UCHL-95907 PREPRINT

DIFFUSION IN A P E R I O D I C LORENTZ GAS

Bill Moran William G. Hoover

This paper was prepared f o r submittal to t h e Journal of Statistical Physics.

CJRCUlATlON COPY SUBJECT TG RECALL

January, 1987 IN TWO WEEKS \

This i s a preprint of a paper intended for publication in a journal or proceedings. Since changes may be made before publication, this preprint is made available with Ihe understanding that it will not be cited or reproduced without the permission of the author.

7

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DIFFUSION I N A PERIODIC LaRENTZ eASI@)

Blll b r a n end William C. Hoover

Laurence Livennore National Laboretory Livermore, California 94550, 8nd Department of Applied Science

University of California at Devis-Livennore

and

Stronzo Bestiale Institute for Advanced Studies et Palenno,

Sicily, Italy

ABSTRACT

We use a constant Wiving Force” Fd together with a Gaussian thenno- stetting “Constraint Force” Fc to s h l a t e a nonequilibriun steady-state current (particle velocity) in a periodic two-dimensional classical Lorentz gas. The ratio o f the average particle velocity to the driving force (field strength) is the Lorentz-gas conductivity. A regular %alton-BoardW lattice of fixed particles is arranged in a dense triangular-lattice structure. The moving scatterer particle travels through the lattice, et constant kinetic energy, making elastic hard-disk collisions with the fixed particles. At low field strengths the nonequilibriun conductivity is statistically indistinguish- able from the equilibrlun Green-Kubo estimate of Wachta and Zwanzig. low-field conductivity varies smoothly but in a complicated way with field strength. For moderate fields the conductivity Oenerally decreases nearly linearly with field, but is nearly discontinuous at certain values where interesting stable cycles of collisions occur. As the field is increased the phase-space probability density drops in apparent fractal dimensionality from 3 to 1. #Ye campare the nonlinear conductivity with similar zero-density results from the tw-particle Boltzmann equation. We also tabulate the variation of the kinetic pressure as e function of the field strength.

The

Key words: Gaussian themstat, two dimensional , periodic Lorentr gas, hard disks, conductivity, fractal.

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I. Introduction

The simglest know nonequilibriun steady-state problem rith reversible equations of m t l m is the Galton Board[1,2) or Lorentz Gas problem illustrated in Figure 1. A single point particle of mass m moves with velocity Y and momentum p=mv through a regular lattice of fixed scatterers of diameter Q

under the influence of two external ffelds, Fd and Fc. An equivalent alternative view of the dynamics results if identical particles with mass 2m, diameter d 2 , velocities 2 v/2, and munenta f p move with appropriate periodic boundaries. The constant driving field

F d = E ,

generates a net velocity in the field direction. time-varying themstatting or constraining force Fc maintains the system at a constant %mperatureW (kinetic energy) and thereby makes a steady state possible:

= - Cp ; C = E(pX/m)/(p /m) .

At the same time the

2 FC

It is necessary to introduce a constraint force in order to observe a steady nonequilibriun state. In the absence o f a constraint the t h e m 1 velo- city p/m, by conservation o f the total energy (p /Zm)-Ex, would have to vary as the square root of ths x displacement. Ufth growing x dlsplacement the acceleration due to the field, which is independent o f velocity, becomes negligible relative to the scattering accelerations, which are proportional to the velocity. inversely as the velocity for hard disks, might be expected to describe the

-1/2 motion. At long times the drift velocity <dx/dt> would then vary as x and the resulting net displacement varies as the two-thirds power of t h . fact, the motion resulting in the absence o f constralnts 1s so irregular that we could not determine a quantitative powr-law dependence of t dx/dt 1 on x.

steady diffusion outside the linear r e g h . The friction coefffcient C could be based OCI Nose mechanics, a modlflcatlon 12-41 of Hamiltonian mechanics readily applicable to the Galton hard. But the resulting phass-space des- cription would become more complex, involving five time-dependent variables,

2

In this long-time limit the low-field conductivity, which varies

In

Thus a constraint force o f sane kind must be lmposed in order to observe

#

(x,y,pX'py,~) rather than the simpler set of three rewired by Causslrn dynamics. The present work Is based an the simpler dynamics, Gaussian dym-

a h , the special case of Nose/mechanlcs ln which the nlexatlon time asso- ciated with the frictional themstettlng force q p approaches zero.

It 1s certainly not obvious, but it is at least plausible that this system vi11 reach a nonequillbrlun stete with a nonvanishing current approximately e w a l to the linear-response-theory prediction worked out by Machte and Zwanzlp[ 5 J . But the nunerlcal calculetions sumrarized here show consistency with those earlier results together with topologically interesting behavior at higher field strengths &re the mean velocity becomes of the order of the thermal velocity.

The problem reduces to a three-dimensional one, with two coordinates { x , y ) specifying the location of the scattering particle within a unit cell o f the scattering lattice and the third coordinate 0 giving the angle corresponding to the direction of motion. This dimensionality can be further reduced, fran three to two, by taking advantage o f the integrability o f the equations of motion between collisions. Thus the steady-state probability density in the three-dimensional phase space is slmply related to the steady- state two-dlnensional probabi lity density for collisions. Section 11 w describe the equations of motion which constitute the mathemati- cal statement of this problem. The numerical solution is described in Section 111, with some of the analysis relegated to the Appendix. Our conclusions are contained in Section IV.

In the following

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11. Mathematical Formulation

Ue use Cartesian space coordinates, locating the moving point particle at x and y with a polar velocity coordinate 8 giving the direction of motion, relative to the x axis. Thus the velocity cosrponents i n the x and y directions are proportional to the cosine and sine o f 8, respectively:

&?noting the drfving force, or fl8ld Fd as the vector (E,O) parallel t o

the x axis and th8 isokinetic constraint force Fc as -Cp, the first-order equations o f motion for the scatterfng particle are the folloufng:

0

x = p x / m = (p/m)cos@ ;

= py/m = (p/m)sine ;

ix = E + Fx - CPX & = Fy - CPy * 6 = -(E/p)sine

I

For more details the closely-related references by Hoover[2,6) and by Morriss[f] can be consulted.

Figure 1 shows the Galton-Board geometry. The trlangular lattice can be conveniently broken up into either parallelogram or hexagonal cells. Heme we use hexagons. The hexagonal cells have an axis o f symnetry provided that the field direction is chosen to lie at any angle m/6 relative to a lattice direction. Here we arbitrarily choose the field perpendicular to a row of fixed scatterer particles. Provided that the density o f these PartlCleS, relative to their close-packed density, exceeds 3/4 there is no possiblity that the moving point particle can avoid scattering. arbftrarfly choose a density equal to 4/5 the maximun clore-packed density. In the Machta-Zwanzlg system o f units, with scatterers o f unit radius separated by a distance 2tU, our densfty choice corresponds to a value for their W o f 6 - 2 = 0.236068.

In the present work we

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The equations o f motion can be 1nteDrated malytlcally [S] to calculate the displacements, bx and Ay, parallel and perpendicular to the Cleld directlon during the time t:

n . I

t = -(p/E)ln[ tan(B/2)/tan(80/2)] . With these results the ncolllsionsa with the horizontal straight-line

boundaries o f the split-hexagonal cell shown in Figure 1 can be calculated analytically. A highly accurate calculation of the collisions with the curved disk surface shown in the figure can be based on a rapidly convergent series expansion described in the Appendix. about lS,w)o,OOO collisions per hour using a Cray-1 computer.

In this way it is possible to treat

The collision history can then be used to generate either of two probabi- lity density functions, the phase-space density f(x,y,B) or the simpler collision probability density g(o,6) giving the probability of a collision with configurational angle of -act u and post-collisional direction of motion,

relative to head on, described by the angle 6. These angles are defined in Figure 1 and cover the ranges 0 ( a ( v and -.I2 < B 1 r/2. For transient problems the full phase-space density f is required. For a steady state the function g 1s sufficient. This is because the density at any point in phase space can be calculated from the density at the previous collision

0 f(o)/f(-t) = exp[(E/m)l cosedt] = exp[E(x(o) - x(-t))/mv2] ,

-t

or the next collision

t f(o)/f(t) = exp[(-E/mv)l cosedt) = exp[-E(x(t) - x(o))/mv2] .

0

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These useful results follow from the analog o f Liouville's equation in the three-dlmensional phase space:

The various derivatives i n the nonequilibriun steady state for x and y in the streaming region, between collisions, are:

From these

which gives directly

2 f(t)/f(O) = exp[EAx/(mv )].

The relationship between the probability density ln the full space f(x,y,B) and the probability density at collisions g(a,8) follows from the observatlon that the collision rate at any angle a is proportional to the velocity comQonent i n the radial direction, (p/m)cos(B). Thus we have

The steady-state pressure tensor can likewise be expressed in terms of a and 6, taking lnto account that the colllslons are isokinetic rather than lsoenergetic, as is explained in references 8 and 9. The kinetic contribution to PxxV(m/p )= <cos 8) 1s given ln Table I . 2 2

111. Nunerlcrl Results & Wonequlllbriun Molecular Oynamics

Calculetlons spanning two orders of magnltude i n the f i e ld s t rength E are surmarired i n Table I. The results can be checked I n several ways. First, the l lml t lng conductivity a t vanlshing f ie ld s t r eng th can be conpared wlth the purely equlllbrlun calculation of Uachta end Zwanzig[ 51. Within the f i v e percent uncertaint ies of the extrapolatlon and the epui l lbr lun calculat ions the two approaches agree.

Next, at zero field, the fraction of c o l l l s l o n s w i t h the disk is given by the r a t i o of the half-disk perimeter m/2, where u 1s the disk diameter t o the exposed area of the half hexagon. This value is 0.4136. Because the system is a a x i n g system, the co l l i s ion ra te r can likewise be calculated analyt ical ly , w i t h the resu l t S.MSSv/a. Both numerical estimates were reproduced t o three-figure accuracy i n runs of one mill ion col l is ions, as would be expected from the central l i m i t theorem.

The probabili ty density g is extremely in te res t ing ; i t varies from a cos% d i s t r i b u t i m spanning the two-dimensional 0-6 space et zero field, t o a set of 20 discrete equally-weighted dots a t a f i e l d of 3.69 p /(mu). Thus the dimensionality of the probability density g must vary between two and zero, suggesting .fractaln nonintegral values i n between. A fractal object is typi- c a l l y one ln r h i c h the nunber of pa i r s of points l y ing within a range dr about a point r d t h i n the object varies as a nonintegral power of r. We found that t h i s nonintegral power i s 1.8 for a f i e l d of 2 p /(ma) and 1.6 for a f ie ld of 3.6 p /(Bu). dimensionalities together wi th i l l u s t r a t i o n s suggesting their s t ructure , can be found i n References [ lo) and [ll). corresponding t o the chain of col l i s ions shown i n Figure 2, can be seen i n the series o f probability density functions shown ln Figure 3.

2

2 2 The def ini t ions o f various nearly-equivalent f r ac t a l

The development of the 20 spots,

We found 13 values o f the field s t rength for which stable co l l i s ion cycles T h e f i e l d strengths, the nunber n of co l l i s ions per cycle and the occur.

corresponding conductivit ies are sumnarized in Table XI.

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2 The "eyem, visible just to the right o f the center of the E=* /mu

portion of figure 3 (see figure Ja), corresponds to a stable Kolmogorov- Arnold-Hoser-like nearly-periodic orbit shown in Figure 4. that more-canplex nearly-periodic orbits occur at other field strengths. Most of the phase space comprises a chaotic region which, after a feu collisim t h s , condenses onto the strange attractors shown in Figure 3.

It seems likely

figure 5 shows the dependence of the conductivity K on field strength. In that figure we have scaled the results in terms of the collision rate, as was done for the low-density case in reference [a). In both cases the conductivity Is a generally decreasing function of field strength, but with considerable structure at low fields. consequence of the change in relative importance of flrst-, second-. and third-neighbor collisions as a function of field strength.

The structure is the complicated

Figure 6 shows a different representation o f the probability density g(a,B), as a coarse-grained surface over a 180 x 180 grid in 0-8 space. It is certainly suggested by the Figures that the probability density is a fractal [4,11]. If we index the grid with two subscripts, 1 and J, the coarse-grained probability density g steady-state matrix equation relating the probability of successive collisions:

can be found as the solution of a ij

The matrix M = (Mijkl] is a sparse %ormain matrix, with elements exp(Et/p) where t is the time required to strean from the collision specffied by i and J to that specified by k and 1. m e largest such matrix we considered has 180 = 1,(349,760,000 distinct matrix elements. The tlme-reversibility of the equations o f motion has the consequence that the "transposem of the matrix is equal to its inverse :

4

' i j k l 2 1 1 J = 'ik 'J1

An lttrrted rolutlon of tht S2400 x 12100 mattlx W, kglnnlng wlth e unlfom 0 , where g is 160 x 180 artmy, Is shown I n Flmurt 6. mi5 crlculrtlm requlrcd rbout an hour on e CRAY-1. This problm Is one on h l c h COnSlkr8blt progress could be nude wlth the developnent of conputerr only fer orders of rwronltude faster than present machines.

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IV. conclusims

mis simple periodic Lorentr gas, or Galton board, equivalemt t o an iso- kine t ic periodic system o f two disks with a driving field, although simple deterministic, and dynamically time-reversible, displays many of the in te res t ing features o f dissipat ive i r r eve r s ib l e many-body systems far frm equilibrlun:

(1) mixing in t he phase space due t o the Lyapunov i n s t a b i l i t y [10,12-14] o f trajectorfes--the mixing tlme is o f t h e order o f t h e time between collisions.

(ii) nonlinear conductivity, generally decreasing with increasing f ie ld .

(ill) ctmplex phase-space geometry, with condensation onto a s t ranue a t t r ac to r with a f r ac t a l dimensionality.

( iv) effective irreversibility and dissipation with revers ib le equations of motion.

The nunerical work has shown tha t t h e phase space probabi l i ty density has a f r a c t a l character, making it extremely unlikely that any e f f i c i e n t analyt ic description is possible. A t the same time the general dependence of conducti- v i ty on f i e ld strength is r e l a t ive ly simple, being roughly l i nea r . o f the dependence are re l a t ive ly complex.

The de ta i l s

The reasm for the complexity i n t he phase space densi ty can be traced to the various kinds o f sca t te r ing co l l i s ions . A t the conclusion o f a colllslon, described by the angles a and 6, the moving par t ic le can h i t any o f its six nearest neighbors o r any o f its six second neighbors. A t f i n i t e fields even suna o f the third neighbors can be reached--see Figure 7, where t h i s effect is perceptible for a field strength E 1 0 . 5 p /(mu). The types o f c o l l i s i o n s which occur are shown for 12 field s t rengths in the Figure. These co l l i s ions result In relatively-complicated probabi l i ty densities, a s shown in Figure 3. The structure o f the boundarles becomes more complex when the curved t raJec tor les induced by a f i n i t e f ield a r e Included.

2

The presence o f a field also destroys

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the rotational symnetry of the rcettering as a function of a. The iteration of these boundaries, corresponding to the actual sequence of colllslons a particle undergoes, ~eneretes a highly-compllcated structure in the phase space. Because the net motion is primarily in the dfrection of the field, the volune occupied in the space decreases exponentially in time, thereby establishing that the dimensionality of the probability density is less then that of the space in which it is embedded.

Wachta and Zwanzig[5] camnented on the very irregular structure they discovered in the velocity autocorrelation function for this s m rnodel. fhat irreoular structure no doubt has its origin in the superposition of relatively smooth distributions for particles constrained to collide with particular choices antong the neighboring disks available for scattering.

The decrease in occupied phase-space volune is, in the steady state, balanced by the contfnual spreading and bifurcation doe to the harddfsk collisions. Thus the phase-space density contraction from the thennostat can be directly related to the compensating Lyapunov instability o f the trajectories .

In the unstable reversed motion the velocity (or mcmentun) and the friction coefficient C both change sign. Thus any solution of the equations o f motion with a positive conductivity would correspond t o a solution with a negative rnphysical conductivity in the reversed motion. Because the reversed motion 1s subject to Lyapunov instabflity and bas a relative probability o f zero, it m o t actually be observed.

1’ l2’ The motion, described in tens of the Lyapunov exponents A and A may seem paradoxical. These exponents[l2,13] describe the rate at which a three dimensional phase-space sphere o f points, infinitesimal in radius, deforms into an ellipsoid, with principal axes varying as expIAit]. Directly fran the equations o f motion the volune of the sphere must shrink, on the average, so that

3’

AI + A2 + 5 = < 0 .

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On the other hand the equations of motion are t h e reversible, with 8

becoming T-8 in the reversed motion. In the reversed motlon one might expect that the three Ai would simply change sign, with the results

But this is not consistent with the existence of a positive conductivity. What really happens is that the prfncipal directions of growth and decay i n the motion are sensitive to initial conditions and differ in the direct and reversed motion. It is unfortunate that a direct measurement of the Lyapunov exponents is very difficult for singular potentials, despite the existence o f fairly efficient methods for determining these coefficients in sinpler problems [12,13]. On the other hand there are alternative methods for studying the dynamfcs of constrafned systems and it is worthwhile to study their relative merits. Experience with ~ose/mechanics[3,6,~1 suggests that the Nosdfonnulation for the friction coefficient C does little more than complicate the mathematical description, by adding two to the dimensionality of the corresponding phase space.

On the other hand Lagrangian mechanics[2,14] could be applied to this problem. In that case the Lagrangian has the form

where A is a the-dependent Lagrange multiplier chosen such that the instan- taneous kinetic energy, T=mv /2, has the fixed value K. energy + includes both the scattering contributions with the fixed lattice and a linear term - Ex to drive the current. equations o f motion include an effective MSS, m(l + A) rather than a frictlcm coefficient:

2 The potential

In thls Lagrangian case the

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where F describes the scatterlnp lnteractlon with the fixed lattice of disks. h varies wlth time to ensure that T and K are equal. 8ut it should be noted that in the the-reversed equations o f motion 1s unchanged. This suggests that the Lagrangian descrlptlon is not useful i n far-frm-equlllbriun problems in which nonlinear dlssipatlon must be considered. We have investigated these LagrenQian equations in order t o characterize their mode of failure. We found that the parmeter h and the momentun p increase without bound and the correspondirrp conductfvity approaches zero independent of the fleld strength.

Between collisions the Legrange equation for iy can be lntegrated directly:

where we have arbitrarily chosen to set 1 equal to 0 et the initial time, corresponding to 8 velocity v(cosOo,sin80). From this relation the equation of motion for the angle results:

2 6 = Xsin e/sineo . The equation of motion is identical to the Gaussian version provided that E is sufficiently small so that A9 can be ignored. For Em/p equals 1, 2, end 8, by using the choice A=D at each co~llsion, we found rp/m to be 0.098, 0.106. and 0.087.

2

The linking of successive collisions to the probability density 9 suggests the formulation o f the Galton Board problem as a mapping problem, with the collisions corresponding to the iteration process linking g - , go, and g+. 8ut we have not seen a way to take advantage o f this analog. Likewise the field- driven diffusion problem also bears a qualitative but barren resemblance to B

one-dimemima1 random walk with stochastic collisions.

V. Acknowledgnent

We thank Brw Boghosian, Ray Chin, Uilly Moss, and 3. S. Pack for good advice. S. B. thanks the National Science Foundation for a generous travel grant.

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Table I. Conductivity a as a function of field strength for a triangular- lattice Lorentz gas at 4/S the close-packed density. N is the rimer of disk collisions, r is the collision rate, and to is the mean t h e between successive collisions at zero field. The conductivity is given by the average value of the velocity i n the field directfon divided by the field strength. In the two-particle view, with velocities 5 half thg one-particle value, W > / € would have half the tabulated value.

2 Em /p N rp/m DE P <cos2@>

0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 .o 1.1 1.2 1.5 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2 -4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 4 .O 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5

10.0

0.030 0.059 0.089 0.119 0.149 0.178 0.208 0.238 0.267 0.297 0.327 0.357 0.306 0.416 0.446 0.475 0.505 0.535 0. %5 0.594 0.624 0.654 0.683 0.713 0.743 0.773 0.802 0.032 0.862 0.891 0.921 0.951 0.891 1.010 1.040 1;189 1.337 1.486 1.634 1.783 1.931 2.080 2.228 2.377 2.526 2.674 2.823 2.971

4x107 4x107 4x107 4x107 3x107 3x107 3x107 3x107 3x107 3x107 3x107 3x207

3x107 3x107 3x107 3x107 3x107 3x107 2x106 2x106 2x106 2x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 1x106 l x l $ 1x106 1x106 1x106

3x107

0.0977 0.0954 0.0914 0.0888 0.0875 0.0885 0.0897 0.0904 0.0904 0.0924 0.0930 0.0936 0.0920 0.0912 0.0892 0.0855 0.0815 0.0799 0.0788 0.0771 0.0747 0.0739 0.0729 0.0715 0.0721 0.0729 0.0739 0.0746 0.0738 0.0734 0.0730 0.0725 0.0735 0.0717 0.0752 0.0654 0.0589 0.0575 0.0546 0.0827 0.0492 0.0364 0.0414 0.0395 0.0624 0.0485 0.0333 0.0484

3.365 3.364 3.367 3.370 3.375 3.380 3.388 3.392 3.391 3.394 3.397 3.398 3.3% 3.406 3.409 3.412 3.419 3.434 3.446 3.456 3 A74 3 A95 3.512 3.536 3.550 3.561 3.554 3.559 3.561 3.554 3.550 3.514 3.542 3.522 3.525 3.396 3.192 3.159 3.067 2.562 2.808 2.892 2.976 3.124 3.137 2.704 3.084 3.000

0.500 0. mi 0.502

0 . m 0.505

0.503

0.506 0.508 0.510 0.512 0.513 0.520 0.525 0.527 0.531 0.535 0.536 0.536 0.538 0.539 0.542 0.543 0.545 0.547 0.550 0.555 0.560 0.565 0.567 0.574 0.582 0.588 0.595 0.609 0.618 0.665 0.691 0.706 0.723 0.744 0.753 0.764 0.771 0.776 0.782 0.782 0.795 0.796

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Table 11. Limlt cycles o f n collisions each found at field strength E with the corresponding conductlvities u .

h U / p 3.69 6.00 6.17 6.56 0.29 8.90 9.05 9.24 9.29 9.47 9.66 9.78 10.0

10 8 32 IO 6 n 20 5 14 6 8 6 19 8 ~~ ~~ ~~~ - ~ - - ~

100 R ~ / I I W 13.4 8.27 7.32 6.03 S.64 5.02 5.06 4.20 3.30 3.79 3.93 3.52 4.84

fiQure 1. Ceometry o f the Lorentr Cas. A fixed rcet terer pa r t i c l e , r i t h diameter 8 , lies i n a hexagonal cell of volune (3/4) Vo is the close-packed volkm, (314) 'l2u2. A point par t ic le , with speed v, mass m, and k ine t ic energy m2/2 = p2/(Zm), sca t te rs elastically from the curved surface of the f ixed particle. The two angles a and 8 define a collisim, a being measured re la t ive t o the f ie ld direction end B being measured r e l a t i v e t o the normal following the co l l i s ion .

112 (V/V$ t W0/4 *re

2 f igure 2. Cycle of 20 co l l i s ions s t ab le a t a f i e ld strength of E = 3.69 p /mu. This pattern corresponds t o the 20 spots which develop i n the probabili ty density function shown i n Figure 3. Collisions 1, 6, 11 end 16 i n the cycle are labelled i n Figure 2 and correspond t o the rightmost dot i n Figure 3.

Figure 3. Field-strength dependence of the probability densi ty g(a , B)/cos 6 with increasing f ie ld s t rength. This density function is unity for zero f i e l d and approaches 20 4 functions ( 5 c lus t e r s o f 4) a t a f ield s t rength o f 3.69 p /mu. 2

Figure 3a. ing i n Figure 3 wi th E = 4p /mu.

Enlarged view with three times as many points o f the '@eyea appear- 2

Figure 4 . Trapped t ra jec tory corresponding t o the meye" seen i n Figure 3 a t a f ie ld E = 4p /mu. The motion is quasi-periodic and stable. 3

Figure 3. Nonlinear dependence o f conductivity K (mean veloci ty divided by f i e l d strength) on f ie ld s t rength E. The present work, shown 8s a ser ies o f f i l l e d circles is compared t o the low-density 8oltmnn-equation r e su l t s [6 ]

by introducing io = 0.29713 ma/p, the mean time between successive co l l i s ions a t zero field. the present high-density results a t zero f ie ld . Uachta and Zwanzig's zero f ie ld conductivity is indicated by X.

The Boltmnn-equation results are sceled t o match

Figure 6 .

face. The figure compares the first (left) end second (right) approximations based on matrix i t e r a t ion w i t h a 180 x 180 grid i n a 4 space.

Representation o f the probability density g(a,b)/cus 6 as a sur-

- 18 -

Figure 7. The regions corresponding to colllsfons with particular neighbors of the half-disk shown at the bottom of the three sketches. A t zero field strength, the boundaries of these regions exhibit three-fold symnetry, corres- ponding to the rotational symrretry of the trfangular lattice. mis symnetry Is lost at finite fields. Iteration o f the boundaries results in the fractal structure evident i n Figures 3 and 6. The shaded regions correspond to trajectories starting on the half disk and ending on the same particle. Collisions with the third neighbors at a field of 0.5 p2/m form a small, barely perceptible 3 sided region near a = 3r/4 and 8 = - r/8. neighbor collisions are m r e visible at a field o f p /mu near a = 3 d 4 and 6 = 0. a and 6 describe the half-disk post-collfsion configuration and direction of motion.

nfrd 2

References :

[*I This work was carried out under the auspices of the United States Department of Energy at the University of California's Livermore National Laboratory under Contract U-7405-Eng-48.

[l] M. Kac, "Probability., Scientific American 211, 92 (September, 1964).

[2] W. C. Hoover, Molecular Dynamics, Lecture Notes in Physics, Volume 258 (Springer-Verlag , Heidelberg , 1986)

/ [3] S. Nose, "A Unified Formulation o f the Constant-Temperature Molecular Dynamics Method', Journal of Chemlcal Physics 81, 511 (19841, and "A Molecular Dynamics Method for Simulations i n the Canonical Ensemble", Molecular Physics - 52, 255 (1964).

/ (41 H. A. Posch, W. C. Hoover, and F. 3. Vesely, Wynamics of the Nose Oscillator: Order, Chaos, and Instability', Physical Review A a , 4253 (1986).

(51 3. Machta and R. N. Zwanzlg, mDiffusion In a Periodic Lorentt Gas", Physical Review Letters a, 1959 (1983).

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16) W. C. HOOYer, "Nonlinear Conductivity 8nd Entropy l n the Two-gOdy bltzmann Cas., Journal of Stetisticel Physics 42, 587 (1986).

[7] C. P. C(orrlss, %hear Flow i n the f w o - b d y Bolttmann Gas", Physics Letters m, S 9 (1985).

[SI 1. C. Hoover and K. W. Kratky, "Heat Conductivlty of Three Perfodic Hard Disks y& N o ~ l l l b r i u n Molecular Dynamics", Journal of Stetisticel Physics 42, 1103 (1986).

[9] K. W. Kratky end W. G. Hoover, 'Hard-Sphere Heat Conductivity via Nonequilibriur: Molecular Dynamicsm, Journal of Stat istical Physics (to appear i n 1987).

[lo] 3. 0. Fanaer, E. OttD and 3. A. Yorke, "The Dimension o f Chaotic Attractors", physica 70, 153 (1983).

I l l ] B. 8. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, San Francisco, 1982)

(121 W. G. Hower and H. A. Posch, 'Direct Measurement of Lyapunov Exponentsm, Physics Letters m, 82 (198s).

[13] W. 6 . Wver, H. A. Posch, 8. L. Holian, and S. Bestiale. "Dense-Fluid Lyapunov Spectra American physical Society (1987).

Nonequilibriun Molecular Dynamicsm, Bulletin of the

[14] W. G. Hawer, aAtomistic Nonequilibriun Computer Simulations", Physica - 118A. 111 (1983).

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APPENOIX: Collisions with t h e disk and sides of the hexagonal cell based on a Taylor series expansion i n be .

The exact t ra jectory is described i n Section If o f the text . The equations of motion can be wrl t ten as

x = xo - (p 2 /M) i n (sinWsin8,)

Y = Yo - (P2/M)(9 - eo) (1)

Expanding around eo i n a Taylor series t o second order i n de = 8 - 8 we have: 0

x = xo - (p 2 /R+)[cote, ae - csc2eo(m 2 1 2 ) )

Solving t o second order t h e in te rsec t ion of the t ra jectory with t h e disk

x 2 + y 2 = u / 4 = 2 .

1x0-(p 2 /mE)rcote,,Ae - CSC 2 2 /2)1i2 + (yo-(p 2 /~E)AoJ~.

2 Keeping terms o f order be o r lower we have the quadratlc equation:

s o ~ u t l o n o f t h e quadratic equatlm gives an approximate value o f A6 mich is then used i n equation (1) to find an exact point on the t r a j ec to ry close t o the intersection with the disk. The process is repeated to convergence by

calculatlng new coefflclents o f the quadratic equatlm. Me found that typi-

ca l ly , for flelds E 5 p /(mu), three LteratLons were su f f i c i en t to ca l cu la t e t h e coordinates at t h e Intersect ion t o seven s ignif icant figures.

2

To f ind the intersect ion with the left and t l g h t rldes of the helcagwral cell, we note that the equation of the left ride 1s glven by the stralght llne

1 I2 - 1/2 y = 3 x + p

where p is the densl ty r e l s t i v e t o the maxlmun close-pecked density. In the calculat ions descrlbed here p = 415.

Solving t o second order in A0 for the intersect ion with the traJectory we have

1/2x + p 2 [ (3/4)1’2 C s ~ ~ e ~ / ( p ~ / f l € ) ] Ae2 + [(l-31/2c~teo)/(pz/,) J A8 + 13 - Yo) = 0

Similarly for t h e r lght s ide of t he hexagonal cell, the s t ra ight l i n e 1s Qlven by

112 - 1/2 y = - 3 x + p

and the quadratic equation becomes

- 22 -

00 O0 OO E-

-

- 23 -

3

2

1

0

- 3

-2

-3

tb \ r

- 4

I I I I I t

- -r

- 1 0 1 2 3 4 5 6 1 xi I ’ -

I

E = 3.00 p2/ma

L E = 5.00p2/ma i 7 j 1

0

- 1

I

- -

I

- 25 -

E = 4.00p2/ma 1

- 1- 0 n

- 26 -

1

0

I I

2 Trapped trajectory at E = 4p /ma I I

-0.5 0.5

v

0

- 27 -

0 0

- LY - E = 0.0p2/ma E = 0.6p2/ma E = 1.0 p2/ma

E = 2.0p2/rna -- E = 2.5 p2/rnu

!

I

. . . . . .

8 \

\

E = 3.0p2/ma

E = 7.0 ~ ~ / m a E = 10 p2/ma

A,