field and density dependence of the lyapunov spectrum for the driven random lorentz gas

ELSEVIER Physica D 112 (1998) 241-249

PHYSICA

Field and density dependence of the Lyapunov spectrum for the driven random Lorentz gas

Ch. Del lago, H.A. Posch * Institut fiir Experimentalphysik, Universitiit Wien, Boltzmanngasse 5, A-1090 Wien, Austria

Abstract

We report on extensive numerical computations of full Lyapunov spectra and related Kaplan-Yorke dimensions for the two- and three-dimensional random Lorentz gases subjected to an external field. This model is one of the simplest nonequilibrium steady-state systems displaying macroscopic irreversibility in spite of its time-reversible microscopic dynamics, in accordance with the Second Law. Nonequilibrium steady states are maintained by Gaussian constraints, keeping the kinetic energy constant. We study the variation of the Lyapunov exponents with the applied field in a wide range of scatterer densities. For dilute systems and small fields our numerical results are in perfect agreement with recent analytical expressions based on Boltzmann equation methods.

PACS: 05.45.+b; 02.70. Ns; 0.20.-y

1. Introduct ion

One of the outstanding problems of statistical physics is the reconciliation of the irreversible behavior of particle systems with the time reversibility of their underlying dynamical equations. Through the development of time-reversible computer thermostats it has become possible to study such model systems in

nonequilibrium steady states by simulation techniques [1,2]. The systems are driven away from equilibrium by mechanical or thermal forces. The dissipated heat is removed by thermostatting or ergostatting con- stralnts, which have the desired property of leaving the ensuing equations of motion invariant with respect to time reversal. The rate of heat removal is proportional to ( d l n 6 V / d t ) < 0, where 6 V is an arbitrary volume element in phase space, and (...) is a time average. As

* Corresponding author.

a consequence of this continuous shrinking process of phase-volume elements, the Gibbs entropy diverges and a multifractal invariant phase space measure is generated in this class of nonequilibrium steady-state models [3,4], which is responsible for the displayed macroscopic irreversibility in spite of the underlying

time-reversible microscopic dynamics [5,6]. Although such phase space objects have an information dimension strictly less than the phase space dimension, a typical trajectory is space filling and ergodic in the sense that it comes arbitrarily close to every point of the accessible phase space [5,7]. Furthermore, in

any small neighborhood of a multifractal attractor point there are points belonging to the corresponding mnltifractal repellor, constructed from the attractor by a time-reversal transformation (which leaves all coordinates unchanged and reverses the sign of all momenta and momentum-like thermostat variables). Only a trajectory fully on the (unstable) repellor could

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242 C. Dellago, H.A. Posch/Physica D 112 (1998) 241-249

violate the Second Law, which is prevented by the fractal geometry of the repellor.

The multifractality of the invariant measure is a result of continuous stretching and folding operations in phase space. Thereby the exponential divergence of nearby trajectories in some directions of the phase

space is overcompensated by an even stronger contraction in some other direction. The exponential expan-

sion and contraction characteristics of the phase flow are described by the so-called Lyapunov exponents.

Consider a dynamical system in an L-dimensional phase space. The time evolution of a state vector F is given by F(t) = ~ t (F(0)), where qst is the phase flow.

An infinitesimally small perturbation 6F of the trajec-

tory evolves according to the linearized equations of motion ~F(t) = (Oq~t/OF) • ~F(0). If a typical pertur-

bation 3F grows exponentially, the system is sensitive to initial conditions and is referred to as chaotic. The average logarithmic growth rate of a perturbation 3F/

is measured by the Lyapunov exponent

1 I~Fz(t)[ Xl = lim - log (1)

where {SFl(0)}; l = 1 . . . . . L, is a specific set of orthonormal vectors in tangent space. In an L- dimensional phase space there are L Lyapunov ex-

ponents, which we take to be ordered such that

X1 _> X2 _> ---XL. The whole set {Xl}; l = 1 . . . . . L, is referred to as the Lyapunov spectrum. The sum

n X ~ / = 1 I describes the logarithmic expansion rate of an n-dimensional object in phase space. For Hamil-

tonian systems ~L=Â ~'l = 0 due to the conservation of phase volume. The symplectic nature of Hamilto- nian dynamics also causes the Lyapunov exponents

to be paired, such that )~l + XL-I+I = 0 [8]. How- ever, for the isokinetic equations of motion used to describe nonequilibrium steady states, this symmetry is replaced by the conjugate pairing )~1 + XL-/+I = c, where c is a negative constant [9]. It is related to the transport properties of the system, linking the microscopic phase space contraction with the macroscopic irreversible entropy production [1,2].

With the relation of Kaplan and Yorke the Lyapunov spectrum can be related to the information dimension of the underlying strange attractor. The dimension of

a phase space object, neither shrinking nor growing, is given by

2n l=1 Xz

DKy = n + I)Vn+l[ , (2)

n X where n is the largest integer for which Y~-/=I l > 0.

All these properties are not the consequence of many degrees of freedom, but are already present in simple models with a very low-dimensional phase

space, such as the externally driven Lorentz gas. This model consists of a point mass m moving in a static ar- ray of spherical hard scatterers of radius R, on which it is elastically reflected. A homogeneous external field E, constant in time and with magnitude E, acts

on the particle. To force the system into a steady state and to prevent its heating up, a Ganssian thermostat keeps the kinetic energy of the particle constant.

Because of its simplicity, the driven Lorentz gas is considered a paradigm for the study of transport

processes in particle systems [10]. Most of the recent studies are concerned with two-dimensional regular systems, for which the scatterers are located on

the sites of a regular lattice. For low densities the horizon is infinite. This means that there are trajecto- ties, for which the length of the free particle motion between collisions is unbounded. Time correlations decay asymptotically ~ c ( R ) / t , and the diffusion coefficient diverges logarithmically [11-13]. Here,

c ( R ) is a function of the scatterer radius only. For large scatterer densities, say 4 /5 of the close-packing density on which most of the recent numerical work is concentrating [4,7,14-16], the horizon of the system

is finite, i.e. no collision-free trajectories are possible. This leads to exponential correlation decay and finite

diffusion coefficients [10]. In the present study we treat the random Lorentz gas

in two and three dimensions, for which the scattering bodies, disks and spheres, respectively, are randomly arranged in space and do not overlap. The horizon is always finite. Earlier work with this model was mainly concerned with very dilute systems [17-19]. Here we close this gap and investigate numerically the field dependence of the Lyapunov spectrum over the whole range of densities. In Section 2 we introduce the model and give a short overview of the method used for the

c. Dellago, H.A. Posch/Physica D 112 (1998) 241-249

computation of Lyapunov exponents. Our algorithm is

based on the method of Benettin et al. [20], for which it is necessary to determine the time evolution of the

reference trajectory F(t) together with a complete set of tangent space vectors {gF/(t)}. We present and dis-

cuss our results in Section 3. It is interesting to note that in two dimensions the qualitative variation of the

maximum and minimum exponents with the applied field is different for large and small densities. At large

scatterer densities, an increase of the field lowers the maximum exponent, whereas the minimum exponent increases. At low densities, both exponents decrease with increasing field [17]. In three dimensions, how-

ever, this cross-over of the field dependence at intermediate densities is absent, and all exponents decrease

with increasing field [21], regardless of the density. Section 3 is concluded with a short summary.

2. M o d e l a n d M e t h o d

Since the scatterers are randomly distributed, the

system is isotropic and all directions of the field are equivalent. For practical reasons we take the field to point into the positive x-direction in the two-

dimensional case and into the positive z-direction in three dimensions. Due to the random arrangement of the scatterers, the horizon of the system is finite for

all densities and no infinite straight paths are possible. The randomly located scatterers are not allowed to overlap.

During the free streaming between collisions with the scatterers the particle moves according to

(t = p / m , (3)

p = E - ffp, (4)

~" = (E . p ) / p2 , (5)

where q and p are the position and the momentum

of the wandering particle, respectively; E is an external homogeneous field coupled to the particle, and is a thermostat variable representing a Gaussian thermostat, which keeps the kinetic energy p2/2m and, hence, the momentum norm p constant. Both in two and three dimensions the equations of motion (3)-(5)

243

can be solved analytically during the free streaming between collision [7,21]. The time evolution of a tan-

gent vector 3F ----- {3q, 3p} is obtained by differenti- ating these solutions with respect to the phase space

coordinates.

Next, we need to determine the effect of a collision of the wandering particle with a fixed scatterer on the reference trajectory and the tangent space vectors.

When the particle hits a scatterer, the position of the wandering particle is unchanged, whereas its momen-

tum suffers a specular reflection on the surface of the scatterer:

Pf =- Pi - 2(pi . n). (6)

Here, n is a unit vector pointing from the center of the scatter to the collision point. Throughout, the sub-

scripts i and f indicate the state of the system imme- diately before and after a collision, respectively.

For the derivation of the collision rules for the tangent vectors we refer to a recently developed for- realism [22]. Here we just summarize the results. The

spatial component 3q of an arbitrary tangent vector 8F changes according to

~qf = 3qi - 2(8qi • n)n. (7)

At collisions the momentum component ~p of a tangent vector is affected by the curvature of the scatterers and the presence of the field:

~Pf = ~Pi - 2(~pi • n)n + ~Pc + ~PE- (8)

The term ~Pc is due to the curvature of the scatterer surface,

gPc = - 2 [ ( p i • ~n)n + (Pi • n) tn] , (9)

where 6n ---- 8n/~q • ~qc is the change of the normal vector n caused by the displacement

(~qi. n) ~qc = 6qi (10)

(Pi" n)

of the collision point of the perturbed trajectory with respect to the collision point of the reference trajectory. The term ~PE is a consequence of the applied field and the thermostat:

(pi. n) q ~PE -= 2m(n . E) ~qi ' n n + -------w--Pf/• (11)

Pi • n


Expressions (7)-(11), together with the analytical so-

lutions of the l inearized equations of motion, yield

the exact t ime evolution of an arbitrary tangent-space

vector 8F, and Benett in 's method can be applied. We

mention that Lyapunov spectra of the Lorentz gas may

be computed also by using small but finite perturba-

tions [21], instead of infinitesimally small ones as in

our approach. However, this method is less accurate

and computat ionally far less efficient than ours.

At very low densities, configurations of nonover-

lapping scatterers can be generated by sequentially

putting the scatterers into the simulation box. For each

new scatterer random coordinates are selected such

that no overlap occurs with the scatterers already in

the box. At higher densities, however, this method

provides nonuniform distributions. Furthermore, it be-

comes increasingly difficult to find a free volume, large

enough to accommodate a new scatterer. Therefore we

resorted to hard-body molecular dynamics simulations

to generate random configurations of non-overlapping

scatterers in two or three dimensions. Such a simula-

tion of N hard spheres or disks started with the parti-

cles located on the sites of a regular lattice (a triangular

lattice in two dimensions, and an fcc-lattice in three).

The initial momenta were drawn from a M a x w e l l -

Boltzmann distribution. After every 10 single-particle

collision times the actual configuration of the hard

spheres or disks was saved and subsequently used as

scatterer configurations for the Lorentz gas. An inter-

val of 10 collision times was sufficient for successive

configurations to be uncorrelated [19].

3. Results and Discussion

4 i i i i

3.5

3

2 .5

2

1.5

1

0.5

0 t 1 I I

0 0.2 0.4 0.6 0.8

P/Po

Fig. 1. )~+ for the 2D random Lorentz gas in equilibrium as a function of the reduced density P/PO (PO = 1/(2~,/3)R-2) - The dotted line refers to )~+ for the regular Lorentz gas, where the scatterers are located on the sites of a triangular lattice [23]. The exponents are given in units of p/mR.

All results presented in this section are obtained as

an average over four different scatterer configurations.

The density of the scatterers is defined as p =-- N / V ,

where N is the number of the scatterers, and V is the

volume of the simulation box, which is a rectangle

with aspect ratio 2/~,/3 in 2D, and a cube in 3D. This

choice makes it possible to study also very dense sys-

tems near the close-packing density. Periodic bound-

ary conditions were used. We measure distances in

units of the scatterer radius R, Lyapunov exponents

and collision rates in units of m R ~ p , and the exter-

nal field in units of p 2 / m R . In the following we de-

note the Lyapunov spectrum of the two-dimensional

(2D) model by {)~+, 0, 0, )~1}, and that of the 3D case

by {)~+, )~+, 0, 0, )~2, ~1-}, where + and - indicate

positive and negative exponents, respectively, and the

subscript emphasizes conjugate pairs. Two of the ex-

ponents vanish due to energy conservation and the

regular behavior in flow direction.

The positive Lyapunov exponent )~+ for the two-

dimensional random Lorentz gas in equilibrium is

shown in Fig. 1 as a function of the reduced den-

sity P/Po (symbols, and connecting smooth line).

P0 = 1 / (2~ /3 )R-2 is the close-packing density of the

disks. These data are also included in Table 1, which

lists in the first column the reduced scatterer density,

and in the second and third the collision frequency

v and the two nonvanishing Lyapunov exponents for

the field-free equilibrium case, respectively. As re-

quired by the pairing symmetry, )~+(0) = - )~1(0) .

For comparison, also )~+ for a two-dimensional reg-

ular Lorentz gas, with the scatterers on a triangular

lattice, is shown by the dotted line in Fig. 1 [23]. The

difference between the random and the regular case is

small and vanishes completely in the dilute-gas limit.

(2. Dellago, H.A. Posch/Physica D 112 (1998) 241-249 245

Table 1 Simulation results for the 2D random Lorentz gas for various reduced densities p/po,(where P0 = 1/(2~/3) R-2 is the close-packing density; v is the collision frequency for the equilibrium systems (E = 0))

P/PO v E 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.087 0.054 0.191 0.172 0.147 0.126 0.110 0.095 0.082 0.070 0.059 0.047 0.038 0.173 0.118 0.337 0.326 0.303 0.276 0.250 0.224 0.199 0.176 0.153 0.132 0.110 0.260 0.196 0.480 0.474 0.456 0.431 0.403 0.372 0.342 0.311 0.280 0.247 0.212 0.346 0.290 0.626 0.625 0.611 0.592 0.566 0.535 0.503 0.469 0.433 0.393 0.352 0.433 0.411 0.788 0.788 0.781 0.766 0.745 0.717 0.688 0.652 0.616 0.571 0.527 0.520 0.567 0.968 0.966 0.963 0.951 0.933 0.910 0.884 0.852 0.815 0.776 0.725 0.606 0.779 1.180 1.178 1.176 1.168 1.159 1.143 1.119 1.089 1.058 1.018 0.973 0.693 1.079 1.442 1.438 1.436 1.433 1.422 1.408 1.389 1.364 1.336 1.297 1.250 0.779 1.533 1.784 1.781 1.780 1.776 1.769 1.752 1.733 1.706 1.676 1.635 1.591 0.866 2.332 2.280 2.275 2.273 2.278 2.261 2.247 2.220 2.196 2.171 2.141 2.098 0.953 4.049 3.078 3.075 3.070 3.066 3.059 3.038 3.027 3.005 2.979 2.941 2.912

0.087 0.054 -0.191 -0.214 -0.269 -0.338 --0.415 -0.496 -0.581 --0.667 -0 .754 -0.844 -0.935 0.173 0.118 -0.337 -0.349 -0.379 -0.423 --0.475 -0.535 -0.599 --0.668 -0.738 -0.809 -0.879 0.260 0.196 -0.480 -0.488 --0.506 -0.533 --0.569 -0.609 -0.655 -0.706 --0.759 --0.813 -0.866 0.346 0.290 -0.626 --0.634 --0.646 -0.664 -0.687 --0.713 -0.744 -0.778 -0.814 --0.849 --0.883 0.433 0.411 -0.788 --0.794 -0.805 --0.817 -0.832 -0.848 -0.867 -0.888 -0.910 --0.929 -0.946 0.520 0.567 -0.968 -0.970 -0.979 --0.987 --0.996 -1.006 -1.017 -1.028 -1.037 -1.046 -1.047 0.606 0.779 -1.180 -1.181 -1.187 --1.193 --1.202 -1.209 -1.214 -1.216 -1.219 -1.218 -1.211 0.693 1.079 -1.442 -1.440 -1.445 -1.451 --1.454 -1.457 -1.459 -1.459 --1.457 -1.448 -1.432 0.779 1.533 --1.784 -1.782 --1.786 -1.790 -1.792 --1.788 -1.785 -1.776 -1.767 --1.750 --1.732 0.866 2.332 -2.280 -2.276 -2.276 --2.286 -2.276 --2.270 --2.254 -2.243 -2.229 --2.215 -2.189 0.953 4.049 -3.078 --3.075 -3.072 --3.070 -3.065 -3.048 --3.041 -3.026 -3.005 -2.974 -2.953

In columns 3 - 13 the nonvanishing Lyapunov exponents are listed according to the applied field E, the maximum exponent £+

in the top half of the table, and the minimum exponent L 1 in the bottom half. These data are the result of an average over four different random configurations of 1600 scatterers, generated by a hard-disk molecular dynamics simulation as detailed in the text. The standard deviation varies between 0.3% for E = 0, and 0.8% for E = 1. )~+ and v are measured in units of p /mR.

+ ~

2 i i 1 i i i i i i

1.8 3d

1.6

1.4

1.2

1

0.8

0.6

0.4 J A~ o 0.2 l e t

0 t i F i t i v i i

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

P/PO

Fig. 2. )~+ and )~+ for the 3D random Lorentz gas in equilibrium

as a function of the reduced density P/Po (P0 = ~/2/8R-3) - The exponents are given in units of p /mR.

In the 3D case the re are two pos i t ive exponen t s ,

)~+ > )~+ > 0. T h e i r r e d u c e d - d e n s i t y d e p e n d e n c e

in e q u i l i b r i u m is s h o w n in Fig. 2 w h e r e this t ime

,o 0 = ~ " 2 / 8 R -3 deno te s the c l o s e - p a c k i n g dens i ty o f

spheres in 3D. T h e s e da ta are a lso c o n t a i n e d in Table 2,

in w h i c h the s e c o n d and th i rd c o l u m n s l is t the co l l i s ion

f r e q u e n c y v and the four n o n v a n i s h i n g L y a p u n o v ex-

p o n e n t s in e q u i l i b r i u m ( E ----- 0) , respect ively . A l so in

this case the pa i r ing s y m m e t r y )~- (0) = - ) ~ - ( 0 ) ; l =

1, 2 is c lear ly obeyed . A c o m p a r i s o n o f )~+ (0) and

)~+ (0) revea ls a su rpr i s ing symmet ry , as is s h o w n in

Fig. 3, w h e r e w e h a v e p lo t t ed )~+(0)/~.~-(0) as a func-

t ion o f the r e d u c e d densi ty. Th i s ra t io has a m a x i m u m

nea r P/Po = 0.5. A c c o r d i n g to k ine t ic theory [18]

and r e c e n t s imu la t i on resu l t s [19], ) q (0 ) / )~2 (0 ) - + 1

for p - + 0. Fig. 3 sugges t s tha t the s ame l imi t is ap-

p r o a c h e d also at h i g h densi t ies . Th i s dens i ty depen-

dence for e q u i l i b r i u m L y a p u n o v spec t ra di f fers f r o m


Table 2 Simulation results for the 2D random Lorentz gas for various reduced densities P/PO, (where P0 = C'2/8R -2 is the close-packing density, v is the collision frequency for the equilibrium systems (E = 0))

P/PO v E 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.170 0.107 0.306 0.287 0.260 0.235 0.214 0.196 0.180 0.165 0.152 0.140 0.130 0.283 0.198 0.469 0.457 0.429 0.399 0.370 0.343 0.319 0.296 0.275 0.255 0.237 0.396 0.310 0.633 0.623 0.600 0.569 0.538 0.506 0.476 0.447 0.419 0.393 0.368 0.509 0.453 0.809 0.802 0.781 0.755 0.723 0.690 0.658 0.623 0.591 0.559 0.529 0.622 0.641 1.009 1.002 0.985 0.964 0.938 0.908 0.875 0.841 0.808 0.775 0.744 ~.+ 0.735 0.896 1.241 1.235 1.222 1.203 1.180 1.156 1.127 1.095 1.064 1.036 1.007 0.849 1.267 1.521 1.517 1.505 1.490 1.470 1.448 1.424 1.396 1.369 1.343 1.318 0.962 1.856 1.869 1.866 1.855 1.844 1.828 1.809 1.786 1.763 1.740 1.714 1.687 0.984 2.016 1.945 1.941 1.933 1.921 1.907 1.887 1.866 1.843 1.820 1.794 1.768

0.170 0.107 0.277 0.258 0.231 0.206 0.185 0.166 0.149 0.134 0.120 0.106 0.094 0.283 0.198 0.423 0.410 0.383 0.353 0.324 0.297 0.271 0.247 0.224 0.201 0.179 0.396 0.310 0.570 0.560 0.537 0.507 0.476 0.443 0.412 0.381 0.351 0.321 0.290 0.509 0.453 0.729 0.722 0.702 0.676 0.645 0.612 0.578 0.542 0.506 0.470 0.433 0.622 0.641 0.908 0.902 0.886 0.867 0.842 0.812 0.779 0.744 0.708 0.669 0.630 )~2 + 0.735 0.896 1.119 1.114 1.103 1.086 1.065 1.041 1.013 0.982 0.948 0.914 0.876 0.849 1.267 1.379 1.375 1.365 1.350 1.333 1.312 1.290 1.264 1.236 1.204 1.169 0.962 1.856 1.711 1.707 1.698 1.687 1.673 1.656 1.636 1.613 1.588 1.558 1.522 0.984 2.016 1.786 1.784 1.776 1.765 1.752 1.734 1.715 1.692 1.667 1.637 1.601

0.170 0.107 -0.277 -0.302 -0.355 -0.421 -0.494 --0.572 --0.653 -0.736 -0.821 -0.907 -0.993 0.283 0.198 -0.423 -0.438 -0.474 -0.523 -0.580 --0.644 -0.712 -0.783 -0.856 -0.931 -1.006 0.396 0.310 -0.570 -0.579 -0.604 -0.640 -0.685 --0.735 -0.791 -0.850 -0.911 --0.975 -1.039 0.509 0.453 -0.729 -0.735 -0.752 -0.779 -0.812 --0.851 -0.895 --0.941 -0.991 --1.043 -1.095 0.622 0.641 -0.908 -0.912 -0.923 -0.944 -0.970 --1.000 -1.032 -1.069 -1.108 -1.146 --1.185 L~ 0.735 0.896 -1.119 -1.121 -1.130 -1.144 -1.162 -1.184 -1.209 -1.236 -1.264 -1.292 -1.316 0.849 1.267 -1.379 -1.380 -1.385 -1.394 -1.406 -1.420 -1.438 -1.455 -1.472 -1.486 -1.497 0.962 1.856 -1.711 -1.711 --1.712 -1.716 --1.722 -1.731 -1.738 -1.746 -1.754 -1.758 -1.758 0.984 2.016 -1.786 -1.787 -1.788 -1.792 -1.798 -1.802 -1.809 -1.814 -1.819 -1.820 -1.820

0.170 0.107 -0.306 --0.331 -0.384 -0.450 -0.524 -0.602 -0.684 --0.768 -0.854 -0.941 -1.030 0.283 0.198 --0.469 -0.485 -0.520 -0.569 -0.627 -0.690 -0.759 -0.832 -0.907 -0.985 -1.064 0.396 0.310 -0.633 -0.642 -0.667 -0.703 -0.747 -0.798 -0.855 -0.916 -0.980 -1.048 -1.117 0.509 0.453 -0.809 --0.815 -0.831 -0.858 -0.890 --0.929 -0.975 ,1.023 --1.076 -1.132 -1.191 0.622 0.641 -1.009 --1.011 -1.022 -1.041 -1.066 --1.095 -1.128 --1.167 -1.208 -1.252 -1.299 ~1 0.735 0.896 -1.241 -1.243 --1.250 -1.261 -1.278 -1.299 -1.323 -1.350 -1.380 --1.413 -1.447 0.849 1 .267 -1.521 -1.522 --1.525 -1.533 -1.543 -1.556 -1.572 -1.587 -1.605 --1.625 -1.647 0.962 1.856 -1.869 -1.869 --1.869 -1.873 -1.878 -1.883 -1.889 -1.896 ,1.906 -1.914 -1.924 0.984 2.016 --1.945 -1.945 --1.946 -1.948 -1.953 --1.955 -1.960 --1.965 -1.972 -1.977 -1.986

In columns 3 - 13 the nonvanishing Lyapunov exponents are listed according to the applied field E, the maximum exponent ~.+ in

the top quarter of the table, the intermediate positive and negative exponents ~.2 + and )~2 in the second and third quarters, respectively,

and the minimum exponent ~'7 in the bottom quarter. The data are the result of an average over four different random configurations of Scatterers. For the reduced densities P/PO = 0.170 and 0.283 the scatterer configurations were generated by randomly distributing 108.000 spheres in the simulation box. For all other densities the scatterer configurations are obtained by molecular dynamics simulations of 6912 hard spheres as detailed in the text. The standard deviation for the Lyapunov exponents is about 0.2% in most

± cases, with the exception of a few points for reduced densities 0.509, for which it is 0.6%. )~1,2 and v are measured in units of p/mR.

the case o f a gas o f ha rd disks in 2D [22] or hard

spheres in 3D [24], w h e r e the posi t ive exponen t s t end

to b e c o m e equal only at very h igh densi t ies , whereas

they are quite d i f ferent for low densi t ies .

N o w we turn our a t tent ion to the dr iven case ( E

0). Due to the act ion o f the field, the Lyapunov ex-

ponen t s )~{(E) deviate f r o m their equi l ib r ium value

)~/i(0). For smal l fields the devia t ion A)~/~(E) =

C. Dellago, H.A. Posch/Physica D 112 (1998) 241-249 247

1.110

1.105

+~ 1.100

"< 1.095

1.090

1.085

1.080

[ ]

I I I I

0 0.I 0.2 0.3 0.4 I I I I I

0.5 0.6 0.7 0.8 0.9

P/Po

Fig. 3. Ratio )~+/)~2 + for the 3D random Lorentz gas in equilibrium as a function of the reduced density P/PO

2d

0 -

L -0.2 -0.4 -0.6

~ -0.8

.81

0 1 E

Fig. 5. Deviation of L~- from its equilibrium value as a function of the density and the field strength E for the 2D driven Lorentz gas. P0 = 1/2v/~R-2 is the close-packing density. The exponents are given in units of p/mR, and the field in units of p2/mR.

(P0 = #-2/8R-3).

~q

+ " o

~ , i ~ ~ I -0,1 0

-0.1 ~<~ -0.3 .....

-0.2 8 I

-0.3 0

0 0.2 0.4 0 . 6 " 1 "20"4p/P0 Fig. 6. Deviation of )~+ (solid lines) and )~+ (dotted tines) from E their respective equilibrium values as a function of the density

Fig. 4. Deviation of )~1 + from its equilibrium value as a function of the reduced density mad the field E for the 2D random driven Lorentz gas. #90 = 1/(2C'3) R-2 is the close-packing density. The exponents are given in units of p/mR, and the field in units of p2/mR.

)~/~(E) --)~/~(0) is proport ional to E 2. Figs. 4 and

5 show A)~I + and AJ,~- as a funct ion of the reduced

densi ty P/Po and the field strength E for the two-

d imens iona l case. As can be inferred f rom Fig. 4,

AJ~ + is a decreasing funct ion of E at all densities. At

low densit ies the curvature of A)~+(E) changes sign,

leading to a local m i n i m u m as a funct ion of P/Po for fixed E. A different behavior is observed for the

deviat ion A$C 1 of the negat ive Lyapunov exponent

f rom its equ i l ib r ium value (Fig. 5). Whereas at low

and the field strength for the 3D driven random Lorentz gas. PO = ~/2/8R3 is the close-packing density. The exponents are given in units of p/mR, and the field in units of p2/mR.

densit ies A)~]-, l ike A)~ +, is a decreasing funct ion of

the field strength, it increases with E for h igh densi-

ties. This implies that the coefficient of the E2- te rm

in the field-strength expansion o f )~- changes sign at a

densi ty P/Po ~ 0.8. For very high fields (outside the

range covered by Fig. 4) the mot ion tends to become

regular with )~+(E) -+ 0. For conven ience we have

l isted these f ie ld-dependent Lyapunov exponents for

the 2D r andom Lorentz gas also in Table 1.

Ana logous results for the two posit ive and the two

negative exponents of the 3D r andom Lorentz gas are

shown in Figs. 6 and 7, respectively. The difference


3d

001 o 403+ -0.4 3.6 -0.6 ~ 3.23"4 08 30

.81 .8 1

0 0.2 ~ O , 2 0 - 4 p / p o 0 0.2 0.4 0 . 6 ~ 0.20.

E • 1 E - 1

Fig. 7. Deviation of )~1 (solid lines) and )~2 (dotted lines) from their respective equilibrium values as a function of the reduced density and the field strength for the 3D driven Lorentz gas. Po = V~/8R3 is the close-packing density. The exponents are given in units of p/mR, and the field in units of p2/mR.

between A)~ + (smooth lines) and AL + (dashed lines)

can hardly be distinguished in Fig. 6. The same is true

also for the two negative exponents in Fig. 6. At low

densities and weak fields, this observation is in per-

fect agreement with results obtained from kinetic the-

t r y [18] and very accurate numerical simulations [19].

However, at higher fields not covered by these figures,

the field dependence of A)~, 2 and A~,L2 becomes sig-

nificantly different [21]. Our numerical results for the

field- and density-dependent Lyapunov exponents for

the 3D random Lorentz gas are listed in columns 3 -

13 of Table 2. Columns 1 and 2 contain the reduced

density (P0 = ~ /2 /8R -3) and the collision frequency

(for the field-free case), respectively.

The field-induced deviations for the positive expo-

nents in three dimensions, A)~2 , in Fig. 6 behave

similarly as A)~ + for the 2D case• A similar corre-

spondence is found also for the negative exponents in

Fig. 7, which resemble the behavior of AL 1 in two di-

mensions, except that they continue to decrease with

the field even at the highest densities shown in the fig-

ure. The same qualitative field dependence has been

reported in [21] for a 3D regular Lorentz gas, with

the scatterers on a hexagonal lattice, and for a den-

sity P/Po = 0.658• It is, however, possible that such

a cross-over of the field dependence still occurs also

in three dimensions, but only for densities comparable

to close packing.

Fig. 8. Kaplan-Yorke dimension of the 2D driven random Lorentz gas as a function of the density and the field strength. P0 = 1/(2~/-3) R-2 is the close-packing density. The field is measured in units of p2/mR.

5.5 5

4.5 4

°-81 o 0.2

• 1 E

Fig. 9. Kaplan-Yorke dimension of the 3D driven Lorentz gas as a function of the density and the field strength. PO = "v"2/8R3 is the close-packing density. The field is measured in units of p2/mR.

We have mentioned already that the Lorentz gas is,

perhaps, the simplest model which displays all the rel-

evant properties of a dynamical ly t ime-reversible sys-

tem in nonequilibrium steady states: the existence of

a multifractal invariant natural measure with an in-

formation dimension strictly smaller than the phase

space dimension, which is supported by the full phase

space. We show in Figs. 8 and 9 the Kaplan-Yorke

dimension, computed with Eq. (1), for the 2D and 3D

models, respectively, and for the full range of densities

and fields studied here. As expected [25,26], for small

fields DKy decreases quadratically with the field. In

both two and three dimensions this decrease is stronger

for low than for high densities. A quantitative theory

for this effect is still missing.

C. Dellago, H.A. Posch/Physica D 112 (1998) 241-249 249

In summary, we have computed the Lyapunov

spectra of the 2D- and 3D-Lorentz gas with random,

nonoverlapping scatterers, as a function of the re-

duced density P/Po and of the applied external field

E over a wide range of densities. In the equilibrium

3D-Lorentz gas the ratio ~+/)~+ has a maximum at

the density P/Po ~ 0.5 and tends to 1 at low and high

densities. An interesting cross-over has been found in

the field dependence of )~]- in the 2D model. At low

densities )~1- decreases with the field, whereas it in-

creases at high densities. There is no theory at present

capable of explaining these properties. A listing of all

our numerical data is therefore given in Tables 1 and 2.

Acknowledgements

We thank Professor Wm.G. Hoover for many lively

and interesting conversations on this and related sub-

jects. We gratefully acknowledge the financial support

from the Fonds zur Fiirderung der wissenschaftlichen

Forschung, Grants P09677 and P11428, and the gener-

ous allocation of computer resources by the Computer

Center of the University of Vienna.

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2123.

field and density dependence of the lyapunov spectrum for the driven random lorentz gas

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