Incomplete relaxation in a two-mass one-dimensional self-gravitating system

PHYSICAL REVIEW E 68, 056120 ~2003!

Incomplete relaxation in a two-mass one-dimensional self-gravitating system

Kenneth R. Yawn and Bruce N. MillerDepartment of Physics, Texas Christian University, Fort Worth, Texas 76129, USA

~Received 1 July 2003; published 24 November 2003!

Due to the apparent ease with which they can be numerically simulated, one-dimensional gravitationalsystems were first introduced by astronomers to explore different modes of gravitational evolution. Theseinclude violent relaxation and the approach to thermal equilibrium. Careful work by dynamicists and statisticalphysicists has shown that several claims made by astronomers regarding these models were incorrect. Unusualfeatures of the evolution include the development of long lasting structures on large scales, which can bethought of as one-dimensional analogs of Jupiter’s red spot or a galactic spiral density wave or bar. Theexistence of these structures demonstrates that in gravitational systems evolution is not entirely dominated bythe second law of thermodynamics and also appears to contradict the Arnold diffusion ansatz. Thus it is correctto assert that the one-dimensional planar sheet gravitational system is the nonextensive analog of the Fermi-Pasta-Ulam model of dynamical systems. This paper is an extension of a preliminary study where we conclu-sively showed mass segregation and equipartition of kinetic energy in a two-mass planar sheet system for thefirst time. Here we employ both mean-field theory and dynamical simulation to more thoroughly probe thestatistical and ergodic properties of these systems. Valuable information is obtained from local and global timeaveraging, and temporal and spatial correlation functions. Using these tools we show that the system appearsto approach the equilibrium distribution on very long time scales, but the relaxation is incomplete.

DOI: 10.1103/PhysRevE.68.056120 PACS number~s!: 05.90.1m, 05.70.2a, 05.20.2y, 95.10.Fh

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I. INTRODUCTION

The study of one-dimensional models has historicabeen motivated by the simplicity of the physical systemMany of these systems have analytical or nearly analytsolutions. In other cases, although the system is physicsimple, the equations do not admit an analytical solutionare nevertheless straightforward to set up and dynamicsimulate on a computer. One-dimensional systems wamong the first simulations done on computing machinethe late 1940s and early 1950s. One of the first and mfamous numerical experiments was conducted by FePasta, and Ulam in 1955 on a one-dimensional arraycoupled oscillators with force terms that were small pertbations from linearity@1#. The unexpected nonergodic resuof the experiment would cause an explosion in dynamphysics and chaos theory several years later. Odimensional models of gases have been studied extensin statistical mechanics and one-dimensional lattice modhave been used in solid state physics to study metal allmagnetic spin systems, glasses, phonon propagation,tronic bands, and a host of other systems. Due to the adcomplexity of the physics in higher dimensions, ondimensional systems are often used as simpler modelactual three-dimensional systems.

A central problem for stellar dynamics is the determintion of the time scale for the relaxation of an isolated, gratationally bound system, such as a galaxy or a globular cter. A primary reason for the difficulty is the extremecomplicated nature of physics in three dimensions. Since1960s, the one-dimensional~1D! self-gravitating system habeen used as the simplest model for studying the dynamproperties of gravitational systems, such as relaxationdiffusion. The study of 1D gravitational systems is motivatby two primary factors. The first is the hope that some of

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general features of the 1D system may translate to 3D asphysical systems, perhaps giving some unique insightthe underlying physics. The second is that 1D systems apon the surface to be relatively simple but often exhibit intesting and unexpectedly complex behavior. This hastainly proved true over the many years that these systhave been studied.

Correlating results derived from one-dimensional gravitional systems to three dimensions is a difficult task, sincis well known that dimensionality plays a critical role imany physical phenomena. Despite this difficulty, seveimportant similarities do exist between one- and thredimensional gravitational systems and these are summarbelow.

~1! Both 1D and 3D systems have long range unscreegravitational forces.

~2! For systems containing a large number of particexisting in a stationary state, the distribution of particle psitions and velocities is governed approximately by the Vsov equation for incompressible fluid flow@Eq. ~1!#.

~3! Evolution of the systems from a highly nonstationastate appears to go through a period of violent relaxatafter which the distribution quickly settles down to a quaequilibrium state.

It is worth noting that three-dimensional gravitational sytems suffer from several problems that do not exist in odimension.

~1! Through three body or higher-order interactions, pticles can gain enough energy to escape~or evaporate! fromthe system.

~2! Through three body or higher-order interactions, pticles can lose enough energy to form gravitationally boupairs ~binaries!.

~3! A singularity exists in the gravitational force atr50.

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K. R. YAWN AND B. N. MILLER PHYSICAL REVIEW E 68, 056120 ~2003!

To manage these problems in practical computer simtions, two- and three-dimensional programs require elabocutoff schemes for the gravitational potential. In contrasince the gravitational field in one dimension is uniform, tphase space is compact, and the equations of motionparabolic and easily solvable on a computer without theof slow integration techniques. This allows the completionvery long simulations of systems with moderately largeN ina few days or weeks on desktop workstations.

Large astrophysical systems, such as galaxies, maytain upwards of 1011 stars. Gravitational systems are unlikgaseous systems~driven by true short-range interactions! orplasmas~where the range of the electric force is limited bDebye screening! in that each particle continuously feels theffect of all the others. As the number of particles in tsystem is increased, however, the effect of any single parrelative to the background composed of all other particdecreases as 1/N. In the limit that N→`, while the totalmass is kept constant, only the long-range interaction ofparticle with the continuous background field remains. In t‘‘mean-field’’ approximation, the evolution of large gravitational systems is exactly described by the Vlasov, or cosionless Boltzmann equation, Eq.~1! @2#,

] f

]t1v•

] f

]x2“F•

] f

]v50, ~1!

whereF is a solution of the Poisson equation. For systewith finite N, the solution to Eq.~1! provides a short-timeapproximation to the single-particle distribution function.one dimension, the Vlasov equation describes the flow oincompressible fluid inm5(x,v) space, where them spaceis a projection into the (x,v) plane of a point in the2N-dimensional phase space. Them space then containsNpoints, each of which represents the position and velocitya particle in the system. In the limit that the number of pticles,N, approaches infinity while constraining the total sytem energy and mass to constants, the system can bescribed as a continuous fluid inm space. This limit is oftenreferred to as the Vlasov limit. Liouville’s theorem states ththe area of an element in them space is constant under thaction of the Hamiltonian. Therefore, the flow of this fluidm space is incompressible and obeys the Vlasov equatio

Following a suggestion by Oort@3#, the one-dimensionaself-gravitating system~OGS! was first considered asmodel for the motion of stars perpendicular to the plane ohighly flattened galaxy by Camm@4#. Camm’s analysis wasthe first to apply the Vlasov equation to the 1D system,assuming that the long-range gravitational forces washany effects of stellar encounters, making the system estially collisionless. With the assumption of separability of tdistribution function, Camm derived the stationary isothmal solution to the Vlasov equation. In 1967, Lynden-B@5# published a theory describing a rapid relaxation proctermed ‘‘violent relaxation,’’ in which large changes in thgravitational potential are driven by rapid fluctuations in tmass distribution. Cohen and Lecar@6# used the 1D systemto study Lynden-Bell’s theory of violent relaxation and founthat the systems do relax to a stationary state after a s

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time, but not necessarily to those predicted by Lynden-Betheory. Later, Cuperman, Hartman, and Lecar@7# investi-gated violent relaxation and the applicability of Vlasov dnamics. A breakthrough came when Rybicki@8# first derivedthe exact single-particle equilibrium distribution function fthe discrete one-dimensional self-gravitating system in bthe canonical and the microcanonical ensemble for arbitrN. In the Vlasov limit, these distributions were shownconverge to the isothermal solution of the Vlasov equatfound earlier by Camm.

In principle, additional information can be obtained frothe statistical distribution of particle pairs. Monaghan@9#,and Fukui and Morita@10# showed that for the OGS in thVlasov limit, the two-particle distribution function is thproduct of the single-particle distribution functions, and crections are of order ofN21. In contrast with the single-particle distribution, at the present time an analytical exprsion for the pair distribution is not known, and therefothere is no exact method for computing the effects of twparticle or higher-order correlations in a system with a finpopulation.

Computer simulations of finite population 1D systemshow that they tend to progress through several quasiequrium ~approximately stationary! states as they evolve fromarbitrary initial conditions. These quasiequilibrium statesten last for extremely long times. When a simulation is intiated from a nonstationary distribution, fluctuations causby the changes in the mean-field potential rapidly decaysulting in a state of microscopic relaxation. This processfrequently referred to as violent relaxation@5#. In this state,the system is virialized~i.e., 2̂ KE&/^PE&.1), and the sys-tem energy is distributed approximately equally betweenthe particles in what is called equipartition of total energThe time scale for microscopic relaxation is distinguishfrom the much longer time scale for macroscopic relaxatto thermal equilibrium.

For anN-particle 1D gravitational system, a single poiin the 2N-dimensional phase space (G space! defines thestate of the system. The system trajectory in this spacgoverned by Hamiltonian dynamics as the state of the syschanges. Because of inexact knowledge of initial conditioa probability density can be defined that indicates a rangpossible states that the system can occupy. For a conservsystem, such as the OGS, the total energy is a constantdefines an isolating integral that will restrict the trajectorythe system to the energy hypersurfaceSE defined by theHamiltonian.

In order for a system to approach equilibrium fromarbitrary initial condition, it must exhibit the properties oboth ergodicity and mixing@11#. A system is ergodic if thephase-space average of a dynamical quantity is equal totime average. Ergodic flow can exist only if there areother isolating integrals that will prevent the system frosampling the entire energy surface. Ergodicity implies tall areas of the energy surface are equally accessible andall states on the energy surface are equally probable, thussystem will spend equal time in equal areas of the phspace. Qualitatively, mixing is described as the spreadingof the phase-space probability density as the system evo

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INCOMPLETE RELAXATION IN A TWO-MASS ONE- . . . PHYSICAL REVIEW E 68, 056120 ~2003!

If a system is initially far from equilibrium, then the probability density may fill a localized area on the energy surfaAs the system evolves under the action of the Hamiltonithis area will move over the energy surface. If the systemmixing, the probability density will begin to spread itself oueventually covering the entire energy surface equally whpreserving its initial area.

When investigating gravitating systems it is importantkeep several things in mind. Often, computer simulationsgravitating systems~which are necessarily finite in the population! are studied with, and compared to, Vlasov dynamiThe Vlasov limit is a singular limit in the sense that infinitime averages of dynamical quantities cannot yield thepected equilibrium results if the limitN→` is taken prior totheT→` limit. This follows from the fact that a true Vlasosystem has an infinite number of stable stationary solutiand Casimir invariants, and therefore does not convergthe unique maximum entropy state; i.e. Vlasov dynamdoes not obey the second law of thermodynamics. Thisbeen demonstrated by numerical integration of the Vlaequation for the OGS where it was shown that structurethem space appear to persist indefinitely without any signdissipation@12#. On the other hand, a finite population sytem may sample an approximately stationary stateclosely resembles a stationary Vlasov solution for a lotime. However, for the system with finite population, it is ntruly stationary and will slowly drift away due to discreparticle effects. Therefore an infinite time limitT→` of adynamical quantity in the discrete system followed by tVlasov limit N→` will give the equilibrium solution pro-vided the system is mixing.

Prior to 1982 the prevailing thought was that the OGshould relax to equilibrium from an arbitrary initial state intime scale on the order ofN2tc , wheretc is the approximatetime it takes a particle to make one complete crossing~fulloscillation! in the system@13#. However, Wright, Miller, andStein @14# showed that there was no relaxation in a time2N2tc . Subsequent studies by Leuwel, Severn, and Roseeuw@15# showed relaxation in a timeNtc from specialinitial states chosen close to equilibrium. From that wothey predicted relaxation on this same scale for any inisystem chosen close to equilibrium. Reidl and Miller@16#refuted that claim by showing that the relaxation time dpends very sensitively on the initial quasiequilibrium stachosen. Comparison of the position distribution of compusimulations with the prediction of Rybicki’s discretN-particle distribution function continued to show major dcrepancies even after long run times. Particles were mconcentrated toward the center and toward the outer ethan predicted. In addition, Reidl and Miller@17,18# did er-godic studies of 1D systems containing from 2 to 20 particlooking for evidence for strong ergodic behavior whiwould result in eventual thermal equilibrium. They foundtransition atN511 particles where previously stable regioof the phase space became ergodic and mixing. In theirsequent study of early relaxation, Reidl and Miller@19#showed a maximum mixing in phase space when the syspopulation was on the order ofN530.

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During this same time period, Tsuchiyaet al. @20,21# be-gan studying the approach to equilibrium of the single-mOGS using a measure of the equipartition of total systenergy among all the particles. This measureD(t) is thedeviation of the average energy per particle from the thretical equipartition value given by the virial theorem. Thestudies showed that after an initial short period of micscopic relaxation in whichD(t) is approximately constantD(t) steadily declines in a linear fashion on a log-log scindicating a power-law decay. After some period of timelarge peak appears which was interpreted as the onsethermal equilibrium. Shortly afterward however, using bosmall and large-N systems@22# ~Paper I!, we demonstratedthat this phenomena was not the onset of equilibrium butit might be evidence for the system becoming trappedlong periods of time in restricted regions of the phase spaTypically, if the system is followed long enough, many peaare seen to appear after the first initial large peak, indicaperiods where the system is restricted from equally explorthe entire energy hypersurface. Following up on their earwork with a study of the distribution of particle energieTsuchiyaet al. @23# showed evidence for continual transtions back and forth between a state closely mimickingisothermal distribution~isothermal-like! to a far from equi-librium distribution~transient or itinerant state!. This will bereferred to as the transitional distribution period.

It was recognized by a number of investigators thatalternative approach for studying convergence to equilibricould be found by employing two different mass speciesthe OGS@24,25#. The eventual occurrence of equilibriumwould then be characterized by the equipartition of thenetic energy as well as the spatial segregation of the heand light particles. However, perhaps due to algorithm ahardware limitations, early attempts failed to obtain a covincing result. To complement Tsuchiya’s investigationsthe single-component system, we investigated the evoluof a two-component system consisting of equal populatiof each species with a 3:1 mass ratio@26# ~Paper II!. Weobserved that both equipartition of kinetic energy and msegregation occurred after approximately 106 system cross-ing times. To our knowledge, this was the first time that tphenomenon was conclusively demonstrated for this systThe time scale is on the same order as the occurrence otransitional distribution period discussed above for a singcomponent system.

In other recent work dynamicists have studied the averdivergence of phase-space trajectories of the OGS. PosLyapunov exponents are attributes of strong ergodic propties, such as mixing. Earlier, Tsuchiyaet al. @20# and morerecently Milanovicet al. @27# and Tsuchiya and Gouda@28#investigated the Lyapunov spectra of the OGS. At this tithere are unresolved questions concerning the convergof the spectra to a universal scaling function asN becomeslarge. Power-law dependence on population was foundboth maximum and minimum exponents, both of which dcrease with increasingN. This suggests an approach tomore integrable system as the population increases. Orecent work has attempted to relate this model more direto physical stellar systems such as globular clusters@29#.

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While the preceding review has focused on studies ofisolated one-dimensional gravitational system, it shouldnoted that they have also played a useful role in studiestructure formation in cosmology. By including the Hubbexpansion in the dynamical evolution it is possible to deonstrate the formation of clusters and ‘‘voids’’ in them space@30,31#. A few models have been considered and, in commwith galaxy observations, they demonstrate a bifractal disbution of matter. They also provide the only model that eactly follows the Zeldovich ansatz@32,33# between crossings@34#.

This paper continues the investigation of multiple-macomponent systems started previously by us in PapeThere, as a preliminary study, a single system size and mratio was investigated. The results conclusively showedthe first time mass segregation and equipartition of kineenergy. Here the complete mean-field theory is developedan equilibrated two-mass system in the Vlasov limit usthe maximum entropy principle. Coupled differential equtions for the resulting density functions in them space foreach mass species are derived and can be integrated nucally for any given temperature and mass ratio. Theyused to estimate the particle distribution of systems withnite population in equilibrium.

Dynamical simulations of systems with varying popution (N516, 32, 64, 128! and three different mass ratio~2:1, 3:1, and 5:1! were carried out on the computer. Thobserved equipartition of kinetic energy~the ratio of the ki-netic energies of the heavy to light masses! and mass segregation were compared with the theoretical predictions.addition, an extensive range of statistical measures calated from the numerical simulations are used to investigproperties that cannot be predicted from mean-field theThese include the spatial correlation between particle ptions, the time correlation of kinetic energy fluctuations, aa measure of the rate of relaxation toward equilibrium.shown below, surprising results are obtained, some of whappear to contradict the conventional wisdom concerngravitational systems.

II. DESCRIPTION OF THE SYSTEM

The two-mass component 1D self-gravitating systemcollection ofN labeled, planar sheets, each of constant mdensity, infinite in they-z plane, which are constrained tmove along thex axis under their mutual gravitational attration. It is convenient to introduce an auxiliary labeling whicincreases in the direction of increasingx. Therefore, in speci-fying the auxiliary particle number, sayj, the acceleration ofthat particle is uniquely defined and vice versa.

Only gravitational forces are considered~i.e., no short-range interactions!. Therefore, on crossing, they merely pathrough one another. Since the sheets are infinite in exthe gravitational field in thex direction is uniform. Clearly,the system can be thought of as a collection of particles ewith massmj constrained to move in one dimension. Tgravitational acceleration is found through Poisson’s eqtion

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2]A~x,t !

]x54pGr~x,t !, ~2!

wherer(x,t), the mass density, is given by

r~x,t !5(j 51

N

mjd„x2xj~ t !… ~3!

and G is the universal gravitational constant andd is theDirac delta function. Therefore,

A~x,t !52pG(j 51

N

mjS„x2xj~ t !…, ~4!

whereS(x) is defined as

21, x.0

S~x!50, x50

1, x,0. ~5!

The acceleration experienced by thej th particle from the lefttherefore depends only on the difference in the mass ofticles on the right and on the left and can be expressed sply as

Aj52pG~MR2ML!. ~6!

At the instant when two particles cross, the velocities dochange. Therefore the velocity of each particle variescon-tinuously in time while, at a crossing, each particle expeences a discrete jump in the acceleration. Between crossthe particle positions evolve quadratically in time. Thchange in the acceleration of thej th particle during a crossing is

DAj564pGmj 71 ~7!

where the top signs correspond to the case where thej thparticle has a crossing with a particle coming from the land the bottom signs to a crossing from the right. The Hamtonian of the system is

H5(j 51

N pj2

2mj12pG(

j , imimj uxi2xj u, ~8!

wherepj , xj , andmj are the momentum, position, and maof the j th particle, respectively. It is customary to define tcharacteristic period of a particle in the system astc5(Gr0 /p)21/2, wherer0 is the equilibrium mass densitevaluated at the origin andG is the universal gravitationaconstant. This represents a typical period of oscillation oparticle in the system.

Since the potential energy is a linear homogeneous fution of the coordinates, all dependence on parameters cascaled away by introducing convenient units as follows:

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pMGGF E

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whereL is the length,V is the velocity,T is the time,G is theuniversal gravitational constant, andE and M are the totalsystem energy and mass, respectively. The dimensionunits of acceleration, velocity, position, and time are thgiven by

A→a5A

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X→x5F3pM2G

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T. ~10!

For simplicity set

M51 and 2pG51, ~11!

making the total energy of the systemE5 34 , and the charac-

teristic period of oscillationtc52pt. These units will beadopted throughout the remainder of this paper.

III. VLASOV „MEAN-FIELD … THEORYFOR THE TWO-MASS SYSTEM

Here we derive the equilibrium solution for the probabity distribution and mass density of the two-mass systemthe Vlasov~or mean-field! limit. In this limit all correlationsbetween particles vanish and, therefore, the derivation bewith the assumption of statistical independence. Coupthis assumption, Eq.~12!, with the Hamiltonian Eq.~8! anexpression, Eq.~19!, is obtained for the entropy. Subjectconstraints on the energy and normalization of probabithe entropy is maximized using the method of Lagrange mtipliers, Eqs.~21! and ~22!. The solution yields the reducepartition functions, Eqs.~36!, and the form of the particledensity functions, Eq.~37!. With these and the boundarconditions at the origin and at infinity, Poisson’s equatigivesa first-order differential equation for the potentialw inthe two-mass mean-field theory, Eq.~49!. The remainingconstantsA and B are solved with the help of Eq.~53!. Afinal integral forx as a function ofw is obtained and solvednumerically. Inversion of the solution yieldsw as a functionof x. All of these elements can be applied to Eq.~37! tocalculate the density of each species as a function of posthat can be compared with results obtained from simulatio

Consider theN-particle distribution functionf (N)(xi ,v i)for a system with two mass types. In the Vlasov limit

f (N)5)a

f a)b

f b ~12!

is the product over the single-particle distribution functioof each mass type, where

)a

f a5)j

f a~xj ,v j !, etc. ~13!

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The two distribution functions,f a and f b , will independentlysatisfy the Vlasov or Collisionless Boltzmann equation,

] f a~x,v !

]t1v~ t !

] f a~x,v !

]x2

]w~x!

]x

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]v50, ~14!

] f b~x,v !

]t1v~ t !

] f b~x,v !

]x2

]w~x!

]x

] f b~x,v !

]v50, ~15!

where the acceleration

a5]v]t

52“w~x!. ~16!

For a stationary distribution, the functions are independentime so the first terms on the left are expressly zero. Tentropy of the system is given by

S52k Tr@P ln P#

52kE dG f (N)ln f (N), ~17!

whereP is the probability that the system is at a particupoint in the phase space andk is Boltzmann’s constantTherefore, from Eq.~12!,

S52kF E dGa)a

f a ln )a

f a1E dGb)b

f b ln )b

f bG~18!

which reduces to

S52kFNaE dmaf a ln f a1NbE dmbf b ln f bG . ~19!

Let aa5Na /N andab5Nb /N whereNa andNb are thenumbers of particles of mass typesa andb respectively, i.e.,aa andab are the fractions of the total number of particleseach type. Then the total entropy per particle is

s[S

kN52aaE dmaf a ln f a2abE dmbf b ln f b . ~20!

Expressed in this form,s survives the Vlasov limit. Next weconstruct the extremum ofs given the constraints

E5^H& ~21!

and

E dmaf a5E dmbf b51. ~22!

The energy, given by the statistical average of the Hamtonian, is

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2MbE dmbf bvb

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1pGMa2E dmaE dma8uxa2xa8u f af a8

1pGMb2E dmbE dmb8uxb2xb8u f bf b8

12pGMaMbE dmaE dmbuxb2xau f af b , ~23!

wherema85(xi8 ,v i8), f a85 f (ma8), andMa is the total mass ofparticles of type a, such that Ma5Nama . IntroducingLagrange multipliersb,ga ,gb , let

I 5s1b~E2^H&!1gaS 12E dmaf aD1gbS 12E dmbf bD .

~24!

We need to findf a and f b such that

daI 5dbI[0, ~25!

where da represents the variation with respect tof a , andsimilarly for db . Linearizing ind f a andd f b , we find

daI 52aaE dmad f a~11 ln f a!2bMa

2 E dmad f ava2

2pGbMa2E dmad f aE dma8uxa2xa8u f a8

2pGbMa2E dma8d f a8E dmauxa2xa8u f a

22pGbMaMbE dmad f aE dmbuxb2xau f b

2gaE dmad f a , ~26!

and similarly fordbI . Collecting terms and using the symmetry of the dummy variables gives

daI 5E dmad f aF2aa~11 ln f a!2bMa

2va

2

22pGbMa2E dma8uxa2xa8u f a8

22pGbMaMbE dmbuxb2xau f b2gaG ~27!

and finally the first variation with respect to typea is

daI 5E dmad f a@2aa~11 ln f a!2bMah~va ,x!2ga#50,

~28!

where

05612

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21w~x! ~29!

and

w~x!52pGFMaE dx8ux2x8una~x8!

1MbE dx8ux2x8unb~x8!G1const ~30!

is the gravitational potential. In Eq.~30!

na~x8!5E f a~x8,v !dv ~31!

is the single-particle density function for particles of massaand similarly for massb. The term in@ # in Eq. ~28! mustequal zero; therefore,

2aa~11 ln f a!2bMah~va ,x!2ga50. ~32!

Solving for f a(va ,x) gives

f a~va ,x!5e2(11ga /aa)e2(bMa /aa)h(va ,x) ~33!

and similarly for f b(vb ,x).A solution for w(x) is needed to substitute back into th

equations forf a and f b . If the equation forw(x) cannot besolved in an obvious manner analytically, it can be donumerically. First note that Poisson’s equation for the todensity is

d2w~x!

dx254pG@ra~x!1rb~x!#

54pG@Mana~x!1Mbnb~x!#. ~34!

It is clear from Eq.~33! that in order to normalize the probability density, the term multiplying the exponential mustthe partition function

Za5E dma2e2(bMa /aa)[(1/2)v21w(x)] . ~35!

To obtainna(x) andnb(x) thev dependence inZa andZb isintegrated out to get reduced partition functions for bomass types,

za5E dx8e2(bMa /aa)w(x8)

and

zb5E dx8e2(bMb /ab)w(x8) ~36!

and therefore

na~x!51

zae2aw(x) and nb~x!5

1

zbe2bw(x) ~37!

where

0-6

396333030


TABLE I. ConstantsA, B, za , andzb .

Mass ratio (b/a) A B za zb

2:1 0.1962125883 0.3037874117 2.4502466167 1.58258443:1 0.1639079303 0.3360920697 2.9331663805 1.43046885:1 0.1238967947 0.3761032053 3.8804008776 1.2782907

m

l

-ra-

d.

a5bMa

aaand b5

bMb

ab. ~38!

Differentiating with respect tow(x) and rearranging gives

na521

a

dna

dwand nb5

21

b

dnb

dw. ~39!

Substituting this into Poisson’s equation, Eq.~34!, leads to

d2w

dx214pGFMa

a

dna

dw1

Mb

b

dnb

dw G50, ~40!

giving

d2w

dx214pG

d

dw FMana

a1

Mbnb

b G50. ~41!

It is possible to show that

1

2 S dw

dxD 2

14pGFMana

a1

Mbnb

b G5const ~42!

is an integral invariant of Eq.~41! thus reducing it to a first-order differential equation. Substituting Eq.~37!, the expo-nential forms forna andnb , into Eq. ~42! gives

1

2 S dw

dxD 2

1Ae2aw(x)1Be2bw(x)5const, ~43!

whereA andB are also constants. The constantsA andB canbe determined by analyzing the boundary conditions. Symetry considerations atx50 give

w~0!50 ~44!

and

Udw

dxUx50

50. ~45!

With Eq. ~43! this implies thatA1B5const. The boundarycondition at infinity,w(uxu→`), implies

w→`⇒Udw

dxU5const. ~46!

From the Hamiltonian Eq.~8! it is seen that far away fromthe mass distribution

w~x!52pGMuxu1const. ~47!

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-

This then gives

1

2 S dw

dxD 2

5A1B51

2~2pGM!2. ~48!

M52pG51 implies thatA1B5 12 . The constant in Eq.

~43! is therefore equal to12 . SubstitutingA1B for the con-stant in Eq.~43! gives the following first-order differentiaequation forw(x):

dw~x!

dx5@2A~12e2aw(x)!12B~12e2bw(x)!#1/2. ~49!

Equations~42! and~43! can be used to find an equation forAindependent ofB giving

A54pGMa

a F E2`

`

e2aw(x)

dxG21

. ~50!

Using Eq.~49! and the fact thatdx5dw(dx/dw) in Eq. ~50!gives

A54pGMa

a F E2`

` dwe2aw

@2A~12e2aw!12B~12e2bw!#1/2G21

.

~51!

Rearranging and using Eq.~11! produces

E0

` dwe2aw

@A~12e2aw!1B~12e2bw!#1/25

A2Ma

aA. ~52!

The simple substitution ofu5e2aw and du52ae2aw thengives

E0

1 du

@A~12u!1B~12ub/a!#1/25

A2Ma

A. ~53!

Equation~53! contains two known quantities,Ma ~the to-tal mass of particles of typea) and b/a ~the ratio of themasses of typeb to type a). This equation was solved numerically for various mass ratios using a Romberg integtion technique to obtain the value of the constantA and thenusingA1B5 1

2 to obtain the constantB. OnceA andB weredetermined, the reduced partition functions, Eq.~36!, wereintegrated in a similar form as Eq.~53!. Table I shows thevalue of A, B, za , andzb for the three mass ratios studieNote that for a single-mass systemA5B50.25.

0-7

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In

ilibfo

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The potential functionw(x) is required to obtain the complete form for the density functionsna(x) andnb(x). Rear-ranging Eq.~49! and integrating gives the following equatiofor x as a function ofw:

uxu5E0

w dw8

@2A~12e2aw8!12B~12e2bw8!#1/2. ~54!

Again, the integral can be converted to a more manageform by making the substitution

u5e2aw8⇒e2bw85ub/a. ~55!

This gives the following form for the integral definingx as afunction of w for a particular mass ratio:

uxu521

aE1

e2awdu

u$122@Au1Bub/a#%21/2. ~56!

The integral was evaluated numerically using an adapstep size integration technique and inverted to obtainvalue of w over a range ofx. A cubic spline interpolationalgorithm was used to obtain values for the function inintegrand for any value ofx where the integral was evaluatenumerically. Figure 1 shows the one-sided potential futions for all three mass ratios investigated.

The parameterb defines the temperature of the system.these units, for a system with an energy of3

4 , b52. Thevalue ofb, the fitted functions forw(x), the reduced parti-tion functionsza and zb , and Eq.~37! provide all the ele-ments needed to calculate the density of particles in equrium and compare them with the results of simulationssystems of varying mass ratio and particle number.

FIG. 1. Potential function from the mean-field theory for t2:1, 3:1, and 5:1 mass ratio OGS. All units are dimensionless.

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ee

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IV. TWO-MASS DYNAMICAL SIMULATIONS

The dynamical simulations of two mass-species 1D gratational systems were run on high-speed workstations rning under the Linux operating system. Since, betweencounters, the particles fall freely toward each other untheir mutual gravitational attraction, the solutions to simpquadratic equations are all that is required to determineparticle positions and velocities at future times. Insteadupdating the entire system at each encounter, by takingvantage of the simplicity of the system clever schemesbe devised to efficiently update the positions and velociof only the encountering particle pair and its nearest neibors. The algorithm first calculates the time when eaneighboring pair of particles will have a crossing. The smaest crossing time is found and that pair of particles is allowto evolve up to the time of crossing. The new crossing timfor that pair and their nearest neighbors is calculated, andsmallest crossing time in the list of pair crossing is fouagain. This process continues as the system is alloweevolve and statistical data are taken at specific time intervIn addition, at regular intervals the entire set of coordinaand velocities is updated to a common time and the crostime table is recalculated as another measure in reducumulative numerical errors which might cause particle psitions to get out of order. For systems larger than 64 pticles, it is more efficient to use a binary tree to store atrack the particle information rather than traditional linestoring and sorting techniques. These systems were alloto evolve for a total time of 108t.

Two types of data files were maintained as the numerexperiments progressed. First, ‘‘snapshots’’ of all the partipositions and velocities were taken at fixed intervals1000t for the entire length of the simulation. These snashots are a record of the position and velocity of all systmembers at a particular time in the system evolution. Tadvantage of snapshot data is that all averages can be clated after the completion of the simulation to save calcution time, and both forward and backward averages cancalculated if desired. In addition, calculation of particle desities and correlations is simplified with the snapshots. Insecond type of data file, running averages were calculaand saved independently. These data were taken at inteof much shorter duration than the snapshots during the esystem evolution, and at longer time intervals than the snshots during the late stages of evolution. Thus the short-tdynamics could be studied more accurately, while effectuse of storage was permitted during the later stages.

For this study, the ratio of the mass of the heavy to ligparticles was varied as well as the total number of particin the system. Five different system populations with thrdifferent mass ratios were investigated. The specific pareter values used are displayed in Table I. Initial conditiowere generated by uniformly sampling points inside a boxfixed size inm space for both heavy and light particles. Thsize and shape of the box are determined by the total enand the desired virial ratio, respectively. A virial ratRv ir ial '2 was chosen to reduce the effects of the initviolent relaxation phase. These conditions were chosen

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cause they provide an initial state that~1! is far from equi-librium, ~2! has a kinetic energy ratio which is easily chaacterized, and~3! will rapidly approach a quasiequilibriumVlasov state@23,26#. Since both the heavy and light particleare sampled identically inside the fixed box inm space, thereis no equipartition of kinetic energy or mass segregationthe initial state and it is far from equilibrium. In addition, duto this sampling, the initial value ofRkinetic will be approxi-mately equal to the ratio of the heavy to light masses.Fig. 2 for a view of a typical initial state of the system inmspace for a 5:1 mass ratio, and Fig. 3 for a view of the fistate of the system inm space for a 5:1 mass ratio followina complete run.

V. STATISTICAL MEASURES

In general, the ergodic properties of a system are difficto ascertain. Ergodicity has been proven rigorously for onhandful of dynamical systems. There are, however, somedicators that are useful in determining whether a systemergodic and mixing. The decay of correlations in time is

FIG. 2. Initial state (m space! of a 128-particle two-mass OGSwith a 5:1 mass ratio. All units are dimensionless.

FIG. 3. Final state (m space! of a 128 particle two-mass OGSwith a 5:1 mass ratio. All units are dimensionless.

05612

n

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necessary consequence of mixing behavior and is one oferal measures used to analyze gravitational systems. In PII, equipartition of kinetic energy and mass segregation wtaken as direct signs that the system was sampling the elibrium distribution. In this study, we use dynamical simultion to test the predictions of Vlasov equilibrium theory ffinite size systems. Specifically, we compare the ratio ofkinetic energies of the two mass types, the average distafrom the system center of mass for the heavy and light scies (DH andDL), and the average particle densities of eamass type with the mean-field theory presented in Sec. IIIaddition, theN-body simulations can also be used to stuproperties not addressed by mean-field theory. In thisegory we include both spatial and temporal correlation futions, and the rate of relaxation of the system towards eqlibrium. We also compare the relaxation time scales oftwo-species systems with those studied earlier for a sinmass component.

A. Equipartition of kinetic energy „Rkinetic…

The equipartition theorem states that every coordinateis represented by a simple quadratic expression in the Hatonian of a conservative system~e.g., p2/2m) will, on theaverage, contributekT/2 to the energy.Rkinetic , which is theratio of the average kinetic energy of the heavy mass pticles to the light mass particles (Rkinetic5^KEH&/^KEL&),is used to measure equipartition. As a thermodynamic sysrelaxes, the system members begin to share kinetic enequally on the average. For a system containing two typemasses, this sharing should be easily observed. Thereforthe system relaxes, the ratioRkinetic should approach a valuof unity. This is a measure of macroscopic relaxation to eqlibrium. In an isolated system with a finite number of paticles such as the OGS~microcanonical ensemble!, however,the equipartition theorem may be exact only in the limitlargeN.

B. Mass segregation of heavy and light particles„DH and DL…

As the system evolves in time and equipartition of kineenergy begins, the heavy particles are transferring somtheir kinetic energy to the lighter ones. While the systecontinues to relax, the light particles move farther out frothe center of the system forming a halo, and the heavyticles move in toward the center forming a core, in a proccalled mass segregation. As the system evolves we can tthis process by computing the time average of the distafrom the origin for both the heavy (DH) and light (DL)masses. The separation of these quantities is a anotherindication that the system may be macroscopically relaxto an equilibrium configuration and is closely relatedRkinetic .

C. Particle density averages†ŠNm,j„t…‹‡

Throughout the time of the simulations, data are takenthe density of particles of each mass type (m5H or L forheavy or light! in the configuration space. For each matype, bins of equal probability were created based on

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mean-field theory discussed previously. These bins varsize but are scaled so that each bin will contain 5% ofparticles for a system in equilibrium. Since the potentialsymmetric about the origin, the statistics of the correspoing bins on each side of the origin are combined duringdata collection of the simulations. To create the bins,respective density functions, Eq.~37!, are integrated by stepping in position until the integral is equal to 0.05. This prcess is continued until all ten bins for the positive configration space have been calculated. Since the systemsymmetric, these bins are then mirrored to the negaspace. During a simulation, at regular snapshots in time,number of particles in each bin@Nm, j (t)# is counted and theresults are averaged up to that simulation time. The valuthe end of a complete simulation will give a good long-timaverage for the population of each bin. This can thencompared to the theoretical mean-field prediction to estimthe applicability of the mean-field theory to the actual dnamical system.

D. Relaxation measure†sm2„t…‡

To track the relaxation of the density to the equilibriuprediction, we compare the average density in each bin atime t and 2t. To quantify the difference, we define the sttistical function of time,sm

2 (t), as follows:

sm2 ~2t !5

1

Nbin(j 51

Nbin

@^Pm, j~2t !&2^Pm, j~ t !&#2 ~57!

where

^Pm, j~ t !&52Nbin

N^Nm, j~ t !& ~58!

is the time averaged population of binj up to timet normal-ized to the theoretically determined number of particlesbin, and them5(H,L) indicates either the heavy or lighmasses. This measure is similar to a variance if we consthe value at the later time, 2t, as the true mean. Alternativelwe can think ofsm

2 (t) as a metric between vectorsPm, j (t8)at different times. The rate at which this measure decreamay give useful insights into how these systems relax.example, linear decay on a log-linear plot would indicaexponential decay in time and a characteristic decay timrelaxation could be defined. On the other hand, linear deon a log-log plot would imply a power-law decay and ncharacteristic decay time.

E. Bin correlation

Useful information concerning the spatial correlationparticle pairs can be obtained by computing the correlaof populations in different bins. The bin correlation for eamass type is calculated in the following manner: The averand the variance for each bin is first calculated usingentire dataset covering the full duration of the simulatiusing

sn, j2 5Š~Nn, j2^Nn, j&!2

‹ ~59!

05612

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at

ete-

he

r

er

esr

toay

fn

ee

where n indicates the mass types, andj indicates the binnumber. Next, the correlation between mass types in spebins at a specific timeCm,i ;n, j (t) is calculated using

Cm,i ;n, j~ t !5Š@Nm,i~ t !2^Nm,i&#@Nn, j~ t !2^Nn, j&#‹t

sm,isn, j,

~60!

whereNn, j (t) is the population of mass typen in bin j attime t of the simulation and the average^ & t is taken up totime t.

F. Time correlation †C„t…‡

We can learn about the duration of memory in the systby studying correlations in time. In order to investigate tpossibility of the presence of long-time oscillations or nomal modes that might develop as the system relaxes, colations of the macroscopic kinetic energy ratioRkinetic werecomputed for a steady progression of time delays. Sincelier studies of the single-component system suggested thexists in one of several macrostates for periods of ab106t, the data were averaged over sequential blocks60 000t dimensionless time units. The time correlation funtion Ck(t) essentially gives the average of the product offluctuations of a variable measured at time separationt. Thusthe time correlation inx is defined

Ck~ t !5F 1

N2k (j 51

N2k

xjxj 1kG2^x&2, ~61!

where N is the total number of data points,j labels eachblock of duration t560 000kt, and xj is the average ofRkinetic within block j.

G. Equipartition measure †D„t…‡

In their studies of a single-component system, Tsuchet al. @20,21# proposed a test for the equipartition of totenergyD(t) as a measure of phase-space ergodicity. Fosystem that is completely ergodic over the energy surfathe value of a macroscopic observable is simply the tiaverage of the corresponding microscopic operator oveinfinite time period. Equipartition of total energy is the equdivision of energy among the members of a dynamical stem over an indefinitely long time period. In the conclusiowe will explain that this is a necessary, but insufficiemanifestation of ergodicity@22#. In the infinite time limit, theaverage of the energy per unit mass~specific energy! « i willachieve a unique value for alli,

« i5 limT→`

1

TE0

T

« i~ t !dt5«0[5E

3~62!

for a system with a single-mass species.Convergence to equipartition can be quantified by int

ducingD(t) defined by

0-10

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ns

n-Wheef

hi

vet-to

Insin

niteim

emthede-

ngn.

ini-m

eaupa-Inhe

the

ed

srolyn-cy

ass.

ssriodsta-5inhe

a

mu-


D~ t ![1

«0S 1

N (i 51

N

@« i~ t !2«0#2D 1/2

, ~63!

where« i(t) is the averaged value of the specific energy upa timet. Therefore,D(t) measures the deviation of the aveage energy per particle from the theoretical equipartitvalue. If a system is sampling the equilibrium ensembD(t) will trend monotonically toward zero. However, thvanishing ofD(t) is not a guarantee of equilibrium, and caoccur for other ensembles.

VI. SIMULATION RESULTS AND COMPARISONTO THEORY

Using the algorithm discussed in Sec. IV, simulatiowere carried out for three different mass ratios~2:1, 3:1, and5:1! and four different populations (N516, 32, 64, 128! fora duration of 108 time units. For all runs, the statistical quatities defined in the preceding section were computed.first consider the behavior of the kinetic energy ratio of ttwo species,Rkinetic . In all cases we found that as timprogressed,Rkinetic approached the equilibrium value ounity. Important insights can be gained by studying how tlimit is approached.

Figure 4 shows a lin-log plot ofRkinetic for a 128-particlesystem with 2:1 mass ratio. This graph is a continuous aage of the last 903106t in the negative time direction staring at the end of the run, i.e., a backward average. By sping the backward average after 903106t we reduce theeffect of the initial transients due to violent relaxation.Paper II this backward average was also calculated udata averaged over windows of 60 000t. In that caseRkineticappeared to rapidly approach the expected value of uHere we seeRkinetic approaching unity more slowly as thsimulation progresses since the effect of fluctuations in t

FIG. 4. Linear plot ofRkinetic for a 128-particle systems with2:1 mass ratio backward averaged over the last 903106t of thesimulation. All units are dimensionless.

05612

o

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is not reduced by local averaging. The results for all systpopulations and mass ratios were similar. Of course, assystem populations increase the statistical fluctuationscrease.

Rkinetic was also averaged in the forward direction usithe complete dataset from the initiation of the simulatioFigure 5 is a lin-log plot of a forward averagedRkinetic for a64-particle system with a 2:1 mass ratio. Notice that thetial value will closely, but not exactly, represent the systemass ratio. Also note the presence of a long lasting platfrom about 23102t until 106t where the curve begins to didown toward unity. We should be mindful that this is a logrithmic plot, so early times are emphasized in the figure.fact, the remainder of the evolution is also very gradual. Tdip at 106t occurs at approximately the same time asonset of mass segregation.

To quantify the long lasting memory effects discussabove, a time correlation functionCk(t) was also calculatedfor Rkinetic @see Eq.~61!# for a 64-particle system with a 3:1mass ratio using the 60 000t windowed data. Figure 6 showthe time correlation function rapidly descending toward zeearly, then slowly rising to a local maximum approximate107t later, and finally dropping back down to near zero idicating the possible existence of long- time low frequenmodes in the system.

Mass segregation, the spatial separation of different mspecies, is quantified byDH andDL and occurred in all runsFigures 7, 8, and 9 show lin-log plots ofDH andDL versustime for 128-particle systems with 2:1, 3:1, and 5:1 maratios, respectively. Here as well we see a transient pefollowed by a long plateau where the segregation seemstionary ~but see the comment above concerning Fig.!.Then, after about 106t, a period of rapid divergence is againitiated, and then tapers off towards the end of the run. Tmean-field predictions for the equilibrium values ofDH and

FIG. 5. Lin-log plot ofRkinetic for a 64-particle system with a2:1 mass ratio forward averaged over the entire length of the silation. All units are dimensionless.

0-11

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ndzon-


DL are also indicated on the graphs. Note that the timwhere the curves for the heavy and light masses begidiverge are approximately equal for all three mass ratios.see that by the end of the run the time averaged quanthave not yet achieved their equilibrium values. We alsothat there is no apparent dependence of the relaxationon the mass ratio for this process.

Comparison of time scales for mass segregation betw

FIG. 6. Linear plot of the time correlation function ofRkinetic

for a 64-particle system with a 3:1 mass ratio using the slidwindow of width 60 000t. All units are dimensionless.

FIG. 7. Mass segregation for a 128-particle systems with amass ratio. The vertical axis shows (DH andDL) the average of theabsolute value of the distance from the origin for the light aheavy masses. Predictions from theory are indicated by the horital lines. All units are dimensionless.

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systems of different populations did not reveal any obviopattern either. However, the segregation time scales fortwo smaller systems~16 and 32 particles! were significantlyshorter than for the larger systems~64 and 128 particles!with segregation for the 32-particle 3:1 mass ratio systbeing the shortest clear transition. Although it may be ser

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dipitous, it is intriguing that this result matches well withose of Reidl and Miller@19# for a single-component system, where maximum mixing in phase space was seensystem sizes of approximately 30 particles.

To directly compare mean-field~Vlasov! theory with thesimulations, the configuration space was partitioned into bof equal probability according to the theory~see Sec. III!. Inthe simulations, the time averaged population of eachwas calculated over the course of the run. The results128-particle systems with mass ratios of 2:1, 3:1, and 5:1shown in the summary chart, Fig. 10.

In Fig. 10 we label the bins according to their distanfrom the system center. A general trend is seen in whichbin population increases with the distance from the syscenter. Thus, when compared with the theoretical pretions, the inner bins are slightly underpopulated whileouter bins are overpopulated. It is important to recognizethe outer boundary of the last bin is at infinity. In generagreement with theory is better for both the lighter macomponents and the smaller mass ratios.

The relaxation measuresm2 (t) was calculated by compar

ing the complete dataset of bin populations at the timest and2t @Eq. ~57!#. Thus, as a run progresses, the relative seption in time between the data points is ever increasing. Fure 11 showssm

2 (t) for a 128-particle system with a 2:mass ratio plotted on a log-log scale. Linear decay witslope of approximately21 is observed both before and aftthe peak. The occurrence of this peak coincides withtransitions of bothRkinetic and mass segregation. The finlinear decay indicates that the system is asymptoticallyproaching a stationary or quasiperiodic distribution whileslope indicates power-law decay and the lack of a characistic decay time to equilibrium for the system.

The results forsm2 (t) are very similar to the equipartition

measureD(t) from Paper I for a system with a single-macomponent. A typical log-log plot ofD(t) for a single-mass

FIG. 10. Bin average data for 128-particle systems with 2:1, 3and 5:1 mass ratios. The vertical axis is the average numbeparticles in each bin and the horizontal axis labels the bin numData for both the heavy and light masses are shown. The theocally predicted value is shown as the horizontal line at the value6.4. All units are dimensionless.

05612

or

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system of 64 particles with similar ‘‘waterbag’’ initial conditions is reproduced in Fig. 12. Take special note of the lindecay with slope approximately20.5 both before and aftethe peak. However, the time scales for the peaks are appmately an order of magnitude different.

The correlation of bin populations,Cm,i ;n, j , is displayedin Figs. 13 and 14 in 3D plots. They show the correlatiofor the light and heavy particles, respectively, of a 12particle system with 3:1 mass ratio after a simulation

,ofr.ti-f FIG. 11. Log-log plot ofsm

2 (t) for a 128-particle system with a2:1 mass ratio. Note the linear decay with slope equal to21 bothbefore and after the peak indicating the lack of a characteridecay time. All units are dimensionless.

FIG. 12. D(t) for a 64-particle system with the waterbag initiconditions showing an ‘‘equilibrium’’ peak as defined by TsuchiyKonishi, and Gouda followed by a general trend toward zero.units are dimensionless.

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108t. The plot for the light particles shows a large dowward spike indicating a high degree of correlation betwethe outermost bin and its nearest neighbors. As the separfrom the last bin increases, the correlation moves raptowards zero. For the inner bins, the correlation is somewflat. In the case of the heavy masses, there is a large negcorrelation between the outermost bin and all other bins.magnitude of this negative correlation increases as the sration from the outermost bin increases, i.e., moving towathe center of the system and away from the outermost bi

FIG. 13. Correlation of bin populations after 108t for the lightmasses of a 128-particle system with a 3:1 mass ratio. Notelarge negative correlation between the outer bins and their neneighbors. All units are dimensionless.

FIG. 14. Correlation of bin populations after 108t for the heavymasses of a 128-particle system with a 3:1 mass ratio. Notelarge negative correlation between the innermost and outermbins ~center to edge of the system!. All units are dimensionless.

05612

nionlyativeea-sA

significant negative correlation occurs between the outermand innermost bins indicating system wide influence.

The cross correlation between the heavy and light mbin populations was also calculated for each run. Figureshows the cross correlation for a 3:1 mass ratio 64-partsystem. A large negative correlation is seen between theferent mass types in the outermost bins, while the correlabetween the outer and inner bins tends to grow more posas the separation between bins increases. Plots of the 2:15:1 mass ratios~not shown! have similar trends with theexception that the magnitude of the correlations increasethe mass ratio becomes larger. Since the decay of corrtions is closely related to relaxation, this may indicate thlarger mass ratio systems have longer relaxation timAnalysis of other system sizes also showed similar trenWhen comparing the cross correlations of different systsizes, the magnitude of the correlations steadily increasethe system population increases. This in turn may indicthat the relaxation time increases as a function of the syspopulation.

VII. SUMMARY AND CONCLUSIONS

As an extension of our earlier investigation of the twmass OGS, this paper included a larger variety of syspopulations and mass ratios than the study initially begunPaper II. The original investigation was the first to defintively show mass segregation and equipartition of kineenergy in a two-mass OGS. Here, by expanding the rangstatistical tools used for the analysis the system evolutionbeen probed more deeply.

As a basis for the investigation, a theoretical mean-fimodel was developed in the Vlasov limit using the maximuentropy principle. Potential functions were derived for ea

heest

hest

FIG. 15. Cross-correlation~light to heavy masses! of bin popu-lations for a 64-particle system with a 3:1 mass ratio after 108t.Note the large negative correlation between the outermost binsthe increasing positive correlations between the outermost binthe rest of the system. All units are dimensionless.

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mass ratio and from these predictions for the particle deties. These predictions were compared with results fromdynamical simulations. Additional statistical measures uincluded spatial correlation functions and a time-dependrelaxation measure.

The following statements summarize the main resultsthe two-mass investigation.

~1! Dynamical simulations of all mass ratios and systsizes demonstrated that the macroscopic relaxation proties of mass segregation and equipartition of kinetic eneconverge to their predicted equilibrium values.

~2! Positional correlation functions calculated from dnamical simulations showed large correlations existing eafter 108t ('107tc). This represents an extremely long timin the system evolution.

~3! The 1/t behavior of the relaxation measure,s2(t),indicates the lack of a characteristic relaxation time forsystem, and possible fractal structure in the phase spacaddition, it is contradictory to the exponential decay of crelations one would expect from a normal thermodynamsystem exhibiting ergodic behavior. A peak in the relaxatmeasure occurs at the transition from the long-lived quequilibrium state to another state more closely representaof equilibrium.

~4! Time correlations of the ratio of kinetic energies insystem containing two-mass types showed long-time colations on the order of 1.43107t. This is over 23106 char-acteristic times. In that period of time, an individual particin a 128-particle system will have had over 53108 crossingswith other system members. Clearly relaxation to equilrium, if it occurs in this system, occurs on an extremely lotime scale and weak, but significant, correlations continueexist.

The simulations showed that mass segregation and epartition of kinetic energy occur for all system populatioand mass ratios at roughly similar times with all of thebeing on the order of a few milliont. These macroscopictransitions occur at significantly later times than virializatiand the rapid microscopic relaxation seen in the single-msystems, but it is of the same order as the large first transpeaks seen later in time inD(t) for the single-mass system@20–22#. By themselves, these two simple measuresstrong indicators of a transition to a final equilibrium stafrom a quasiequilibrium state. However, other measures sas analysis of the density and velocity distribution functioindicate that the system has not completely relaxed toequilibrium distribution even after very long time@20,21,27#. For the limited number of runs attempted, anasis of the time scales for mass segregation and equipartshow that the macroscopic relaxation time is independenboth the system population and the mass ratio. However,bin correlation functions indicate that the relaxation timmay indeed be correlated with both the mass ratio andsystem population.

An interesting feature of the relaxation measuresm2 (t) is

the linear decay~on a log-log plot! that occurs both early anlate in the system evolution. This 1/t decay indicates thathere is no characteristic time for relaxation of the systand may indicate a fractal structure to the phase sp

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Klinko and Miller have shown similar results in a systemconcentric spherical shells with angular momentum@35#.However, the decay for that system was strictly monotocally decreasing on average. In this case, the regions otdecay were separated by a large peak whose timing cosponded with other features of the system dynamics sucmass segregation, and equipartition of kinetic energy. Tinitial decay is representative of the long-lived quasiequilrium state. As the system transitions from this state to omore closely representative of equilibrium, the peak insm

2 (t)begins to form as it measures differences in the two staover the 2t2t time interval. This peak insm

2 (t) was veryreminiscent of those that occurred inD(t) for the single-mass system. Following this transition,sm

2 (t) is then mea-suring the approach to the new state at botht and 2t, and the1/t decay is again evident.

As a direct comparison with the mean-field theory, tconfiguration space was divided into bins of equal probaity calculated from the theory. During the simulation a runing average of the population of each bin was kept. Atconclusion of the simulation the average population of ebin was compared to the theoretically predicted value. It wnoted that in all cases the simulations followed the predicdensities fairly well with the inner bins being slightly undpopulated and the outer bins slightly overpopulated. Thresults hold for the larger 128-particle systems as well. Twould also be consistent with results found by Tsuchet al. in which an overabundance of higher energy particwas found to be present in the distribution for a single-msystem that otherwise looked very similar to the isothermdistribution@21#. These higher energy particles tended toside, on average, farther out in the tail of the density disbution. This is reflected in the overshoot ofDL seen in themass segregation plots.

Another interesting result came from the bin correlatioThese results reinforced to some degree the results of thepopulation averages. The correlation measure showed lanegative correlations between the outermost bin and all obins. In addition, the plots of the bin correlation for the ligand heavy masses were quite dissimilar with the light masshowing an unusual negative correlation spike betweenoutermost bin and its nearest neighbor. This spike wasseen in the heavy mass correlations where there waroughly consistent negative correlation between the oumost bin and all other bins.

The bin cross correlation of the different mass types incated a correlation between relaxation time and both mratio and system population. The magnitude of the crossrelations increased as the mass ratio and the system potion increased. This implies that the relaxation time~relatedto the time scale of the decay of correlations in the syste!increases as the mass ratio and the system populationcrease. A large negative correlation between the lightheavy masses in the outermost bins was evident in alltems as well as large positive correlations between theermost bin and all other bins.

Substantial work has been done on the OGS over mthan 30 years; however, the question of ergodicity and reation is still an open issue. This paper has shown that t

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mass systems exhibit several features normally assocwith systems either approaching or that have already attaequilibrium, however, substantial correlations still exist afsimulation times exceeding 108t. This suggests that collective motion may take place over very long times. This tyof behavior has been previously observed in Vlasov simutions of the OGS@12#. In addition, some of the differencenoted between the Vlasov theory and simulation results~e.g.,in the bin population studies! must be due to finite size effects, although this was not indicated in other measuresthe range of system populations studied here~e.g., the timescale for mass segregation!. Finite size effects may bemasked by differences in the initial conditions.

In these numerical experiments consideration of the elution of both average quantities and correlations suggesfollowing scenario: The system rapidly relaxes to a quasitionary state. We conjecture that this state is ‘‘close’’ tostationary Vlasov state, of which there are many~since forthe latterN→` they cannot be identical!. At present we lackthe ability to predict this state from the set of initial condtions. The system then appears to slowly, but continuouevolve through a sequence of quasistationary states unfinally approaches thermal equilibrium. As in highedimensional systems we expect that, asN increases, the timeto approach equilibrium will also increase and thus becomore difficult to observe in simulations. Thus we could sthat in gravitational systems relaxation itself is a discreteneffect. It is especially hard to observe in one-dimensiosystems because, at crossings, only the acceleration ocrossing pair changes so the mixing is very slow. Thushave shown here that there is a window in population whwe can observe that the system approaches the equilibstate. For smaller values ofN, as was demonstrated for th

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single-mass system, the phase space may be segmentestable and unstable regions in which it will be trapp@17,18#. For larger values ofN we expect equilibration onlonger times, but to see it will require an extensive comptational effort. We anticipate that for sufficiently largeNthere is a scaling region where theN dependence of the evolution is more pronounced. However, we have not seendence for this for the system sizes we were able to simulIt would be interesting if in fact the scaling behavior coube determined from theory.

Clearly, based upon these results as well as those of oresearchers, it is still not possible to draw a definitive coclusion on the question of relaxation in the OGS. Howevthe general results of these studies suggest that equilibriubeing approached. The evolution of global measurementthe macroscopic behavior of the one-dimensional sgravitating system indicates that the system is relaxing toequilibrium state in a finite, but very long time, while mesurements of more local properties of the system, suchtime averaged probability densities, are within a few percof the equilibrium values, but have not completely coverged. What is clear, however, is that these systems exvery weak diffusion, extremely long correlations in time, aweak ergodic properties in the phase space. Future workneed to include more thorough studies of other correlatiin the system as well as an investigation of the possible frtal structure in the phase space.

ACKNOWLEDGMENTS

The authors are grateful for the support of the ReseaFoundation and the Division of Information ServicesTexas Christian Univesity.

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Incomplete relaxation in a two-mass one-dimensional self-gravitating system

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