kolmogorov-sinai entropy and lyapunov spectra of a hard-sphere gas

PHYSICA ELSEVIER Physica A 240 (1997) 68-83

Kolmogorov-Sinai entropy and Lyapunov spectra of a hard-sphere gas

Ch. Dellago, H.A. Posch *

Institut fiir Experirnentalphysik, Universitdt Wien, Boltzmanngasse 5, A-1090 Wien, Austria

Abstract

The mixing behavior of a hard-sphere gas has its origin in the exponential growth of small perturbations in phase space. This instability is characterized by the so-called Lyapunov exponents. In this work, we compute full spectra of Lyapunov exponents for the hard-sphere gas for a wide range of densities p and particle numbers by using a recently developed algorithm. In the dilute-gas regime, the maximum Lyapunov exponent is found to obey the Krylov relation )~ (x p In p, a formula exactly derived for the low-density Lorentz gas by Dorfman and van Beijeren. We study the system-size dependence and the effect of the fluid-solid-phase transition on the spectra. In the second part of this work we describe and test a direct simulation Monte Carlo method (DSMC) for the computation of Lyapunov spectra and present results for dilute hard-sphere gases. Excellent agreement is obtained with the results of the deterministic simulations. This suggests that the Lyapunov instability of a hard sphere gas may be analyzed within the framework of kinetic theory.

1. Introduction

Mixing in phase space is a necessary condition for the relaxation o f a nonequilibrium

state towards equilibrium and therefore for statistical mechanics to apply [1]. Consider

a mixing classical Hamiltonian system and a small initial region A evolving with the phase flow. While according to Liouville's theorem the volume of the region is

conserved, it is spread across the energy hypersphere more and more uniformly. After

a long enough time, the probability to find a trajectory originating from A in any region

B of the energy hypersphere is proportional to the measure of B. Consequently, each

point on the energy surface becomes equally likely and the system is ergodic, i.e. the time average o f a physical observable may be obtained by averaging over the energy

hypersurface.

* Corresponding author. Tel.: 43-1-313673109; fax: 43-1-3102683; e-mail; [email protected].

0378-4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PH S0378-4371 (97)00 131-3

Ch. Dellago, H.A. Posch/Physica A 240 (1997) 68 83 69

As suggested by Krylov [2], mixing in phase space has its origin in the mechanical instability of the system, which is characterized by at least one positive Lyapunov

exponent. The rate at which a nonequilibrium distribution evolves towards equilibrium is related to the Kolmogorov-Sinai entropy hxs, which for Hamiltonian systems equals the sum of all positive Lyapunov exponents [3]. The Kolmogorov-Sinai entropy hxs is the information production rate [4] and may be regarded as the speed with which an initial cell of the phase space spreads across the whole accessible phase-space volume.

Systems of hard bodies interacting via elastic hard collisions, like hard spheres and billiards, are paradigms for the understanding of the foundation of statistical mechanics. A large amount of analytical and numerical work deals with the ergodicity and the approach to equilibrium of such systems [2,5-11]. However, up to now the numerical analysis of the mechanical instability of hard systems has concentrated on billiards and simple two-body problems like the Lorentz gas [12]. In the present paper we compute full Lyapunov spectra and the related Kolmogorov-Sinai entropy for a system of three- dimensional hard spheres using a recently developed method [13]. A different approach

to this problem is followed in [14]. Most many-body systems are chaotic in the sense that initially close trajectories

diverge exponentially fast. Due to this strong sensitivity to the initial conditions the motion becomes unpredictable after a short time. Consider an L-dimensional phase space ,/// and a phase flow {~t}, which maps an initial state vector F (0 ) into the

vector F (t) at time t

F ( t ) = ~' ( r ( 0 ) ) . ( l )

The tangent space to J// a t / " is denoted by ~-( / ' ) and consists of all infinitesimal displacements/)/" of a satellite trajectory from the reference trajectory/' . In the following, we use the same basis for all tangent spaces ~--(F). The phase flow {~t} generates a family of Jacobi maps Jt r =- Oa~t/OF, which describe the time evolution of a tangent

vector 6F:

6 r ( t) = J r " 6F (O). (2)

In the case of a smooth dynamical system, the phase flow {~t} is obtained by integrating the equations of motion

[~ = v ( r ) . (3)

The Jacobi maps J~ are generated by integrating the respective linearized equations of motion

6I" = D ( F ) . 6 F , (4)

where D ( F ) = OF~OF is the stability matrix of the system. Since in a chaotic system the trajectories diverge at exponential rates, the so-called Lyapunov exponents are defined by

) ~ ( F ( 0 ) , f F ( 0 ) ) = lim -1 In 16r( t ) l t~oo t 16r(0)l (5)

70 Ch. Dellaoo, H.A. PoschlPhysica A 240 (1997) 68-83

According to Oseledec [15] there are L orthonormal initial vectors 6Fl yielding a set of exponents 2t referred to as the Lyapunov spectrum of the system. The 2t are indepen- dent of the metric and the initial conditions. We order them such that 21/> 22 >/• • • ~> 2L. Initial states infinitesimally displaced from the reference trajectory and located in a hypersphere in phase space, evolve into a hyperellipsoid under the action of the phase flow. The Lyapunov exponents are the average logarithmic expansion and contraction rates of the principal axes of the hyperellipsoid. In a Hamiltonian system, the phase volume is conserved and the expansion in certain directions must be exactly compen- sated by a contraction in some other directions. Consequently, the sum of all Lyapunov exponents vanishes. Furthermore, due to the symplectic nature of the equations of motion, the Lyapunov exponents appear in pairs of equal magnitude and opposite sign. For each quantity conserved by the phase flow one exponent vanishes. In a system of d-dimensional hard spheres, the conservation of total energy, total momentum, the center of mass, and the neutral behavior in flow direction cause 2d + 2 exponents to vanish.

For the numerical computation of Lyapunov spectra of smooth dynamical systems the method of Benettin et al. [16,17] has become a standard. In its original version it requires the simultaneous integration of the equations of motion of the reference trajectory and of a complete set of tangent vectors. In a hard-sphere gas, however, the time evolution of the system in phase space and in tangent space involves also hard elastic collisions of the particles. To take these collisions into account a generalization of Benettin's method is required [13]. In the following, we give a short overview of this generalization. We refer the reader to reference [13] for a more detailed description.

Consider a smooth dynamical system/~ = F(F), where at the times {zl, z2,'c3,...} the smooth streaming is interrupted by discontinuous events

r f = M ( U ) . (6)

The superscripts i and f indicate the initial and final states of the map M specifying the action of the instantaneous events on the phase vector F. We assume that M is differentiable with respect to the phase space coordinates. During the free streaming, the time evolution of the tangent vectors is governed by the equations of motion (4). We are left with the task to determine how an arbitrary vector 6F changes at a collision.

A linear approximation in time and in phase space yields for the effect of a single collisional event on the tangent vectors [13]

•Ff = OM'6l ' i ÷ [ OM ] O---F - ~ . F(F i) - V(M(Fi)) 6~, (7)

where, 6zc is the time delay between the collision in the reference system and in a satellite system displaced by 6F. OM/OF is the matrix of the derivatives of M. Eqs. (3), (4), (6), and (7) suffice to follow exactly the time evolution of the system in phase and in tangent space. All that is needed for the application of Benettin's method is now at our disposal.

Ch. Dellago, fLA. Posch/Physica A 240 (1997) 68-83 71

The remainder of this paper is organized as follows. In Section 2 we discuss the collision dynamics of hard spheres in phase space and in tangent space. We present

full Lyapunov spectra and the corresponding Kolmogorov-Sinai entropies for various densities and particle numbers. In Section 3 we test and apply a variant of the direct simulation Monte Carlo method (DSMC) to compute Lyapunov exponents in low-density gases. Since the DSMC-method is very efficient, the particle-number dependence of the spectra can be readily studied. We conclude with a short summary in Section 4.

2. Molecular dynamics

In this section we apply the formalism outlined in the Introduction to analyze the effect of the elastic collisions of hard spheres on the tangent space vectors 6F and compute full Lyapunov spectra for systems consisting of N identical elastic hard spheres of mass m and diameter a.

2.1. Method

The state of the system is specified by the 6N-dimensional phase vector F =

{ql,q2 . . . . . qN,Pl ,P2, . . . ,pN} containing the positions qj and momenta pj of all particles. The vector 6F = {6ql, 6q2 . . . . . 6qN, @1,6p2 . . . . . @U } containing all displacements from the reference trajectory F is the corresponding vector in tangent space. For the application of Benettin's method we need the time evolution for the reference system F(t) and for a complete set of tangent vectors 6Ft(t), l = 1,.. . ,6N, both during the free streaming and due to the elastic collisions.

Between collisions the particles move on straight lines and the system evolves according to

q/ ( t ) = qj (0) + p j (O)/mt for j = 1, . ,N (8)

p / ( t ) = p / ( 0 ) " '

The respective time evolution of the tangent vectors is given by

6qj (t) = 6qj (0) + 6pj (O)/mt for j = 1 , . . . ,N . (9)

6pj (t) = 6rj (o)

Whenever two particles k and l collide their position remains unchanged, and their momenta change according to

pg____pik ~- (P " q)q/ fT 2, (10)

P/l' _ pi _ -- t (p • q) q/cr 2, ( 1 1 )

72 Ch. Dellago, H.A. Posch/Physica A 240 (1997) 68-83

where q =_ qt--qk and p =-Pt-Pk. It follows from (7) that the configuration components of the tangent vectors simply suffer a specular reflection [13],

b q ~ = 3qik + (6q . q ) q / a 2 , (12)

6qf = 3q~ - (3q • q ) q / a 2 , (13)

where 3q - 6 q l - 6qk, and do not depend on the surface curvature of the colliding particles. However, a respective term appears in the transformation rules for the momentum components [ 13],

= 6pik + (3p • q ) q / a 2 + ~ [(p • 6qc)q + (p • q)6qc] , (14)

• l 3p f = 3p~ - (6p • q ) q / a 2 - -fi [ (p -3q~)q + (p • q)bq~] , (15)

where 6p = 6Pl - 6pk, and

(6q' q) 6qc = 6q ( p . q) p. (16)

6qc is the displacement vector of the collision point of a satellite system from that of

the reference system. Obviously, only the components of 6F pertaining to the colliding

particles are affected by the collision. The last term in Eqs. (14) and (15) contains the

dispersing effect of the hard collisions. With these expressions the time evolution of F(t) and 6Ft(t) , l = 1 . . . . ,6N, can be

obtained and Benettin's method can be applied for the computation of full Lyapunov

spectra of hard-sphere gases.

2.2. Results

We measure distances in units of a, and time in units of (ma2N/K) 1/2, where K =

~ p 2 / 2 m is the total kinetic energy of the system. Lyapunov exponents are measured in units of (ma2N/K) -1/2. We define the density as p = N/V, where V is the vol-

ume of the simulation box, and measure it in units of a -3. In these units, the close

packed density is p = v ~ a -3. We use a cubic simulation box and periodic boundary conditions. At the beginning of the simulation the spheres are placed on the sites of a

face-centered-cubic (fcc) lattice. To obtain a regular packing, a particle number com- patible with the fcc-lattice is chosen: N = 4n3;n = 1,2 . . . . . The initial momenta of the particles are taken from a Gaussian distribution with zero mean. After setting the total momentum to zero we rescale the momenta to obtain the desired kinetic energy,

K = N . The usual collision-by-collision approach is applied to determine the trajectory of

the system in phase space [ 18-20]. In addition to the trajectory we determine the time evolution of a complete set of tangent vectors by using the transformation rules of the previous section. To avoid a collapse of all tangent vectors into the direction of fastest

Ch. Dellago, H.A. Posch/Physica A 240 (1997) 68-83 73

25

20

15

10

I I I f I I

i I I I i I

0.2 0.4 0.6 0.8 1 1.2 p

1.4

Fig. 1. Maximum Lyapunov exponent (diamonds) and Kolmogorov-Sinai entropy per particle (squares) of the 108-particle system as a function of the density. The density p is given in units of tr 3, and the Lyapunov exponent )q and the Kolmogorov-Sinai entropy per particle in units of (Nma2/K) -1/2.

growth, they are periodically reorthonormalized. The time-averaged logarithms of the

renormalization factors are the Lyapunov exponents.

In Fig. 1, the maximum Lyapunov exponent (diamonds) is shown as a function

o f the density. All exponents have an accuracy of better than +0.5%. At a density

o f p ~ 0.956 -3, which corresponds to 67% of the close packed density, the well

known fluid-solid phase transition occurs [21,22]. At this density the maximum Lya-

punov exponents drops significantly due to the different collision rates in the fluid

and in the solid phase. Near this transition density the exponents are calculated with

higher precision. At least 20 × 106 collisions for each point lead to an accuracy of

±0.1%. We point out that the relatively low-particle number o f N = 108 does not

allow the coexistence o f the two phases and the system is either solid or fluid. Fur-

thermore, the periodic boundary conditions enhance the formation of the solid phase.

Since each simulation is started from a regular configuration with the particles ar-

ranged on an fcc-lattice, the formation of the solid phase is more likely to occur

in the two-phase region. Therefore, no overlap o f the solid and the fluid branches

of the equation o f state is observed. An extension of the fluid branch of 2(p) to-

wards higher densities is to be expected if the simulations are started with an initial

configuration obtained by compression of a fluid state o f lower density. The pluses

in Fig. 1 denote the Kolmogorov-Sinai entropy per particle, hxs/N, where hxs is

the sum of all positive Lyapunov exponents. The accuracy of hxs/N is better than 0. l% for all densities. The drop in the collision rate at the phase transition, also

causes a discontinuity for the Kolmogorov-Sinai entropy. We note that both disconti-

nuities for 21 and hxs/N are much more pronounced in three-dimensions than in two-

dimesions [13].

The discontinuities o f 21 and hKs/N become very small if they are plotted as a function of the collision rate v = l /r , where z is the mean time between collisions of

74 Ch. Dellaoo, H.A. PoschlPhysica A 240 (1997) 68-83

100

40

20

10

/////

/ / / . ~ )q

1 10 100 1000

Fig. 2. Maximum Lyapunov exponent (solid line) and Kolmogorov-Sinai entropy per particle (broken line) of the 108-particle system as a function of the collision rate v of an individual particle. The Lyapunov exponent 21, the Kolmogorov-Sinai entropy per particle and the collision rate v are all measured in units of (Nma2/K) -1/2. The linearity of 2(v) in the log-log plot suggests the dependence 2 = av b. A fit to the measured data in the range v = 30 - 800 gives a = 2.0 and b = 0.46.

an individual particle, as shown in Fig. 2. This means that the discontinuities for 21

and hxs can indeed be attributed to the sharp drop of the collision rate at the phase

transition. The same behavior has been found also in a two-dimensional system of hard

disks [13]. The linearity o f 21(v) in the log-log plot suggests that 21 -- av b. A fit in

the solid range 30 < v < 800 yields a = 2.0 and b -- 0.46.

In the dilute-gas regime the 21 was estimated by Krylov [2]. Considering the diver-

gence o f trajectories starting from the same point in configuration space with a slightly

different direction o f the momenta, he obtained

21 = vlog ( ~ ) , (17)

where v is the collision rate o f an individual particle and l is its mean free path. With l = (v/21ta2p) -1 and v = 4 x / ~ u a 2 p [32], where u = (2K/mN) 1/2 is the mean

thermal speed, we obtain the following relation in the dilute gas limit

(18) ).1 = 4 uaZp log \ x/2 J "

As can be inferred from Fig. 3, this expression gives the correct functional dependence

o f ).l(P). However, the Krylov relation underestimates the true exponent by a factor

o f ~ 2.8. In the low density regime 21 is proportional to his . Since hxs = ~ , > o 2 i , this suggests the existence o f a limiting shape o f the Lyapunov spectrum for p ~ 0.

Full Lyapunov spectra {).t/).l } normalized by their respective maximum exponent

for a 108-particle system are shown in Fig. 4. From bottom to top, the densities are 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4, all in units o f a -3. Some relevant parameters


10 0

10-1

10-2

10-3

10-4

10-5

i i i i i i i

hh's ,3 ...... ~7

.~.'~.,.

I I I I

10 -7 10 -6 10 -5 10 4 10-3 10 2 1 0 - 1

Fig. 3. Maximum Lyapunov exponent (diamonds) and Kolmogorov-Sinai entropy per particle (squares) of the 108-particle system as a function of the density in the dilute gas regime. The solid line refers to the Krylov estimate. The exponents have an accuracy ranging from 5:1% for the lowest densities to J:0.1% for the highest densities. The density p is given in units of a -3, and the Lyapunov exponent zl and the Kolmogorov-Sinai entropy per particle in units of (Nma2/K) -1/2.

and results for these simulations are listed in Table 1. Since the hard-sphere gas is

a symplectic dynamical system, its Lyapunov spectrum is symmetrical with respective

exponent pairs summing up to zero. Therefore, only the positive branch of the spectrum is shown in the figure. On the abscissa the index i numbers conjugate pairs of exponents

such that the maximum exponent corresponds to the index 3N -- 324, the next smaller corresponds to the index 3 N - 1, and so on. Finally, the 4 pairs o f vanishing exponents are associated with the indices 1,2, 3, and 4. Of course, Lyapunov exponents exist only

for integer i and it is only for clarity that we connect them by straight lines. At least 300 000 collisions were computed for each spectrum leading to an accuracy of +0.5%.

This number is estimated from the convergence of the Lyapunov exponent to their final

values.

As has been already observed for the Lyapunov spectra of hard disks [13], the curvature of the spectra decreases with increasing density. As the density approaches

the close-packed density, the positive branch of the spectrum becomes essentially fiat. Moreover, the characteristic step from zero to the first nonvanishing exponent becomes

more pronounced with increasing density. In the opposite limit of low densities this step vanishes, and a large number of exponents are very small leading to a highly curved spectrum. We note that the densities p -- 1.0o "-3, 1.2o "-3, and 1.4o --3 belong to

crystalline phases, whereas the system is fluid for all other densities. As can be inferred from the figure, the shape of the Lyapunov spectrum does not change significantly as the system is driven through the phase transition.

An interesting question is whether the maximum Lyapunov exponent of a many-

body system converges towards a finite value in the limit N ---* c~. Several authors have addressed this question for different systems [23,24]. In Fig. 5 we show the

76 Ch. Dellaoo, H.A. Posch/Physica A 240 (1997) 68-83

1

0.9

0.8

0.7

0.6

-~ 0.5

0.4

0.3

0.2

0.1

0

I I I I I I

I I I I I I

0 50 100 150 200 250 300

i

Fig. 4. Lyapunov spectra, normalized by the maximum exponent 21, of a 108-hard-sphere system for the densities p = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4 (from bottom to top), measured in units of tr -3. The respective 21 are listed in Table 1. The index i labels the Lyapunov exponents, which are defined only for integer i. For clarity a solid line is drown through all exponent points. Only the positive branches of the spectra are depicted.

Table 1 Relevant parameters and results of the simulations corresponding to the Lyapunov spectra shown in Fig. 4

p ~ 23N--4 21 hKs/N

0.2 0.6539 0.951 2.495 3.15 0.4 0.2380 1.910 3.913 6.01 0.6 0.1099 3.117 5.561 9.57 0.8 0.0534 4.774 7.915 14.55 1.0 0.0394 5.732 8.840 17.40 1.2 0.0197 8.527 11.889 25.70 1.4 0.0012 38.137 42.884 113.53

The particle number N = 108. The density p is given in units of tr -3, the collision time in units of (Nma2/K) 1/2, and the Lyapunov exponents 2 and the Kolmogorov-Sinai entropy per particle, hxs/N, in units of (Nmtr2 /K) -1/2.

par t ic le -number dependence for the densit ies p = 0.20 --3 and 0.40- -3 . All systems

have the same energy per particle, KIN = 1. The exponents are plot ted as a funct ion

o f N -1/3. In this empir ica l representat ion the curves are near ly linear. The part icle

numbers for the points shown in the figure range f rom N = 32 to 32768. I f the

relat ion 2 N ---- 2 ~ - aN -U3 is fitted to the measured data, one obtains 2 ~ ( 0 . 4 ) = 4.25

and 2 ~ ( 0 . 2 ) = 2.73. The fitted curves are shown by solid lines. Our results suggest

the exis tence o f a the rmodynamic l imit for the Lyapunov exponents . Some authors

conjec tured a logar i thmic d ivergence o f the m a x i m u m Lyapunov exponent with the


3.5

2.5

I ~ I I I I

p = 0.2

I I I I [ I

0.05 0.1 0.15 0.2 0.25 0.3 0.35

N-1/3

Fig. 5. Maximum Lyapunov exponent with error bars for the hard-sphere system as a function of N -1'3. The straight solid lines are fits of the relation 2 N = 2 ~ - aN -I/3 to the measured data. We obtained )~c = 4.25 and 2 ~ = 2.73 for the densities p = 0.4or -3 and p = 0.2a -2, respectively. The Lyapunov exponent 2L is measured in units of (NmaZ/K) -1/2.

particle number [25,26]. To test this hypothesis for the hard sphere system we fitted the function 21(N)-----a+bN -1/3 +c In N. Using a nonlinear least-squares fit we obtained

a = 2.77, b = -1.27, and c = -0.005 for p = 0.2tr -3 and a = 4.21, b = -1.55, and

c = +0.004 for p = 0.4tr -3. We stress that the uncertainty in the parameter c is of

the order of c itself. Since, in both cases, the magnitude of the logarithmic term is

less than the estimated error for all points, our data do not support the hypothesis of

a logarithmic divergence of the maximum Lyapunov exponent.

3. Direct simulation Monte Carlo method

Recently, van Beijeren and Dorfman succeeded in deriving the Lyapunov exponents for a simple statistical mechanical model from kinetic theory [27]. By solving the

Lorentz-Boltzmann equation for the two-dimensional dilute Lorentz gas they obtained the maximum exponent as a function of the density. The same method has been used

to determine the whole Lyapunov spectrum of the externally driven and thermostatted

Lorentz gas (Galton board) as a function of the density and the applied field [28]. In the spirit of kinetic theory we use the so-called direct simulation Monte Carlo

(DSMC) method to compute the Lyapunov spectra for very low-density hard-sphere gases.

3.1. Method

The DSMC-method, which may be regarded as a numerical solution of the Boltzmann equation, has been successfully applied to a variety of rarefied gas flows,

78 Ch. Dellago, H.A. Posch/ Physica A 240 (1997) 68-83

including Couette flows, shock waves, and flows around complicated obstacles [29- 32]. The main idea of the DSMC-technique, introduced by Bird in 1963 [32,33], is the uncoupling of molecular motion and collisions in a small cell during a short time At. In a typical DSMC simulation the system is divided into small cells into which the particles are placed. Then the particles are advanced by the time At according to their actual velocities taking boundaries properly into account and neglecting pos- sible collisions [32]. Collision pairs are then randomly selected in each cell without consideration of their actual position within the cell, and the collision is performed with random collision parameters. The momenta of the particles are changed according to the collision rules (10) and (11). For each single collision a time counter is incremented by the collision time Ate = 2/(Nrca2pVr), where vr is the relative speed of the colliding particles. Such statistical collisions are carried out until the time counter reaches At for each individual cell. As shown by Bird [32], this collision time gives the correct mean free path and collision frequency. Provided the cells are small compared with the mean free path and the time At is short relative to the mean collision time, this method provides a good description of the gas flow.

Since we are interested in equilibrium properties of a spatially homogeneous system,

we apply a slightly modified method and use only one single cell. We proceed in the following steps. (a) First we draw the time zc to the next collision at random from the

distribution

p(zc) = ~ - exp

where v is the collision rate of an individual particle [34]. v is an input parameter for the simulation. In the original DSMC-method the collision time depends on the relative velocity of the colliding particles. However, this choice does not reproduce the correct distribution of collision times of a hard-sphere system, as has been verified by a comparison with molecular dynamics results. The Lyapunov exponents turn out to be rather sensitive to changes in the distribution of collision times. Therefore, the use of the exponential distribution (19) is crucial for the correct evaluation of Lyapunov spectra. (b) The position and the momentum components of the phase vectors and of the tangent vectors are advanced by the time zc. (c) Two particles i and j are selected at random with a probability proportional to their relative speed vr = Ivj - r i ]

PC,,i)P ("j) 02Vr P(vi, v j ) = , (20)

v

where P(vi) is the probability distribution of vi. For this purpose we select a pair of particles and accept it for the collision if an additional random number ~ > Vr/Vma~. Otherwise the pair is rejected and the operation is repeated with a new pair. The maximum relative velocity Vmax is chosen at the beginning of the simulation and is adjusted if necessary. (d) Since the scattering of hard spheres is isotropic [32], we choose the relative position q = qj - qi of the colliding particles at contact, at random. Then we

Ch. Dellago, H.A. PoschIPhysica A 240 (1997) 6~83 79

calculate the postcollisional values of the phase vectors and the tangent vectors accord-

ing to (10)-(15). Since a collision takes place only i f p • q < 0, q must be limited to a hemisphere around p. This choice corresponds to a completely random choice of

the postcollisional relative momentum p. The procedure (a)- (d) is then repeated until the desired number of collisions is reached. Thus, in our case the time interval At of

the original DSMC-method extends over the whole simulation. We note that the total

energy and the total momentum are exactly conserved. Since in the DSMC-method the positions of the spheres are not relevant, it is not

necessary to follow them in time. However, the position component of the tangent space vectors must be taken into account to obtain the correct Lyapunov exponents.

It is important to note that even if the time evolution of the reference trajectory is

dominated by stochastic events, the motion of the satellite trajectory with respect to

the reference trajectory is completely deterministic.

Since the time-consuming calculation of the collision parameters in molecular dynamics simulations is replaced by a random choice, the DSMC-method is computationally

very efficient. At low densities, the MD-simulations are slowed down because the time of flight of an individual sphere is limited by the finite size of the simulation box. How-

ever, the DSMC-method is not expected to yield accurate results in the high-density regime, where spatial and temporal correlations between successive collisions become

important. Both methods are therefore complementary.

3.2. Resul t s

We adopt the same units as for the MD simulations. Since in a DSMC simula-

tion the spatial configuration is irrelevant, no specific boundary conditions are needed.

Initially, the momenta of the N spheres are chosen at random from an appropriate Maxwell-Boltzmann distribution. The total momentum is set to zero and the momenta

are rescaled to obtain the total kinetic energy K = N. At low densities the collision rate

can be determined theoretically [35], at intermediate densities from molecular dynamics simulations.

Fig. 6 shows the maximum Lyapunov exponent 21 for a 108-particle system as a function of the collision rate v -- 1/z. The range z = 1 to 101° in units of mtrZN/K

corresponds to mean free paths l ranging from l = x/-2tr to x/2 × 10l°tr and densities ranging from p ~ 0.15tr -3 to p ~ 1.7 x 10-11a -3. For each point shown in the figure

5 × 10 7 collisions were followed. The diamonds denote our DSMC-result. For com-

parison also the Lyapunov exponents obtained from the MD-simulations are shown by pluses in the figure. Excellent agreement is found for these sets of exponents computed with the two methods.

Two spectra of Lyapunov exponents computed with the DSMC-method for a 108- particle system are shown in Fig. 7 by solid lines. These were obtained for collision rates corresponding to the densities p = 0.001tr -3 and 0.01tr -3, respectively. As be- fore, only the positive branches of the spectra are shown. Since for each collision both the total momentum, the total kinetic energy, and the center of mass are exactly

80 Ch. Dellago, H.A. Posch/Physica A 240 (1997) 68-83

100

10-2

10-4

10-6

10-s

I I I I I

DSMC ~, f "

I I I I

10-10 10-8 10-6 10-4 10-2 10 0

Fig. 6. Maximum Lyapunov exponent (diamonds) obtained with the DSMC-method of a 108-particle system as a function of the collision rate of an individual particle. The accuracy of the exponents is better than/:0.1% for all points shown in the figure. For comparison also the Lyapunov exponent of the 108-particle system obtained by molecular dynamics is shown in the figure (pluses). The densities of the molecular dynamics simulations range from p = 10-76 -3 to p = 0.1~7 -3. The Lyapunov exponent 21 and the collision rate v are both measured in units of (Nma2/K) -I/2.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

I I I I I I

DSMC MD ............

~ ~ p = o.ool

I I I I I I

50 100 150 200 250 300 /

Fig. 7. Full Lyapunov spectra normalized by their maximum exponent of a 108-particle system obtained by MD (dotted lines) and DSMC (solid lines) for the densities p = 0.01o --3 and p = 0.0016 -3. For all data shown the accuracy is better than 4-0.5%. The corresponding maximum Lyapunov exponents are 2MD(0.01) = 0.463 and 2MO(0.001) = 0.0893. DSMC yields 2DSMC(o.o1) = 0.457 and 20SMC(o.o01) = 0.0881. All exponents are measured in units of (Nma2/K) -1/2.

conse rved , 4 pa i rs o f e x p o n e n t s van i sh . For c o m p a r i s o n spec t ra ob t a ined f rom m o l e c u l a r

d y n a m i c s s imu la t i ons for the same dens i t i e s are s h o w n b y do t ted lines. The a g r e e m e n t

is exce l len t .

Ou r resul t s s h o w that the d e c o u p l i n g o f m o l e c u l a r m o t i o n and co l l i s ions is a va l i d

a p p r o x i m a t i o n for the ana lys i s o f the L y a p u n o v ins tab i l i ty o f h a r d spheres in the


0.030

0.026

0.022

0.018

0.014

0.010

i i i i I I i

r = 1000

I I I I I I I

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

N-1/:~

Fig. 8. Maximum Lyapunov exponent with error bars obtained with the DSMC-method for the hard-sphere system as a function of N -U3. The straight solid lines is a fit of the relation 2 N = 2~ - a N -U3, with 2~ = 0.0262 to the measured data. The mean collision time of an individual particle is r = lO00(ma2N/K) U2, which corresponds to a density of p ~ 1.7 × 10-4a -3. The Lyapunov exponent 21 is given in units of (Nma2/K) -U2.

rarefied gas regime. Kinetic theory, with the assumption o f molecular chaos, is a

suitable method for the evaluation o f Lyapunov spectra of many-body hard-sphere

systems.

Since the DSMC-method is computationally very economical we used it to investi-

gate the N-dependence o f 21 at low densities. Fig. 8 shows 21 as a function o f N-~/3 for

a collision time z = lO00ma2N/K corresponding to a density o f p = 1.7 × 10-4o "-3. The

particle number ranges from N = 16 to N = 65 532. The data shown in the figure sug-

gest the existence o f a thermodynamical limit already observed in our MD-simulations.

The solid line denotes a fit o f 2 N = 2 ~ - a N -1/3 to the measured data with a = 0.0234

and 2 ~ = 0.0262.

4. Summary

In the present work we have calculated full spectra o f Lyapunov exponents and

the corresponding Kolmogorov-Sina i entropies over a wide range of densities and

for various particle numbers. The behavior of these quantities may be summarized

as follows. The maximum Lyapunov exponent was found to obey the Krylov relation

21 ~ p log p in the dilute-gas limit. As the density of the system approaches the close-

packed density, the maximum Lyapunov exponent diverges due to the divergence o f

the collision rate. At the f luid-sol id-phase transition, the maximum Lyapunov exponent

as well as the Kolmogorov-Sina i entropy per particle suffer a sudden drop, which is

caused by the decrease of the collision rate at the transition. However, the Kolmogorov-

entropy per particle is a smooth and monotonous function of the collision rate also in

the transition region. At high densities the Lyapunov spectra are fiat and all exponents

82 Ch. Della9o, H.A. Posch/Physica A 240 (1997) 68-83

tend to become equal as the system approaches the close packed density. At low densities many of the exponents are very small and the spectra have a strong curvature. No significant change of the spectral shape is observed at the phase transition. At very low and at very high densities limiting spectral shapes exist.

In the second part of the present work we used a stochastic method (DSMC) to compute Lyapunov spectra for rarefied hard-sphere gases. It assumes that successive collisions are totally uncoupled. Pairs of spheres are chosen at random and suffer a random collision after a random time interval. If the correct distributions for the random quantities are sampled, the method gives very accurate estimates for both the magnitude and the shape of the Lyapunov spectra in the dilute-gas regime. Consequently, an analysis of the Lyapunov instability of hard spheres by kinetic theory seems feasible.

Acknowledgements

We gratefully acknowledge the financial support from the Fonds zur Frrderung der wissenschaftlichen Forschung, Grant P09677, and the generous allocation of computer resources by the Computer Center of the University of Vienna. We had numerous joyful and enlightening discussions on this and related topics with Professor W.G. Hoover. We gratefully dedicate this work to Bill on the occasion of his 60th birthday.

References

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kolmogorov-sinai entropy and lyapunov spectra of a hard-sphere gas

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