = T sin2 θ ,

wherer is the vector connecting the particles andθ is the angle betweenr and r (cf. Sec-tion 1). It is well known [11] that

Ecos2 θ = 1/d, Esin2 θ = (d−1)/d (32)


for the uniform distribution of the directions of the vectors r andr in Rd. One thus concludesthat forN = 2 andT = 1

ETξ = ET int = ETres= ETK = ETac= EEin = EEin1 = EEin

2 = EEout1 = EEcoupl= 0,

ETρ = ETI = 1/d, (33)

ETΛ = ETrot = EText = ETJ = EEout = EEout2 = (d−1)/d (34)

independently of the massesm1 andm2. This agrees with the formulas (27)–(31) forω =ν = 1. Thus, forN = 2 Conjecture 1 is correct for anyd (and, moreover, the equality of themasses of the particles is irrelevant forN = 2).

The formula (27) forETrot andETI as well as the formula (29) forEText, ET int, andETres were proposed in our previous paper [11]. Moreover, the paper [11] contained alsothe equalities (27)–(31) forall the terms in theparticular dimensiond = 3. The formulas(27)–(31) ford= 3 were confirmed in the paper [11] for 36N6 100 by extensive numericalsimulation. Note that ford = 3, the equalityEEout

2 = 1−ω/d reduces to

EEout2 =

2/3 for N = 2,

1/3 for N = 3,

0 for N> 4.

The equalityEEout2 = 0 for d= 3 andN> 4 (independently of the distribution of the masses)

is obvious [see (35) below]. The numerical experiments of the paper [11] forEout2 were of

course carried out forN = 3 only.

Conjecture 2 The equalityETres= 0 holds for any“ reasonable” distribution of the massesof the particles(not only in the situation of equal masses). Moreover, Tres> 0 with proba-bility 1/2.

Conjecture 2 was confirmed in the numerical simulation of ourprevious paper [11] ford = 3 and 36 N 6 100 in the situations of equal and random masses. Some “theoretical”support for this conjecture was also presented in [11].

7 A “physical” proof of Conjecture 1

Consider a system ofN > 2 classical particles inRd with massesm1,m2, . . . ,mN and withthe center-of-mass at the origin. Such a system can be described either by thed×N positionmatrixZ (in a Cartesian coordinate frame) and its time derivativeZ with the columns subjectto the identities (26) or byd× (N−1) matricesZ andZ with arbitrary columns (see Ex-ample 2). We will follow the second approach and regardZ andZ as matrices of Frobeniusnorm 1 [cf. (25)] withd rows andN−1 = ν columns. Recall thatM = 2 according to theprocedures of Section 6.

The following lemma is well known [11] and almost obvious.

Lemma 1 Let a random pointx be uniformly distributed on the unit sphereSp+q−1 ⊂Rp+q

centered at the origin(p,q∈N), and letΦ be an arbitrary affine plane inRp+q of dimensionp. Denote byPr2Φ x the square of the length of the orthogonal projection ofx ontoΦ . Thenthe mathematical expectationEPr2Φ x of Pr2Φ x is equal to p/(p+q).


Proof Let (x1,x2, . . . ,xp+q) be a Cartesian coordinate frame inRp+q. Without loss of gen-erality, one can assume that the planeΦ is given by the equationsxp+1 = xp+2 = · · · =xp+q = 0. If x = (x1,x2, . . . ,xp+q) ∈ Sp+q−1 then Pr2Φ x = x2

1+x22+ · · ·+x2

p. But x21+x2

2 +

· · ·+x2p+q ≡ 1 andEx2

1 =Ex22 = · · ·=Ex2

p+q by symmetry reasons. Therefore,Ex21 = Ex2

2 =

· · ·= Ex2p+q = 1/(p+q) andEPr2Φ x = p/(p+q). �

By the way, the formula (32) forEcos2 θ is a particular case of Lemma 1 forp = 1,q= d−1.

The following reformulation of Lemma 1 is especially useful: if a random pointx =(x1,x2, . . . ,xp+q) is uniformly distributed on the unit sphereSp+q−1 centered at the origin,thenE

(x2

ι1 +x2ι2 + · · ·+x2

ιp

)= p/(p+q) for any fixed set of indices 16 ι1 < ι2 < · · ·< ιp6

p+q.All the ω = min(d,ν) singular values of a genericd× ν matrix Z are positive and

pairwise distinct, whence (see [10])

dimΠleft(Z) =ω(2d−ω −1)

2, dimΠright(Z) =

ω(2ν −ω −1)2

,

dimΠ (Z) = dimΠleft(Z)+dimΠright(Z) = dν −ω

[note thatω(d+ ν −ω) = ω max(d,ν) = dν ]. Let Z be a random matrix uniformly dis-tributed on the unit sphereSdν−1 ⊂ R

d×ν ∼= Rdν centered at the origin [observe that after

the identificationRd×ν ∼= Rdν , the Frobenius inner product (5) becomes the standard innerproduct]. SinceM = 2,

Text = ‖Zleft‖2, T int = ‖Zright‖2, Trot = ‖Zrot‖2.

On the other hand,Zleft, Zright, andZrot are the orthogonal projections ofZ onto the spacesΠleft(Z), Πright(Z), and Π (Z), respectively, in the sense of the Frobenius inner product.According to Lemma 1,

EText =ω(2d−ω −1)

2dν, ET int =

ω(2ν −ω −1)2dν

, ETrot = 1−ωdν

,

andETI = 1−ETrot =

ωdν

, ETres= ETrot−EText−ET int = 0.

We have verified the formulas (27) and (29) forETrot, ET I , EText, ET int, andETres follow-ing the reasoning in our previous paper [11].

Sinceρ = ‖Z‖, one hasρ = ‖Z‖−1〈Z, Z〉 = ‖Z‖cosϑ , whereϑ is the angle betweenZ and Z in the sense of the Frobenius inner product. Let us treatZ and Z as independentrandom matrices uniformly distributed on the unit sphere inR

d×ν centered at the origin. ForTρ = Mρ2/2= ρ2 = ‖Z‖2 cos2 ϑ = cos2 ϑ we therefore have

ETρ =1

dν

[see (32); the equality‖Z‖= 1 is in fact irrelevant here], and

ETΛ = 1−ETρ = 1−1

dν, ETξ = ETI −ETρ =

ω −1dν

.

We have verified the formulas (27) and (28) forETΛ , ETρ , andETξ .


To calculateETJ andETK , it is expedient to use the expressions (10) and (11) forJ2 andK2, respectively. SinceM = 2 andρ = 1,

ETJ = E

ν

∑α ,β=1

[

Γ (1)αβ Γ (3)

αβ −Γ (2)αβ Γ (2)

βα

]

,

where

Γ (1)αβ =

d

∑i=1


d

∑i=1


d

∑i=1

Ziα Ziβ .

Let againZ andZ be independent random matrices uniformly distributed on the unit spherein R

d×ν centered at the origin. Then

E

[

Γ (1)αβ Γ (3)

αβ

]

= E

[

Γ (2)αβ Γ (2)

βα

]

= 0

wheneverα 6= β . Indeed, if one changes the signs of all the entriesZiα , 16 i 6 d, the sums

Γ (1)αβ andΓ (2)

αβ would change their signs while the sumsΓ (2)βα andΓ (3)

αβ would remain thesame. Consequently,

ETJ = E

ν

∑α=1

{

Γ (1)αα Γ (3)

αα −[

Γ (2)αα

]2}

= E

ν

∑α=1

[

Γ (1)αα Γ (3)

αα sin2 θα

]

=ν

∑α=1

EΓ (1)αα EΓ (3)

αα Esin2 θα =νν2

d−1d

=d−1dν

,

whereθα denotes the angle between theα th column ofZ and theα th column ofZ in Rd

(16 α 6 ν). Here we have used the facts that for eachα , the three random variablesΓ (1)αα

(the square of the length of theα th column ofZ in Rd), Γ (3)αα (the square of the length of the

α th column ofZ in Rd), and sin2 θα are independent, and according to Lemma 1

EΓ (1)αα = EΓ (3)

αα =d

dν=

1ν, Esin2 θα =

d−1d

[see (32)]. Analogously,

ETK =ν −1dν

,

and

ETac= ETrot−ETJ −ETK = 1−d+ν +ω −2

dν.

We have verified the formulas (30) forETJ, ETK , andETac.Finally, considerEout

2 andEin2 . Generically, in the notation of Section 5,

k= d, U0 = 0, Eout2 = 0, Eout

1 = Eout for d6 ν ,

k= ν , V0 = 0, Ein2 = 0, Ein

1 = Ein for ν 6 d.(35)

Let ν > d and findEEin2 . Assume all thed singular valuesξ1,ξ2, . . . ,ξd of the matrixZ to be

positive and pairwise distinct (a generic setup). Then in the SVDZ = Dϒ X∗ of the matrixZ, all the three derivativesD, ϒ , andX are well defined, and

Z = DϒX∗+Dϒ X∗+Dϒ X∗ (36)


according to Theorem 6. SinceZ andZ are independent andEin2 is an instantaneous phase-

space invariant (Theorem 8), one may suppose thatD is the identityd×d matrix, X is theidentity ν × ν matrix, while Z is still a random matrix uniformly distributed on the unitsphere inRd×ν centered at the origin. The equality (36) takes the form

Z = Dϒ +ϒ +ϒ X∗. (37)

The lastν −d columns of the matricesDϒ andϒ vanish. Consequently, the equality (37)implies

Ziα = ξiXα i

for anyd+16α 6 ν and 16 i6 d. Taking into account thatM = 2, the second formula (22)[valid for identity matricesD andX] becomes

Ein2 =

d

∑i=1

ξ 2i

ν

∑α=d+1

X2α i =

d

∑i=1

ν

∑α=d+1

Z2iα .

According to Lemma 1,

EEin2 =

d(ν −d)dν

= 1−dν. (38)

Now observe that the formula (38) forν > d and the equalityEEin2 = 0 for ν 6 d can be

combined into the unified formula

EEin2 = 1−

ων

valid for anyd andν . Analogously,

EEout2 = 1−

ωd

for anyd andν . We have verified the formulas (31) forEEout2 andEEin

2 . �

Unfortunately, all these arguments are not mathematicallyrigorous because even in thesituation of equal masses, the procedure of generating the matricesZ andZ described in Sec-tion 6 doesnot give (after the passage fromRd×N toR

d×(N−1) =Rd×ν ) matrices uniformly

distributed on the unit sphere inRd×ν centered at the origin. This can be easily shown evenin the simplest cased = 1, N = 3. For these values ofd andN, the procedure of Section 6for equal masses (in the case ofZ for definiteness) runs as follows.

Let w1 = a, w2 = b, w3 = c be three independent random variables uniformly distributedbetween−1 and 1 (in the “unit segment” inR). One computes the numbers

g1 =2a−b−c

3, g2 =

2b−a−c3

, g3 =2c−a−b

3

and the vectorC(2a−b−c, 2b−a−c, 2c−a−b)

of length 1 [in the notation of (24),C = C2/3 > 0]. Now one has to choose an arbitrarymatrix

Q=

Q11 Q12 Q13

Q21 Q22 Q23

3−1/2 3−1/2 3−1/2

∈ O(3)


(see Example 1) and calculate the vector

C(2a−b−c, 2b−a−c, 2c−a−b)Q∗. (39)

The first two components of the vector (39) are

Z1 = C[(2a−b−c)Q11+(2b−a−c)Q12+(2c−a−b)Q13

],

Z2 = C[(2a−b−c)Q21+(2b−a−c)Q22+(2c−a−b)Q23

]

(Z21 + Z2

2 = 1), the third component is zero. Then(Z1, Z2) is the 1×2 matrix Z one dealswith.

Do there exist fixed numbersQ11, Q12, Q13, Q21, Q22, Q23 such that the point(Z1, Z2)is uniformlydistributed on the unit circleS1 ⊂R

2 centered at the origin? The answer to thisquestion is negative. Indeed, letλ > 0. The inequalities

06 Z16 λ Z2, (40)

that is

06 (2a−b−c)Q11+(2b−a−c)Q12+(2c−a−b)Q13

6 λ[(2a−b−c)Q21+(2b−a−c)Q22+(2c−a−b)Q23

],

−16 a6 1,

−16 b6 1,

−16 c6 1,

determine a certain polyhedron in the Euclidean spaceR3 with coordinatesa, b, c. The coor-

dinates of the vertices of this polyhedron and its volumeV are piecewise rational functionsof λ , Q11, Q12, Q13, Q21, Q22, Q23 with integer coefficients (see Remark 7 below). Conse-quently, the probabilityV/8 of the inequalities (40) is also a piecewise rational function ofλ , Q11, Q12, Q13, Q21, Q22, Q23 with integer coefficients [the denominator 8 inV/8 beingthe volume of the cube max

(|a|, |b|, |c|

)6 1]. On the other hand, if the point(Z1, Z2) were

uniformly distributed onS1, then the probability of (40) would be equal to12π arctanλ .

Remark 7 Consider a collection ofK affine hyperplanes

N

∑j=1

ai j x j = bi , 16 i 6K, K>N+1,

in RN ={(x1,x2, . . . ,xN)

}. Then the volume of any finite polytopeP bounded by these

hyperplanes is a piecewise rational function ofai j , bi with integer coefficients. Indeed, thecoordinates of the vertices of this polytope are (piecewise) rational functions ofai j , bi withinteger coefficients according to Cramer’s rule. Now triangulate the polytopeP into a setof N-dimensional simplices. The volume of any simplex with vertices (v1ι ,v2ι , . . . ,vNι),16 ι 6N+1, is equal to

±1N!

∣∣∣∣∣∣∣∣∣∣∣

1 1 · · · 1v11 v12 · · · v1,N+1v21 v22 · · · v2,N+1...

......

...vN1 vN2 · · · vN,N+1

∣∣∣∣∣∣∣∣∣∣∣

.


Thus, the volume ofP is a polynomial in the coordinates of its vertices with coefficientswhich become integers after multiplication byN!. Various formulas and algorithms for com-puting the volumes of polytopes in Euclidean spaces of arbitrary dimensions are presentedin the articles [46,47,48,49,50] and references therein.

A genuine proof of Conjecture 1 requires further studies beyond the present paper.

8 Numerical experiments on the plane

At previous stages of the project [10,11], we developed and implementedFortran codesfor computing all the hyperangular momentaJ, K, Λ , andL (defined in Section 3) andthe energy termsT, TΛ , Tρ , Trot, T I , Tξ , Text, T int, Tres, TJ, TK , Tac, Eout, Eout

1 , Eout2 , Ein,

Ein1 , Ein

2 , andEcoupl (defined in Sections 4 and 5) for the physically interesting dimensionsd = 2 andd = 3 of the ambient space. The input data for these codes are the entries ofthe d× n matricesZ and Z and the total massM. For fixed dimensiond, the number ofoperations required in the calculation grows linearly withthe numberN of particles (N = nor N = n+1), cf. Remark 2.

Using the codes prepared, we have verified Conjecture 1 (and,to some extent, Con-jecture 2) for systems of classical particles on the plane (d = 2) by numerical simulations.Within each of the two situations defined in Section 6 (particles with equal masses and par-ticles with random masses) and for each value ofN from 3 through 100 (i.e., for a total of98 values),L = 106 systems ofN particles on the Euclidean plane with the center-of-massat the origin were chosen using a random number generator according to the procedures de-scribed in Section 6. A random pointw uniformly distributed in the unit disc onR2 centeredat the origin was always generated as

w = κ1/2(cosϕ , sinϕ),

whereκ andϕ are independent random variables uniformly distributed inthe intervals 06κ 6 1 and 06 ϕ 6 2π. For each system, we calculated all the energy termsT = 1, TΛ , Tρ ,Trot, TI , Tξ , Text, T int, Tres, TJ, TK , Tac, Eout, Eout

1 , Eout2 , Ein, Ein

1 , Ein2 , andEcoupl defined in

Sections 4 and 5.Similar simulations were performed in our previous paper [11] for d = 3 with 105 sys-

tems for each value ofN from 3 through 100 in each of the two situations.As one expects, for all the systems onR2 in both the situations we foundTJ = Text

(see Theorem 5) andEout2 = 0, i.e.,Eout

1 = Eout [see (35)]. ForN = 3 and all the systems inboth the situations, we also foundTK = T int (see Theorem 5) andEin

2 = 0, i.e.,Ein1 = Ein

[see (35)].The energiesTres, Tac, andEcoupl can be both positive and negative [10]. We will denote

their positive and negative “components” as

Tres+ = max(Tres,0), Tres

− =−min(Tres,0), Tres= Tres+ −T res

− ,

T+ac = max(Tac,0), T−

ac =−min(Tac,0), Tac= T+ac −T−

ac,

Ecoupl+ = max(Ecoupl,0), Ecoupl

− =−min(Ecoupl,0), Ecoupl= Ecoupl+ −Ecoupl

− .

As was pointed out above, the quantitiesEout1 and Ein

1 (and, consequently,Eout, Ein,

−Ecoupl, andEcoupl− ) can be arbitrarily large for any fixed valueT > 0. In our simulations,

for any numberN of particles, we encountered systems for whichEout > 5.42592× 104


(for equal masses) orEout > 5.1008× 104 (for random masses); recall thatEout ≡ Eout1

in our calculations. The maximal values ofEout (over allN) we observed turned out to be2.34534×108 (for equal masses) and 2.07751×107 (for random masses). Similarly, for anynumberN of particles, there were systems for whichEin

1 > 5.41936×104 (for equal masses)or Ein

1 > 5.10724×104 (for random masses), and the maximal values ofEin1 over allN were

equal to 2.34534×108 (for equal masses) and 2.07787×107 (for random masses). Finally,for any numberN of particles, systems occurred for whichEcoupl6 −1.08453×105 (forequal masses) orEcoupl6 −1.0208×105 (for random masses), and the minimal values ofEcoupl over allN were−4.69068×108 (for equal masses) and−4.15538×107 (for randommasses). Moreover, in each of the two situations, the maximal values ofEout, Ein

1 , Ein,and−Ecoupl were attained at the same system—for which the “angle” between the spacesΠleft(Z) andΠright(Z) is very small (cf. [10]). Since the quantitiesEout, Ein

1 , Ein, andEcoupl−

are unbounded, we did not examine their statistics.

Remark 8 Of course, the large number of digits in the data above only reflects the particularset of numerical experiments. The same remark refers to similar data below where onlyseveral first digits are significant and informative.

On the other hand, the energiesTΛ , Trot, Tρ , T I , andTξ do not exceedT = 1 while the

termsText ≡ TJ, T int, Tres, Tres+ , Tres

− , TK , Tac, T+ac, T−

ac, Ein2 , andEcoupl

+ do not exceedTrot

according to Theorems 4, 5, and 8. For each of these quantities, we computed themeanvalues

H =1L

L

∑l=1

Hl ,

whereHl is the value of the energyH in question for thel th system for the givenN withinthe given situation (recall thatL= 106), and thesample variances

s2(H) =(H −H

)2= H2−

(H)2

(see e.g. the manuals [51,52,53,54] and references therein). Note that the sample variance isoften defined ass2

⋆(H) = Ls2(H)/(L−1) [53,54]. The mathematical expectations ofs2(H)and s2

⋆(H) are equal to(L− 1)(VarH)/L and VarH, respectively [51,52,53,54], whereVarH = E

[(H −EH)2

]= E(H2)− (EH)2 is the variance ofH. However, forL= 106, the

difference between the “biased” sample variances2(H) and the “unbiased” sample variances2⋆(H) is of course negligible.

The dependences ofH on N for various termsH and both the mass distributions con-sidered are presented in Figs. 1–4. Along the abscissa axis on each of these figures, the“physical” distancex between the left end point corresponding toN = 3 and the point cor-responding to a givenN is proportional to1

2 −1

N−1. In such a coordinate frame, any depen-dence

y=aν +b

cν, ν = N−1,

is represented by astraight line:

x=12−

1ν

⇐⇒ ν =2

1−2x=⇒ y=

aν +bcν

=2a+b−2bx

2c.

In Figs. 2–4, we also show the fractions of systems for whichTres< 0,Tac< 0, orEcoupl> 0.The “oscillations” on the corresponding dotted lines in Figs. 2 and 4 in the region of largeN are due to some subtle shortcomings of the graphic system we used (gnuplot 4.0).


TΛ

T rot

T I

Tρ = Tξ

TΛ

T rot

T ITρ

Tξ

0

0.25

0.5

0.75

1

1001076543

Mea

n e

ner

gie

s in

th

e u

nit

s o

f T

Number N of particles

1001076543


Fig. 1 The mean values of the terms of the Smith decomposition (14) and orthogonal decomposition (15) ofthe total kinetic energyT ≡ 1, as well as the mean values of the shape energyTξ . The left panel correspondsto the equal mass case and the right one, to the random mass case. The physical space dimensiond is equalto 2.

For d = 2 andN > 3 (and, consequently,ν = N− 1 > 2 andω = 2), the formulas(27)–(31) of Conjecture 1 take the form

ETΛ = 1−1

2ν, ETρ =

12ν

, ETrot = 1−1ν, ETI =

1ν, (41)

ETξ =1

2ν, (42)

EText =1

2ν, ET int = 1−

32ν

, ETres= 0, (43)

ETJ =1

2ν, ETK =

ν −12ν

, ETac=ν −22ν

, (44)

EEout2 = 0, EEin

2 = 1−2ν. (45)

Of course, the equalityEText = ETJ follows from the fact thatText ≡ TJ for d = 2, and theequalityEEout

2 = 0 follows from the fact that genericallyEout2 = 0 for N > d+ 1. These

two equalities hold for any distribution of the masses. Similarly, the equalityEEin2 = 0 for

ν = 2 (i.e., forN = 3) follows from the fact that genericallyEin2 = 0 for N 6 d+1 (again

for any distribution of the masses). On the other hand, the formulas (41)–(43) indicate thatETρ = ETξ = EText for d = 2 andN > 3 in the situation of equal masses (although thehyperradial energyTρ , the shape energyTξ , and the external energyText of an individualsystem on the plane are not related to each other at all).

Thus, what we verified was the formulas (41)–(44) for the 10 termsTΛ , Tρ , Trot, T I , Tξ ,Text, T int, Tres, TK , Tac in the situation of equal masses for 36 N 6 100, the formula (45)


T int

T ext= TJ

T res+ =

T res−

(T res > 0) =(T res < 0)

T int

T ext= TJ

T res+ = T res

−

(T res > 0) =(T res < 0)

0

0.25

0.5

0.75

1

1001076543

Mea

n e

ner

gie

s in

th

e u

nit

s o

f T


1001076543


Fig. 2 The mean values of the terms of the projective partition (16)of the rotational energyTrot for T ≡ 1.The left panel corresponds to the equal mass case and the right one, to the random mass case. The physicalspace dimensiond is equal to 2, so thatText ≡ TJ according to Theorem 5. The mean values ofT res

+ andTres− are indistinguishable on the scale of the figure, so that the mean values of the residual energyTres itself

are indistinguishable from zero. The dotted line labeled as“(T res> 0) = (Tres< 0)” presents the fraction ofsystems for whichTres is positive or the fraction of systems for whichT res is negative (both these fractionsare indistinguishable from 1/2 on the scale of the figure).

for the termEin2 in the situation of equal masses for 46N6 100, and the formulaETres= 0

in the situation ofrandommasses for 36 N6 100 (see Conjecture 2)—a total of

11×98+97= 1175

equalities of the formH = EH where theEH values are given by the formulas (41)–(45).All these equalitieswere indeed obtainedin our calculations. Namely, the minimal value,the maximal value, and the mean value of the difference

∣∣H −EH

∣∣ [over the 1175 triples

(the termH, N, equal/random masses)

considered] were found to be 0, 4.81437×10−4, and 2.7892×10−5, respectively, and theminimal value, the maximal value, and the mean value of the ratio

∣∣H −EH

∣∣

[s2(H)

/L]1/2

, L= 106,

turned out to be equal to 0, 2.41227, and 0.698384, respectively (so that indeedH = EHup to statistical errors [51,52,53,54], see Figs. 1–4). Note also that the minimal value,the maximal value, and the mean value of the “weighted” difference 2ν

∣∣H −EH

∣∣ were


TK

TJ =

T ext

Tac

T+ac

T−ac

(Tac < 0)

TK

TJ =

T ext

Tac

T+ac

T−ac

(Tac < 0)

0

0.25

0.5

1001076543

Mea

n e

ner

gie

s in

th

e u

nit

s o

f T


1001076543


Fig. 3 The mean values of the terms of the hyperspherical partition(17) of the rotational energyTrot forT ≡ 1. The left panel corresponds to the equal mass case and the right one, to the random mass case. Thephysical space dimensiond is equal to 2, so thatTJ ≡ Text according to Theorem 5. The dotted line labeledas “(Tac< 0)” presents the fraction of systems for which the angular coupling energyTac is negative.

0, 2.31975× 10−2, and 1.63148× 10−3, respectively, while the minimal value, the max-imal value, and the mean value of the sample variances2(H) equaled 5.02904× 10−5,8.45973×10−2, and 3.5151×10−3, respectively.

We do not show the standard errorsδH =[s2(H)

/L]1/2

= 10−3s(H) of the mean valuesH [51,52,53,54] in Figs. 1–4 since the numbersH +δH andH −δH are indistinguishableon the scale of the figures.

The minimal value, the maximal value, and the mean value of the number of systemsfor which Tres< 0 (over the 98 values ofN) were equal to 498853, 501308, and 499930(of L = 106), respectively, in the situation of equal masses and to 498691, 501145, and499963, respectively, in the situation of random masses. Therefore, in both the situations,Tres is negative for half of the systems up to statistical errors (see Fig. 2).

We have thus confirmed Conjectures 1 and 2 ford = 2 and 36 N 6 100 (to be moreprecise, Conjecture 2 has been confirmed only in the situations of equal and random masses).Moreover, it turns out thats2(Tres

+ ) = s2(Tres− ) at anyN for both the mass distributions up to

statistical errors.The equalityETres= 0 implies thatETres

+ = ETres− for both the mass distributions. Our

numerical simulations suggest that for sufficiently largeN (say, forN& 30) in the situationof equal masses,

ETres+ = ETres

− ≈5

16ν,

and this approximation is quite good. No similar formula exists forETres+ andETres

− in thesituation of random masses. Recall that in our previous paper [11], we found that for spatial


E in2

Ecoupl+

(Ecoupl > 0)

E in2

Ecoupl+

(Ecoupl > 0)

0

0.25

0.5

0.75

1

1001076543

Mea

n e

ner

gie

s in

th

e u

nit

s o

f T


1001076543


Fig. 4 The mean values of the bounded terms of the singular value expansion (18) [and of its refinement (21)]of the rotational energyTrot for T ≡ 1. The left panel corresponds to the equal mass case and the right one, tothe random mass case. The physical space dimensiond is equal to 2. The dotted line labeled as “(Ecoupl> 0)”presents the fraction of systems for which the tangent coupling energyEcoupl is positive.

systems of particles (d= 3),ETres+ =ETres

− ≈ 5/(12ν) for sufficiently largeN in the situationof equal masses. It would be interesting to figure out how the coefficients 5/16 and 5/12can be generalized to larger dimensionsd. For instance, the conjectural formula

ETres+ = ETres

− ≈5(d−1)

8dν

gives the desired values516ν and 512ν for d = 2 andd = 3, respectively, and agrees with the

fact thatTres≡ 0 for d = 1 (independently of the masses of the particles, see Example3).There are no analogs of the formulas (41)–(45) for the situation of random masses (ex-

cept for the equalitiesETres= EEout2 = 0, of course). However, for largeN (starting with

N ≈ 50, say), the followingapproximateequalities hold in our simulations in the situationof random masses:

TΛ ≈1.88N−4.04

2ν, Tρ ≈

0.12N+2.042ν

,

Trot ≈1.83N−5.2

2ν, T I ≈

0.17N+3.22ν

, Tξ ≈0.05N+1.15

2ν,

Text = TJ ≈0.12N+2.05

2ν, T int ≈

1.71N−7.222ν

,

Tres+ = Tres

− ≈0.03N+0.79

2ν,

TK ≈0.88N−3.03

2ν, Tac≈

0.83N−4.222ν

,


Ein2 ≈

1.66N−8.392ν

.

The coefficientsa andb in these expressionsH ≈ (aN+b)/(2ν) are obtained for each meanenergyH via minimizing the sum

100

∑N=50

[2(N−1)H(N)−aN−b

]2.

In fact, we foundb= 0.78 for Tres+ andb= 0.8 for Tres

− .SinceTJ = Text for anyN (provided thatd = 2) andTK = T int for N = 3, one concludes

thatTac= Tres for N = 3 independently of the masses. Accordingly, in our simulations withN = 3, Tac= 0 andTac is negative for half of the systems up to statistical errors for both themass distributions. For larger values ofN, one findsTac > 0, while the fraction of systemsfor which Tac is negative decreases rapidly asN grows for both the mass distributions (seeFig. 3). However, for eachN > 4, this fraction is greater in the situation of random masses.For particles with random masses, we encountered systems with a negative energyTac forany value ofN. On the other hand, for particles with equal masses forN = 32,N = 33, and376 N 6 100, the angular coupling energyTac of all the 106 systems we sampled turnedout to be positive (for each valueN = 34, 35, and 36, we observed only one system withTac < 0). In fact, systems with the center-of-mass at the origin and with a negative energyTac exist for anyd> 2, N> 3 and any massesm1,m2, . . . ,mN, but the “relative measure” ofthe set of such systems is very small for largeN andm1 = m2 = · · ·= mN. To “construct” asystem ofN > 3 particles with prescribed masses and withTac< 0, one may choose a pair(Z, Z) corresponding toN = 3 andTac< 0 and then augment both the matricesZ andZ onthe right byN−3 zero columns (cf. [11]).

For both the mass distributions and any value ofN, the fraction of systems for whichEcoupl > 0 is much less than 1/2 and decreases fast asN grows for both the mass distri-butions (see Fig. 4). For eachN, this fraction is greater in the situation of random masses.Nevertheless, even forN = 100 in the situation with equal masses, the tangent couplingenergyEcoupl was positive for 32969 systems of 106.

In our numerical experiments for spatial systems of particles [11], the energiesTac andEcoupl exhibited a similar behavior.

In a complete analogy with the cased= 3 [11], the 14 energy termsTΛ , Tρ , Trot, T I , Tξ ,

Text, T int, Tres+ , TK , Tac, T+

ac, T−ac, Ein

2 , andEcoupl+ in the present simulations ford = 2 can be

divided into two classes, cf. Figs. 1–4:

A) The seven termsTΛ , Trot, T int, TK , Tac, T+ac, Ein

2 . For each of these energiesH,

the mean valueH(N) increasesasN grows for both the mass distributions, and

H(N)∣∣equal masses> H(N)

∣∣random masses, 36 N6 100.

(46)

B) The seven termsTρ , T I , Tξ , Text, Tres+ , T−

ac, Ecoupl+ . For each of these energiesH,

the mean valueH(N) decreasesasN grows for both the mass distributions, and

H(N)∣∣equal masses< H(N)

∣∣random masses, 36 N6 100.

(47)

We do not consider here the termsTres andTres− because for both the mass distributions,

Tres= 0 andTres− = Tres

+ for anyN up to statistical errors. We do not consider the termTJ

either sinceTJ ≡ Text for d = 2.


In some cases, the increase/decrease ofH(N) asN grows is slightly non-monotonous forlargeN due to statistical errors. Moreover, in the situation of equal masses, as was pointedout above,T−

ac turned out to be zero for all the systems sampled forN> 37 (in fact,T−ac was

found to be very small forN& 15).The exceptions to the rules (46)–(47) are as follows. First,Ein

2 ≡ 0 for N = 3 indepen-dently of the masses. Second,

Trot(3)∣∣equal masses= Trot(3)

∣∣random masses=

12 and accordingly

T I (3)∣∣equal masses= T I (3)

∣∣random masses=

12

(48)

up to statistical errors. Third,

Tac(3)∣∣equal masses= Tac(3)

∣∣random masses= 0

up to statistical errors. Fourth,

Tξ (N)∣∣equal masses> Tξ (N)

∣∣random masses for 36 N6 5

andTξ (6)

∣∣equal masses≈ Tξ (6)

∣∣random masses

within statistical errors. Fifth,

Tres+ (N)

∣∣equal masses> Tres

+ (N)∣∣random masses for 36 N6 5.

Sixth,

T−ac(N)

∣∣equal masses> T−

ac(N)∣∣random masses and

Ecoupl+ (N)

∣∣equal masses> Ecoupl

+ (N)∣∣random masses for 36 N6 4.

In our previous paper [11], being based on numerical simulations for spatial systems ofparticles (d=3) at 36N6 4 and on the formulas (33)–(34) concerning two-particle systemsfor anyd, we conjectured thatETrot = (d−1)/d andETI = 1/d at all the valuesN6 d+1(andT = 1) for any mass distribution. The relations (48) confirm thisconjecture ford = 2,N = 3.

The larger is the numberN of particles, the greater is the contribution of “kinematicrotations” to the total kinetic energyT of the system (see Remark 3) and the smaller is thecontribution of conventional rotations and changes in the singular valuesξ1,ξ2, . . . ,ξω . Thatis why the mean values of the energiesTΛ , Trot, T int (which include the contribution of“kinematic rotations”), andTK (which is connected with the kinematic angular momentumK) increase asN grows for both the mass distributions whereas the mean values of theenergiesTρ , TI , Tξ , andText = TJ decrease. It is however not clear why the mean values ofthe termsTac, T+

ac, andEin2 increase asN grows while the mean values of the termsTres

+ , T−ac,

andEcoupl+ decrease. Why

H(N)∣∣equal masses> H(N)

∣∣random masses for increasing termsH(N) (49)

andH(N)

∣∣equal masses< H(N)

∣∣random masses for decreasing termsH(N) (50)

is a complete mystery.


Similarly to the cased = 3 [11], the sample variancess2(H) of each of the 15 energytermsTΛ , Tρ , Trot, T I , Tξ , Text, T int, Tres, Tres

+ , TK , Tac, T+ac, T−

ac, Ein2 , andEcoupl

+ for d = 2decreaseas the numberN of particles grows for both the mass distributions [recall thats2(Tres

− ) = s2(Tres+ ) up to statistical errors]. For largeN, this decrease is sometimes slightly

non-monotonous due to statistical errors. Of course,s2(T−ac) = 0 in the situation of equal

masses forN> 37. As one expects, the sample variance of each of these termsin the situa-tion of random masses islarger than that in the situation of equal masses for the same valueof N. For all the termsH, the sample variances2(H) gets very small in the situation of equalmasses for largeN.

The exceptions to these rules are as follows. First, in the situation of random masses,s2(T int) decreases starting withN = 4 (rather than withN = 3): this quantity forN = 4 islarger than forN= 3. Second, for both the mass distributions,s2(Tac) decreases starting withN= 4. Third,s2(T+

ac) decreases starting withN= 4 in the situation of equal masses and withN = 5 in the situation of random masses. Fourth,s2(Ein

2 ) decreases starting withN = 4 inthe situation of equal masses and withN = 6 in the situation of random masses. Moreover,the variance ofEin

2 for N = 3 is of course zero independently of the masses. Apart from this,the inequality

s2(H)∣∣equal masses< s2(H)

∣∣random masses (51)

is violated in the following cases:

for H = T int andTK at N = 3,

for H = Ein2 at N = 4,

for H = T−ac andEcoupl

+ at 36 N6 4,

for H = Tξ , Tres, Tres+ , Tac, andT+

ac at 36 N6 5.

As one sees, there is a strong correlation between the violation of the inequality (51) andthat of the inequalities (49)–(50).

Our simulations confirm that the projective partition (16) ensures a very effective sepa-ration between the conventional rotations and “kinematic rotations” [8,11,15,16] comparedwith the hyperspherical partition (17), not to mention the singular value expansion (18).The mean absolute value|Tres| of the residual energy decreases asN grows for both themass distributions, whereas the mean absolute value|Tac| of the angular coupling energyincreases. ForN = 3 one hasTres = Tac independently of the masses, and consequently|Tres|(3) = |Tac|(3); these mean values are equal to 0.168796 in the situation of equal massesand to 0.146767 in the situation of random masses. ForN > 4, one has|Tres| < |Tac|, andthe largerN, the greater is this difference. AtN = 4

|Tres|(4)∣∣random masses= 0.136738< |Tres|(4)

∣∣equal masses= 0.151223<

|Tac|(4)∣∣random masses= 0.214097< |Tac|(4)

∣∣equal masses= 0.250362,

while atN = 100

|Tres|(100)∣∣equal masses= 0.00640506< |Tres|(100)

∣∣random masses= 0.0381073≪

|Tac|(100)∣∣random masses= 0.400477< |Tac|(100)

∣∣equal masses= 0.489895.


9 Conclusions

The statistical studies of our previous paper [11] (devotedto systems inR3) and those ofthe present paper (devoted to systems onR2) are formal in the sense that they do not takeinto account any interaction potentials between the particles. If one considers kinetic energypartitions for interacting particles with a certain potential energyU, then it is more naturalto average various energy termsH at a fixed total energyT +U (averaging over amicro-canonical ensemble, see e.g. [55]) rather than at a fixed total kinetic energyT, cf. [5,6,8,15,16]. Choosing potential energy hypersurfaces at randomaccording to some distributionin an appropriate infinite dimensional functional space, one would probably obtain entirelydifferent statistics of the energy components. In this case, it seems suitable to average theratiosH/T or H/(T +U) over the initial conditions, the potentialU, and the time. It is alsoof interest to compute the mean values of the energy terms at fixed values of the total angu-lar momentumJ (which is, by the way, customary in quantum mechanics, see e.g. [56]) orkinematic angular momentumK.

There are many ways to generalize the energy partitions treated in the present work.One of them is pointed out in our previous paper [11] and consists in defining the energyterms corresponding to the actions of arbitrary subgroups of the orthogonal groups O(d) andO(N). Another approach recently proposed by Marsden and coworkers [6] for d= 3 is calledthehyperspherical mode analysisby the authors. The 3N−6 internal modes of anN-atomsystem (N > 5) in R

3 are classified in [6] into threegyration-radiusmodes, threetwistingmodes, and 3N−12 shearingmodes. Most probably, Marsden’s theory can be generalizedto the case of an arbitrary dimensiond.

One of the main results of our previous paper [11] and the present paper is that in thesituation of equal masses, the mean valuesEH of various componentsH of the total kineticenergyT are expressed in terms of the dimensiond of the physical space and the numberN of particles in a very simple way. However, it is not clear at all whether thedistributionsof H are “simple” functions ofd, N, andH, not to mention the joint distributions of severalcomponents. For instance, we have not attempted to find any expressions for the variancesVarH of H [or, equivalently, forE(H2)] or, say, for the correlation coefficients [51,52,53,54] between the energy terms. We hope that such detailed statistical properties of the kineticenergy partitions of classical systems will be examined (both numerically and rigorously) infurther research.

The work of MBS was supported in part by a grant of the President of the Russia Fed-eration, project No. NSh-4850.2012.1.

References

1. Eckart C (1934) The kinetic energy of polyatomic molecules. Phys Rev 46(5):383–3872. Smith FT (1960) Generalized angular momentum in many-body collisions. Phys Rev 120(3):1058–10693. Chapuisat X, Nauts A (1991) Principal-axis hyperspherical description ofN-particle systems: Classical

treatment. Phys Rev A 44(2):1328–13514. Littlejohn RG, Reinsch M (1997) Gauge fields in the separation of rotations and internal motions in the

n-body problem. Rev Modern Phys 69(1):213–2755. Yanao T, Koon WS, Marsden JE, Kevrekidis IG (2007) Gyration-radius dynamics in structural transitions

of atomic clusters. J Chem Phys 126(12):124102 (17 pages)6. Yanao T, Koon WS, Marsden JE (2009) Intramolecular energytransfer and the driving mechanisms for

large-amplitude collective motions of clusters. J Chem Phys 130(14):144111 (20 pages)7. Aquilanti V, Lombardi A, Yurtsever E (2002) Global view ofclassical clusters: the hyperspherical ap-

proach to structure and dynamics. Phys Chem Chem Phys 4(20):5040–5051


8. Aquilanti V, Lombardi A, Sevryuk MB, Yurtsever E (2004) Phase-space invariants as indicators of thecritical behavior of nanoaggregates. Phys Rev Lett 93(11):113402 (4 pages)

9. Aquilanti V, Lombardi A, Sevryuk MB (2004) Phase-space invariants for aggregates of particles: Hyper-angular momenta and partitions of the classical kinetic energy. J Chem Phys 121(12):5579–5589

10. Sevryuk MB, Lombardi A, Aquilanti V (2005) Hyperangularmomenta and energy partitions in multidi-mensional many-particle classical mechanics: The invariance approach to cluster dynamics. Phys Rev A72(3):033201 (28 pages)

11. Aquilanti V, Lombardi A, Sevryuk MB (2008) Statistics ofpartitions of the kinetic energy of small nan-oclusters. Khim Fiz 27(11):69–86 (in Russian). English translation: Russian J Phys Chem B 2(6):947–963

12. Aquilanti V, Lombardi A, Peroncelli L, Grossi G, SevryukMB (2005) Few-body quantum and many-body classical hyperspherical approach to the dynamics. In: Semiclassical and Other Methods for Un-derstanding Molecular Collisions and Chemical Reactions.Editors: Sen S, Sokolovski D, Connor JNL.CCP6, Daresbury, pp 1–8

13. Lombardi A, Palazzetti F, Peroncelli L, Grossi G, Aquilanti V, Sevryuk MB (2007) Few-body quantumand many-body classical hyperspherical approaches to reactions and to cluster dynamics. Theor ChemAccounts 117(5–6):709–721

14. Lombardi A, Palazzetti F, Grossi G, Aquilanti V, Castro Palacio JC, Rubayo Soneira J (2009) Hyper-spherical and related views of the dynamics of nanoclusters. Phys Scripta 80(4):048103 (6 pages)

15. Aquilanti V, Carmona Novillo E, Garcia E, Lombardi A, Sevryuk MB, Yurtsever E (2006) Invariantenergy partitions in chemical reactions and cluster dynamics simulations. Comput Mater Sci 35(3):187–191

16. Lombardi A, Aquilanti V, Yurtsever E, Sevryuk MB (2006) Specific heats of clusters near a phase tran-sition: Energy partitions among internal modes. Chem Phys Lett 430(4–6):424–428

17. Calvo F, Gadea FX, Lombardi A, Aquilanti V (2006) Isomerization dynamics and thermodynamics ofionic argon clusters. J Chem Phys 125(11):114307 (13 pages)

18. Castro Palacio JC, Velazquez Abad L, Lombardi A, Aquilanti V, Rubayo Soneira J (2007) Normal andhyperspherical mode analysis of NO-doped Kr crystals upon Rydberg excitation of the impurity. J ChemPhys 126(17):174701 (8 pages)

19. Castro Palacio JC, Rubayo Soneira J, Lombardi A, Aquilanti V (2008) Molecular dynamics simulationsand hyperspherical mode analysis of NO in Kr crystals with the use of ab initio potential energy surfacesfor the Kr-NO complex. Intern J Quantum Chem 108(10):1821–1830

20. Goldman VJ, Santos M, Shayegan M, Cunningham JE (1990) Evidence for two-dimensional quantumWigner crystal. Phys Rev Lett 65(17):2189–2192

21. Shik AYa (1993) Two-Dimensional Electronic Systems. SPbGTU (St. Petersburg State Technical Uni-versity) Press, Saint Petersburg (in Russian)

22. Gomez C, Ruiz-Altaba M, Sierra G (1996) Quantum Groups in Two-Dimensional Physics. CambridgeUniversity Press, Cambridge

23. Shik AYa (1997) Quantum Wells: Physics and Electronics of Two-Dimensional Systems. World Scien-tific, Singapore

24. Efthimiou CJ, Spector DA (2000) A collection of exercises in two-dimensional physics, part I. Archivedas\protect\vrule width0pt\protect\href{http://arxiv.org/abs/hep-th/0003190}{arXiv:hep-th/0003190}

(233 pages)25. Abdalla E, Abdalla MCB, Rothe KD (2001) Non-Perturbative Methods in 2-Dimensional Quantum Field

Theory, 2nd edn. World Scientific, Singapore26. Geim AK (2011) Random walk to graphene (Nobel Lecture). Rev Modern Phys 83(3):851–862; Angew

Chem Intern Ed 50(31):6967–6985; Intern J Modern Phys B 25(30):4055–4080. Russian translation:Uspekhi Fiz Nauk 181(12):1284–1298

27. Novoselov KS (2011) Graphene: materials in the Flatland(Nobel Lecture). Rev Modern Phys 83(3):837–849; Angew Chem Intern Ed 50(31):6986–7002; Intern J ModernPhys B 25(30):4081–4106. Russiantranslation: Uspekhi Fiz Nauk 181(12):1299–1311

28. Castro Neto AH, Novoselov KS (2011) New directions in science and technology: two-dimensionalcrystals. Rep Progress Phys 74(8):082501 (9 pages)

29. Novoselov KS, Castro Neto AH (2012) Two-dimensional crystals-based heterostructures: materials withtailored properties. Phys Scripta T146:014006 (6 pages)

30. Seehttp://www.nobelprize.org/nobel_prizes/physics/laureates/2010/31. Mureika J, Stojkovic D (2011) Detecting vanishing dimensions via primordial gravitational wave astron-

omy. Phys Rev Lett 106(10):101101 (4 pages)32. Anchordoqui L, Dai DCh, Fairbairn M, Landsberg G, Stojkovic D (2012) Vanishing dimensions and

planar events at the LHC. Modern Phys Lett A 27(4):1250021 (11 pages)


33. Stojkovic D (2012) Vanishing dimensions: theory and phenomenology. Romanian J Phys 57(5–6):992–1001

34. Smith FT (1980) Modified heliocentric coordinates for particle dynamics. Phys Rev Lett 45(14):1157–1160

35. Aquilanti V, Cavalli S (1986) Coordinates for moleculardynamics: Orthogonal local systems. J ChemPhys 85(3):1355–1361

36. Golub GH, Van Loan CF (2013) Matrix Computations, 4th edn. Johns Hopkins University Press, Balti-more. Russian translation (1999): Mir, Moscow

37. Horn RA, Johnson CR (2012) Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge. Rus-sian translation (1989): Mir, Moscow

38. Voevodin VV, Voevodin VlV (2006) Encyclopedia of LinearAlgebra. TheLINEAL Electronic System.BKhV-Petersburg, Saint Petersburg (in Russian)

39. Watkins DS (2010) Fundamentals of Matrix Computations,3rd edn. Wiley, Hoboken, NJ. Russian trans-lation (2012): Binomial–Laboratory of Knowledge, Moscow,3rd edn.

40. Dieci L, Eirola T (1999) On smooth decompositions of matrices. SIAM J Matrix Anal Appl 20(3):800–819

41. Hicks JS, Wheeling RF (1959) An efficient method for generating uniformly distributed points on thesurface of ann-dimensional sphere. Commun Assoc Comput Machin 2(4):17–19

42. Muller ME (1959) A note on a method for generating points uniformly onN-dimensional spheres. Com-mun Assoc Comput Machin 2(4):19–20

43. Sibuya M (1962) A method for generating uniformly distributed points onN-dimensional spheres. AnnInst Statist Math 14(1):81–85

44. Tashiro Y (1977) On methods for generating uniform random points on the surface of a sphere. Ann InstStatist Math 29(1):295–300

45. Yang Zh, Pang WK, Hou SH, Leung PK (2005) On a combination method of VDR and patchwork forgenerating uniform random points on a unit sphere. J Multivariate Anal 95(1):23–36

46. Cohen J, Hickey T (1979) Two algorithms for determining volumes of convex polyhedra. J Assoc Com-put Machin 26(3):401–414

47. Lasserre JB (1983) An analytical expression and an algorithm for the volume of a convex polyhedron inRn. J Optim Theory Appl 39(3):363–377

48. Allgower EL, Schmidt PH (1986) Computing volumes of polyhedra. Math Comput 46(173):171–17449. Lawrence J (1991) Polytope volume computation. Math Comput 57(195):259–27150. Bueler B, Enge A, Fukuda K (2000) Exact volume computation for polytopes: A practical study. In:

Polytopes—Combinatorics and Computation. Including papers from the DMV Seminar “Polytopes andOptimization”. Editors: Kalai G, Ziegler GM. DMV Seminar 29. Birkhauser, Basel, pp 131–154

51. Cramer H (1999) Mathematical Methods of Statistics, 2nd edn. Princeton University Press, Princeton.Russian translation (2003): NITs “Regular and Chaotic Dynamics”, Moscow–Izhevsk, 3rd edn.

52. Lagutin MB (2012) Vivid Mathematical Statistics, 4th edn. Binomial–Laboratory of Knowledge,Moscow (in Russian)

53. Van der Waerden BL (1971) Mathematische Statistik, 3. Aufl. Springer, Berlin. English translation(Mathematical Statistics, 1969): Springer, New York. Russian translation (1960): “Foreign Literature”,Moscow

54. Wilks SS (1962) Mathematical Statistics, 2nd edn. Wiley, New York. Russian translation (1967): Nauka,Moscow

55. Ruelle D (1989) Statistical Mechanics: Rigorous Results, 2nd edn. Addison-Wesley, Redwood City.Russian translation (1971): Mir, Moscow

56. Zhang JZH (1999) Theory and Application of Quantum Molecular Dynamics. World Scientific, Singa-pore

Statistics of energy partitions for many-particle systems in arbitrary dimension

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