Lyapunov spectra of periodic orbits for a many-particle system

16 Pages·0.26 MB·English

by  Tooru Taniguchi

Lyapunov Spectra of Periodic Orbits for
a Many-Particle System
Tooru Taniguchi1,2, Carl P. Dettmann1 and Gary P. Morriss2
2
0 1 Department of Mathematics, University of Bristol,
0
University Walk, Bristol, BS8 1TW, United Kingdom
2
2 School of Physics, University of New South Wales, Sydney,
n
New South Wales 2052, Australia
a
J

4
2 The Lyapunovspectrum correspondingto a periodic orbitfor a two
dimensionalmanyparticlesystemwithhardcoreinteractionsisdis-

] cussed. Notingthatthematrixtodescribethetangentspacedynam-
D
ics has the block cyclic structure, the calculation of the Lyapunov
C
spectrum is attributed to the eigenvalueproblemof16 16 reduced
. ×
n matrices regardless of the number of particles. We show that there
i is the thermodynamic limit of the Lyapunov spectrum in this peri-
l
n odic orbit. The Lyapunov spectrum has a step structure, which is
[
explained by using symmetries of the reduced matrices.

1
v
5
4 1 Introduction
0
1
0 JustaslowdimensionalchaosisrevealedbyasinglepositiveLyapunovexponent
2 whichindicates exponentialsensitivity to initialconditions,systems with phase
0 space of very high dimension can be characterized by their Lyapunov spectra
/
n which give information about many possible instabilities in the system. As
i a concrete example, we consider the system consisting of N disks with hard
l
n core interactions and periodic boundary conditions. This is a surprisingly good
v: model of a fluid [1], and yet is sufficiently simple that ergodic properties may
i be established under fairly general conditions [2].
X
The following is a brief review of the study of Lyapunov spectra in such
r
systems; for more details and references in numerical work, see Ref. [3] and in
a
analytical work, see Ref. [4]. It has been known for some time that the Lya-
punovexponents of Hamiltoniansystemscome in plus/minus pairs,that is, the
spectrum is symmetric about zero [5]. Non-equilibrium extensions were shown
to exhibit symmetry about a point other than zero [6, 7], leading to the dis-
covery that these extensions also contain hidden Hamiltonian structure [8, 9].
Also known for more than twenty years is the algorithm for numerical compu-
tation of Lyapunov exponents due to Benettin and others [10, 11]. Later, a
constraint method was introduced [12]. More recently, the effects of the hard
collisionshavebeenproperlytakenintoaccount[3,13]. Theexistenceofather-
modynamic limit in Lyapunov spectra, that is, that the spectrum retains its
1
shape as the number of particles increases, has been put forward using random
matrix approximations [14], numerical evidence [15] and mathematical argu-
ments [16], but recent numerical work has suggested in contrast, a logarithmic
singularity of the largest Lyapunov exponent with the number of particles [17].
Lyapunov spectra for diatomic molecules (represented by dumb-bells) show an
explicit separation of the rotational and translational degrees of freedom if the
departure from sphericity is small enough [18]. Finally, and of particular in-
terest to us, careful simulations of sufficiently large systems have revealed a
step structure in the Lyapunov spectrum for the smallest positive Lyapunov
exponents [3, 13, 18]. The above references give an incomplete description in
termsof“PoschLyapunovmodes”,phasespaceperturbationscorrespondingto
thesesmallLyapunovexponentswhichareapproximatelysinusoidalinposition
space. Recently Eckmann and Gat have suggested an explanation of the Lya-
punovmodesofaonedimensionalsystemusingarandommatrixapproximation
of the Lyapunov spectrum [19]. In this paper we study step structure in the
Lyapunovspectrumofatwodimensionalmanyparticlesystemwithoutmaking
any approximations,however we are restricted to periodic orbits.
Periodicorbittheory[20,21]hasprovenveryusefulforinvestigationsofthe
correspondinglowdimensionalsystem,theperiodic Lorentzgas[22,23,24,25],
computing properties such as the diffusion coefficient. These methods cannot
generally be applied directly to high dimensional systems due to the difficulty
offindingalltheperiodicorbits,althoughanotableexceptionistheKuramoto-
ShivashinskyPDEinaregimewheretheeffective numberofdegreesoffreedom
is small [26]. However, periodic orbit arguments have been used to justify
thermodynamic results such as non-negativity of the entropy production [27]
and the Onsager relations [28] without explicitly finding any periodic orbits.
Periodicorbitsofspatiallyextendedsystemsintheformofcoupledmaplattices
havebeenconsideredpreviously[29,30],leadingtoblockcyclicmatricessimilar
to those observed in this paper.
Here we apply methods similar to those to Ref.[30] to reduce the problem
of computing the Lyapunov spectrum to that of finding the eigenvalues of a
relatively small (16 16) matrix. A step structure is observed,which is related
×
tothesymmetriesofthe system. Ourformalismshowsthat,atleastinthe case
ofthisperiodicorbit,thethermodynamiclimitoftheLyapunovspectrumholds
exactly.
2 Lyapunov Exponents of Periodic Orbits for
the Many-Particle System
The system which we consider in this paper is a two-dimensional Hamiltonian
systemconsistingofN disks,interactingonlybyhardcorecollisions. Wechoose
units such that the mass and radius of the particles are one. We write the
positionand the momentum of the j-thparticle as q andp , respectively,and
j j
foralaterconveniencewerepresentthephasespacevectorΓ asacolumnvector
2
(q ,p ,q ,p , ,q ,p )T with T the transpose operation.
1 1 2 2 ··· N N
The dynamics of such a many-particle system is simply separated into the
free flight part and the collision part, and the tangent vector δΓ of the phase
n
space just after the n-th collisionoccurs is relatedto the tangent vectorδΓ at
0
the initial time by δΓ =M δΓ with the 4N 4N matrix M representedas
n n 0 n
×
M M(c)M(f)M(c) M(f) M(c)M(f). (1)
n ≡ n n n−1 n−1··· 1 1
(f)
Here M is the 4N 4N matrix to specify the j-th free flight dynamics, and
j ×
is given by
M(f) Diag(L(1),L(1), ,L(1)) (2)
j ≡ 1 2 ··· N
whereDiag(X ,X , ,X )meansthematrixonwhosediagonalarethematrix
1 2 l
···
(1)
blocks X ,X , ,X for an integer l, and L is defined by
1 2 ··· l j
I τ I
L(1) j (3)
j ≡ 0 I
(cid:18) (cid:19)
where I and 0 are 2 2 identical and null matrices, respectively, and τ is its
j
×
correspondingfree flighttime. On the other hand, M(c) is the 4N 4N matrix
j ×
to specify the j-th collisionof particles. For the simplest case in which the j-th
collisioninvolvesonlythek -thandthel -thparticlescolliding,thematrixM(c)
j j j
is given as the block matrix
I˜
 ... 
 Fj Gj  ← kj
Mj(c) ≡ ...  (4)
 Gj Fj  lj
 ...  ←
 
 I˜ 
 
 
↑ ↑
k l
j j
where I˜is the 4 4 identity matrix and is put in the part of ... of the above
×
representation,andF (G )isthe(k ,k )and(l ,l )((k ,l )and(l ,k ))block
j j j j j j j j j j
matrixelements,andthe4 4nullmatricesareputintheotherelements. Here
×
4 4 matrices F and G are defined by [13]
j j
×
I L(2) 0
F − j (5)
j ≡ −L(j3) I−L(j2) !
3
L(2) 0
G j (6)
j ≡ L(j3) L(j2) !
with the 2 2 matrices L(2) and L(3) defined by
× j j
L(2) n nT (7)
j ≡ j j
n ∆p(j)T ∆p(j)nT
L(3) nT∆p(j) I + j I , (8)
j ≡ j nTj∆p(j) !(cid:18) − ∆p(j)Tnj(cid:19)
(Note that all vectors in this paper are introduced as column vectors, so for
example, nT∆p(j) is a scalar and n ∆p(j)T is a matrix.) where n is the unit
j j j
vector pointing from the center of the k -th disk to the center of the l -th disk
j j
atthej-thcollision,and∆p(j) isthemomentumdifferencep p justbefore
lj− kj
the j-th collision.
Now we consider the case that the movement of the system is periodic in
time, so that the condition Γ = Γ is satisfied for a integer n . In this case
np 0 p
the Lyapunov exponents λ , j = 1,2, ,4N of the periodic orbit are defined
j
···
by λ = (1/t(n ))ln m (n ) in terms of the absolute value of the (generally
j p j p
complex) eigenvalue m| (n ) |of the matrix M , where t(n ) np τ is the
j p np p ≡ j=1 j
period of this orbit.
P
We put the set of the Lyapunov exponents in descending order, namely
λ λ λ . Itiswellknownthatinthetime-independentHamiltonian
1 2 4N
≥ ≥···≥
systemtheLyapunovexponentssatisfythepairingrule[5],namelythecondition
λ +λ = λ +λ = = λ +λ = 0. Noting this fact, thereafter
1 4N 2 4N−1 2N 2N+1
···
we consider only the first half λ , λ , ,λ of the full Lyapunov exponents,
1 2 2N
···
and refer their set as the Lyapunov spectrum in this paper.
3 Periodic Orbit Model and the Reduced Ma-
trix
By using the method given in the previous section, we calculate the Lyapunov
spectrum for the periodic orbit illustrated in Fig. 1. In this periodic orbit each
particle moves in a square orbit with the same absolute value of momentum
and the same direction of rotation and with a constant free flight time τ (So
the period of the orbit is t(4)= 4τ.). We put 2N (2N ) as the number of the
1 2
particles in each horizontal line (each vertical line) so that N = 4N N . We
1 2
impose periodic boundary conditions, thus requiring the number of particles in
each direction to be even.
Inthis periodicorbitmodel,thereareonlytwotypesofcollisionsillustrated
in Fig. 2, in which the trajectories of two colliding particles are drawn with
their moving directions shown by the arrows. Now we consider the dynam-
ics of particles for a free flight plus one of these two collisions. Such a dy-
namics is described by the matrix multiplications F˜ F Diag(L(1),L(1)) and
1 ≡ 1 1 1
4
Figure1: Periodicorbitofthemanyparticlesystem. Thecirculardotsshowthe
positions of particles at the initial time, and the solid lines give the subsequent
path in the direction of the arrows.
G˜ G Diag(L(1),L(1))(F˜ F Diag(L(1),L(1))andG˜ G Diag(L(1),L(1)))
1 ≡ 1 1 1 2 ≡ 2 2 2 2 ≡ 2 2 2
correspondingto the left (right)orbitin Fig. 2. By using Eqs. (5), (6), (7)and
(8) these matrices are simply given by
0 0 0 0
0 1 0 τ
F˜1 = v v vτ vτ  (9)
− − − −
 v v vτ 1+vτ 
 
 
1 0 τ 0
0 0 0 0
G˜1 = v v 1+vτ vτ  (10)
 v v vτ vτ 
 − − − − 
 
1 0 τ 0
0 0 0 0
F˜2 = v v 1+vτ vτ  (11)
− −
 v v vτ vτ 
 − − 
 
0 0 0 0
0 1 0 τ
G˜2 = v v vτ vτ , (12)
− −
 v v vτ 1+vτ 
 − − 
 
5
π/2
π/2
Figure 2: Two types of particle collisions. The solid lines give the path of the
particles, which move in the direction shown by the arrows.
where one of the components of the collision vector is zero, namely n =(1,0)T
or (0,1)T, and v/√2 is the speed of the particles.
The 4N 4N matrices M(c)M(f), j =1,2,3,4 are represented as
× j j
M(c)M(f) =Diag(P ,P ,P ,P , ,P ,P ) (13)
1 1 1 2 1 2 ··· 1 2
Q Q Q
0 1 2
Q Q Q
1 0 2
 
Q Q Q
2 0 1
M(c)M(f) = Q1 Q0 Q2  (14)
2 2  
 ... ... ... 
 
 Q Q Q 
 2 0 1 
 Q Q Q 
 2 1 0 
 
M(c)M(f) =Diag(P ,P ,P ,P , ,P ,P ) (15)
3 3 2 1 2 1 ··· 2 1
Q Q Q
0 2 1
Q Q Q
2 0 1
 
Q Q Q
1 0 2
M(c)M(f) = Q2 Q0 Q1  (16)
4 4  
 ... ... ... 
 
 Q Q Q 
 1 0 2 
 Q Q Q 
 1 2 0 
 
where the 8N 8N matrices P and Q are defined by
1 1 j j
×
6
F˜ G˜
1 1
G˜ F˜
 1 0 
F˜ G˜
1 1
 G˜ F˜ 
 1 1 
P1 = F˜1 ...  (17)
 
 F˜1 
 F˜1 G˜1 
 
 G˜ F˜ 
 1 1 
 
F˜ G˜
1 1
F˜ G˜
 0 1 
G˜ F˜
1 1
 F˜ G˜ 
 1 1 
P2 = G˜1 F˜1 ...  (18)
 
 F˜1 G˜1 
 G˜1 F˜1 
 
 G˜ F˜ 
 1 1 
 
Q =Diag(F˜ ,F˜ ,F˜ ,F˜ , ,F˜ ,F˜ ) (19)
0 2 2 2 2 2 2
···
Q =Diag(G˜ ,0,G˜ ,0, ,G˜ ,0). (20)
1 2 2 2
···
Q =Diag(0,G˜ ,0,G˜ , ,0,G˜ ) (21)
2 2 2 2
···
In order to get these expression of the matrices the first horizontal row of par-
ticles is numbered 1,2, ,2N from left to right, the second row numbered
1
···
2N +1,2N +2, ,4N andsoonuntilthelastrownumbered(2N 1)2N +
1 1 1 2 1
··· −
1,(2N 1)2N +2, ,4N N .
2 1 2 1
− ···
ThematrixM ,whoseeigenvaluesleadtotheLyapunovspectrum,isgiven
np
by M =M(c)M(f)M(c)M(f)M(c)M(f)M(c)M(f).
np 4 4 3 3 2 2 1 1
It is very important to note that this periodic model is invariant with re-
spect to translations horizontallyor vertically by the distance correspondingto
two particles. This feature is reflected as block-cyclic structures in the matrix
M . By using this fact, as shown in Appendix A, we can show that the eigen-
np
values of the matrix M are equal to the eigenvalues of the 16 16 matrices
np ×
(2πn /N ,2πn /N ), n = 1,2, ,N , n = 1,2, ,N with the matrix
1 1 2 2 1 1 2 2
M ··· ···
(k,l) defined by
M
7
(k,l)
M ≡
S ( k, l) S ( k, l) T (k,l)e−il T (k,l)e−i(k+l)
1 2 1 2
− − − −
S (k,l) S (k,l) T ( k, l)eik T ( k, l)
 2 1 2 − − 1 − − . (22)
T ( k, l)eil T ( k, l)eil S (k,l) S (k,l)e−ik
1 2 1 2
− − − −
 T (k,l) T (k,l) S ( k, l)eik S ( k, l) 
 2 1 2 − − 1 − − 
 
Here, S (k,l), S (k,l), T (k,l) and T (k,l) are defined by
1 2 1 2
S (k,l) (F˜ F˜ )2+(G˜ G˜ )2+(F˜ G˜ )2eik+(G˜ F˜ )2eil (23)
1 2 1 2 1 2 1 2 1
≡
S (k,l) F˜ F˜ F˜ G˜ +G˜ G˜ G˜ F˜ +F˜ G˜ F˜ F˜ eik+G˜ F˜ G˜ G˜ eil (24)
2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
≡
T (k,l) F˜ G˜ G˜ G˜ +G˜ F˜ F˜ F˜ +G˜ G˜ F˜ G˜ eik+F˜ F˜ G˜ F˜ eil (25)
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
≡
T (k,l) F˜ G˜ G˜ F˜ +G˜ F˜ F˜ G˜ +G˜ G˜ F˜ F˜ eik+F˜ F˜ G˜ G˜ eil. (26)
2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
≡
In the next sectionwe investigate the Lyapunovspectrum of this periodic orbit
model by using the eigenvalues of the matrix (k,l).
M
4 Lyapunov Spectrum and its Step Structure
Thereducedmatrix(22)isuseful,notonlytoreducethecalculationtimetoget
the full Lyapunovspectrum, but also to allow us to considersome properties of
theLyapunovspectrumitself. Oneofimportantresultsobtainedbysuchacon-
siderationusingthereducedmatrixisthe existenceofthethermodynamiclimit
inthe Lyapunovspectrum. Itshouldbe notedthattheN N matrices (k,l),
1 2
M
with k = 2πn /N , l = 2πn /N n = 1,2, ,N and n = 1,2, ,N have
1 1 2 2 1 1 2 2
··· ···
the same form of the matrix given by Eq. (22), so in the limit N +
1
→ ∞
and N + the Lyapunov spectrum is given through the eigenvalues of the
2
→ ∞
matrices (k,l), k [0,2π) and l [0,2π). This also implies that the max-
M ∈ ∈
imum Lyapunov exponent takes a finite value in the thermodynamic limit in
this periodic orbit model.
Now we calculate the Lyapunov spectrum in our model. Fig. 3 is the Lya-
punov spectrum in the case of v = 1.8, τ = 2.3, N = 11 and N = 9 (So the
1 2
totalnumberofparticlesisN =4N N =396). Oneoftheremarkablefeatures
1 2
of this Lyapunov spectrum is its step structure. It is important to note that
some steps ofthe Lyapunovspectrum areexplainedby using symmetries of the
reduced matrix (22). Actually we can show (at least numerically) that in the
case of Fig. 3 the Lyapunov exponents calculated using the matrix (k,l) are
M
invariantunderthetransformationsk 2π k andl 2π l,leadingtosteps
→ − → −
in the Lyapunov spectrum. Performing both of these transformations at once
8
Figure 3: Lyapunov spectrum in the case of (N ,N ) = (11,9). (a) Full scale.
1 2
(b) Large j part.
has the effect of taking the complex conjugate of so the result is obvious,
M
however it is not immediately obvious that the spectrum should be invariant if
k and l are transformed separately.
We can investigate the relation between a symmetry of the system and the
step structure in the Lyapunov spectrum another way. For this purpose we
considerthe Lyapunovspectrain the squarecaseandthe rectangularcasewith
the same number of particles. Here, the square system has the symmetry for
the exchange of the vertical and the horizontal directions, which the rectan-
gular system does not have. Fig. 4 is the Lyapunov spectra in the case of
(N ,N ) = (12,12) (the upper two graphs) and in the case (N ,N ) = (24,6)
1 2 1 2
(the lowertwographs)withv =1.8 andτ =2.3. These graphsshow thatthere
is not a remarkable difference in the global shapes of the Lyapunov spectrum
in these two cases, but the square system has (even twice) longer steps in the
Lyapunov spectrum than in the rectangular system, shown in Figs. 4 (b) and
(d). Theselongerstepscomefromanadditionalsymmetryinthesquaresystem.
Actually we cancheck numericallythatthe Lyapunovexponents obtainedfrom
the reduced matrix (22) are invariant under k l, which leads to more degen-
↔
eracy in the case N = N = 12 than when (N ,N ) = (24,6). For N = N
1 2 1 2 1 2
all that is required is that n and n are interchanged. For (N ,N ) = (24,6)
1 2 1 2
degeneracy by this mechanism only occurs if n is divisible by 4 which is much
1
rarer.
It should be noted that some steps of the Lyapunovspectra aretoo close to
be distinguished in Fig. 4. For example, if we can investigate more precisely,
we can see 5 different steps in the long flat part of the Lyapunov exponents λ
j
in j [1018,1097] (in j [1086,1125]) in the Lyapunov spectrum in the case
∈ ∈
9
Figure 4: Lyapunov spectra in the cases of a square system (N ,N )=(12,12)
1 2
((a) Full scale. (b) Large j part.) and a rectangular system (N ,N ) = (24,6)
1 2
((c) Full scale. (d) Large j part.).
of (N ,N ) = (12,12) (in the case of (N ,N ) = (24,6)). Similarly, most of
1 2 1 2
the Lyapunov exponents λ , j [1106,1152] (j [1130,1152])) in the case of
j
∈ ∈
(N ,N ) = (12,12) (in the case of (N ,N ) = (24,6)) are not zero, just too
1 2 1 2
small for the scale of the plot.
5 Conclusion and Remarks
InthispaperwehaveinvestigatedtheLyapunovspectrumofaperiodicorbitof
a two-dimensionalsystem,which consists ofmany disks with hard-coreinterac-
tions. The systemhasarectangularshape(orasquareshapeinaspecialcase),
and we used periodic boundary conditions. By the block cyclic structure of
10