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Dynamical turbulent flow on the Galton board with friction A. D. Chepelianskii(a) and D. L. Shepelyansky(b) (a) Lycée Pierre de Fermat, Parvis des Jacobins, 31068 Toulouse Cedex 7, France (b) Laboratoire de Physique Quantique, UMR 5626 du CNRS, Université Paul Sabatier, 31062 Toulouse Cedex 4, France (January 1, 2001) We study numerically and analytically the dynamics of particles on the Galton board, a regular lattice of disc scatters, in the presence of a constant external force and friction. It is shown that undercertainconditionsfrictionleadstotheappearanceofastrangechaoticattractorinaninitially conservativeHamiltoniansystem. Inthisregimetheparticleflowbecomesturbulentanditsaverage velocity depends in nontrivial manner on friction and other system parameters. We discuss the applications of these results to thetransport properties of suspended particles in a laminar viscous flow streaming through scatters. 1 0 PACS numbers: 05.45.Ac, 47.52.+j, 72.20.Ht 0 2 n Itiswellknownthatdissipationcanleadtotheappear- ical grounds. But we also note that physically such a a ance of strange chaotic attractors in nonlinear nonau- situationcancorrespondto the motionofsmallparticles J tonomous dynamical systems [1,2]. In this case the en- in a viscous liquid flow which laminarlystreams through 1 ergy dissipation is compensated by an external energy largescattersthatinourcasehavetheformofdiscs. The flow so that stationary chaotic oscillations set in on the same model can describe the electron dynamics in anti- 1 attractor. Such an energy flow is absent in the Hamilto- dot superlattice which has been experimentally realized v 6 nianconservativesystemsandthereforethe introduction insemiconductorheterostructures[6]. Insuchstructures 0 ofdissipationorfrictionisexpectedtodrivethesystemto the effects of classical chaos play an important role [7] 0 simple fixed points in the phase space. This rather gen- and the effects of friction we discuss here can appear for 1 eral expectation is surly true if the system phase space relatively strong electric fields. 0 is bounded. However a much richer situation appears 1 0 in the case of unbounded space, where unexpectedly a / strange attractor can be induced by dissipation in an t a originally conservative system. To investigate this situ- m ation we study the dynamics of particles on the Galton - board in the presence of a constant external field and d friction. This board, introduced by Galton in 1889 [3], n o represents a triangular lattice of rigid discs with which c particles collide elastically. For the case of free particle : motion,thecollisionswithdiscsmakethedynamicscom- v i pletely chaotic on the energy surface as it was shown by X Sinai [4]. In this paper we study how the dynamics is r changed if the particles are affected by a friction force a Ff which is assumed to be proportional to the particle velocityvanddirectedagainstit: Ff = γv. Inthefree − space without discs, an external field with an effective force f creates a stationary particle flow with the velocity vf = f/γ. All perturbations decay to this flow with therateproportionaltoγsothatthislaminarflowcanbe considered as a simple attractor. It is interesting to understand how this flow will be affected by the scattering FIG. 1. Chaotic flow on the Galton board. Here the dis- on regularly placed discs. As we will show latter on, in tancebetween discs isR=2.24, f = (-0.5, -0.5) andγ =0.1. this casetheflowcanbecometurbulentandchaotic,and Initially 100 particles are distributed homogeneously along a demonstrate many unusual properties (see Fig.1). The straight line segment in the upper right corner, their color Hamiltonian dynamics of the model has been studied in homogeneouslychangesfromredtogreenalongthissegment. great detail (see [5] and Refs. therein) however, as for our knowledge, the effects of friction were not addressed To study the dynamics of this model, we fix the disc up to now. The investigation of the dynamics in this radius a = 1, and the mass of the particles m = 1 so regime is interesting from the viewpoint of generalphys- that the system is characterized only by three parameters: thedistanceRbetweendiscspackedintoequilateral 1 trianglesalloverthe(x,y)plane;thefrictioncoefficientγ a result the penetration depth for color mixing is not andtheexternalforceofstrengthf. ForR R =4/√3 very large. However for smaller friction, the drift veloc- c ≤ thereisnostraightline,bywhichaparticlecancrossthe ity becomes smaller and the penetration depth increases whole plane without collisions (at f = 0,γ = 0), and we so that the particle makes many turns around discs as start the discussion from this case. The particle dynam- it is shown in Fig. 3. Surprisingly, the drift velocity v f icsissimulatednumericallybyusingtheexactsolutionof increases with the growth of the friction coefficient γ. Newton equations between collisions, and by determin- This dependence is opposite to the case without scatters ing the collisionpoints with the Newton algorithm. This where v drops with the γ growth. f way the trajectories are computed with high precision, and for example for γ = 0 the total energy is conserved with a relative precision of 10−14. A typical example of thechaoticflowformedbyanensembleofparticletrajec- tories is shown in Fig. 1. To illustrate the mixing prop- ertiesofthe flowwe attributedacolortoeachtrajectory that allows to follow their interpenetration and spread- ing. Thisfigureshowsthatthere isacertainpenetration depth of one color into another, however this depth is finite since on averagethe initial color repartition is still visible. FIG. 3. Single trajectory for the same parameters as in Figs. 1,2 but with smaller friction γ =0.004; on a large scale the particle moves with a drift velocity v ≈ 0.05 in the di- f rection of external force. FIG. 2. Example of a single trajectory for the case of Fig.1shownonasmall(mainfigure)andlarge(inset)scales. The drift velocity of theflow is v ≈0.13. f The properties ofcolorpenetrationcanbe understood from the analysis of single trajectory dynamics. Such a typicalexample is presentedin Fig.2. Itshows thattra- jectory flows with an averageconstant velocity v in the f FIG.4. Dependenceoftheflowvelocityv onthefriction directionof the externalforce. This drift velocity is con- f coefficient γ and external field strength f for R=2.24, f/|f| stantonlyonaveragesinceonasmallerscaletheparticle = (-1, -1) and 0.001 ≤ γ ≤ 0.4;0.5 ≤ f ≤ 32. Numerical moves chaotically between scatters (see Fig. 2). Even if results are shown by circles, the straight line shows the de- the average flow is regular, on a small scale the particle pendencev ∝(γf)1/3. Inset shows thedependenceof v on f f dynamics is chaotic and since the dynamics is dissipa- ∆R = R−2 for γ = 0.1 and f = (-0.5, -0.5): points give tive this means that the motion develops on a strange numerical data, lines show theslopes 0.5 and 1. chaotic attractor. This attractor is induced by dissipation in a conservative Hamiltonian system. For the case Thedependenceofthedriftvelocityv onγandonthe f ofFig.2the driftvelocityis relativelylargeandthe par- externalforcef = f isobtainedbyaveragingovermany | | ticle does not have enough time to move around many trajectories. Eachtrajectory is followedfor a sufficiently scatters in the direction perpendicular to the flow. As long time so that the distance from the origin becomes 2 largerthanafewthousandsunits. Uptofluctuations,the Since the dynamics is chaotic the direction of velocity is flow is alwaysdirectedalongthe applied forcef. The re- changed randomly after each collision so that v is ac- f sultingdependence isshowninFig.4accordingtowhich cumulated only between collisions. The time τ is deter- c v (γf)1/3. To understand the origin of this unex- mined by the mean free path l and the average velocity f ∝ pected dependence on parameters we also compute the v : τ = l/v = 2D/v2 and µ = D/(mv2/2). In fact c average velocity square v2 =< v2+v2 > and determine this is the Einstein relation according to which the mo- x y how it varies with f and γ. The data presentedin Fig. 5 bilityisgivenbyratioofthediffusionratetotheaverage giveadependenceoftheform: v2 f4/3/γ2/3. Westress kinetic energy (temperature) [9]. Together with (1) this ∝ that v and v2 are independent of initial velocity of the gives the drift velocity of the chaotic flow: f particle. v l2/3(fγ/m2)1/3 , (2) To explain the observed scaling relations for the dy- f ∼ namical turbulent flow on the Galton board we give the which is in a good agreement with the numerical results followingtheoreticalarguments. Intheregimewithweak inFig.4. Onaveragetheparticlemoveswiththevelocity friction the particles start to diffuse among discs in a v in the direction of the force (see an example in the chaoticmannerwiththe diffusionrateD =vl/2wherev f inset of Fig. 2). The amplitude of fluctuations around is the velocity of the particle and l is the mean free path this direction is ∆r pDm/γ which also determines [8]. For R 1 we have l R 1 while the dependence ∼ ∼ ∼ ∼ the color mixing depth (see Fig. 1). of l on R will be discussed in more detail latter. During Therelations(1),(2)allowtoestimatethevalueofthe thedissipativetimescaleτ =m/γthisdiffusionleadsto γ Lyapunovexponentλ. Indeed,the particlemoveswith a theparticledisplacement∆r pDm/γ alongthedirec- ∼ typical velocity v and as in the case of the Sinai billiard tion of applied field. Here, for generality we introduced withl R 1wehaveλ v/l. Thereforeintheregime the particle mass m. This gives the change in the po- ∼ ∼ ∼ when: tential energy U f∆r fpDm/γ. In the stationary ∼ ∼ regime at time t ≫ τγ, this potential energy should be γ <γc ∼pfm/l , (3) comparablewiththekineticenergyoftheparticlesothat U mv2. As a result we find that the average velocity the value of λ is much larger than the dissipation rate ∼ square is γ/m. As a result for γ γc the strange attractor is fat ≪ and its fractal dimension is close to the maximal dimen- v2 f4/3(l/γm)2/3 , (1) sion 4, which is determined by the number of degrees of ∼ freedom (we remind that contrary to the nondissipative that is in agreement with the data of Fig. 5. case the energy is not conserved). The physical meaning of the critical γ border is the c following: for γ γ the dissipation time τ becomes c γ ≫ muchshorterthanthe time betweencollisionsτ . Inthis ln v2 c 5 case the dissipationdominates chaos and the strange at- tractordegenerates into a simple attractor. Among such simpleattractorsweobservedtrajectoriesthatmovereg- 3 ularly under some angle to the applied force but more often trajectories that simply drop to a fixed point on a disc where the applied force is perpendicular to the 1 discborder. Ofcoursethenumericalsimulationscannot ln(f4/3/ γ 2/3) guarantee that the chaotic attractor at γ < γc will not degenerate to a simple one at very long times. However -1 all our simulations performed with high computer preci- 1 3 5 7 sion give no indications in this direction even when we followed trajectories up to very far distances from the FIG. 5. Dependence of average v2 on f and γ origin (bigger than 105 length units). for R = 2.24, f/|f| = (-1, -1) and parameter range The equation (3), which determines the border for 0.001 ≤γ ≤ 0.4;0.5 ≤f ≤32. Circles show numerical data, disappearance of chaotic flow, is written for the case thestraight line gives theslope 1. l R 1. Therefore,itshouldbemodifiedwhenR 1 ∼ ∼ ≫ and ∆R = R 2 1. Indeed for R 1 the mean − ≪ ≫ The relation (1) allows to determine the mobility µ free path l is given by l = 1/(σn) where σ = 2a = 2 is which relates the velocity of the flow with the applied the disc cross section and n = 2/(√3R2) is the density force: vf = µ f. Indeed during the time τc between of scatters. Then the time between collisions is l/v and collisions the particle is accelerated by force that gives from equation (1) we obtain γ mv/l √fma/R. In c ∼ ∼ the average drift velocity v = fτ /m and µ = τ /m. the other limit ∆R 1 we may assume that D v∆R, f c c ≪ ∼ 3 l ∆R and the escape time from a cell between three can be studied experimentally with viscous liquids and ∼ discs is τ a2/D. Therefore the dissipation be- its investigation can contribute to a better understand- esc ∼ comes dominant and the chaotic attractor disappears at ing of the interplay between dissipation, turbulence and γ >γ m/τ pmf/a3 ∆R. chaos. c esc ∼ ∼ The above changes in the mean free path l also af- fect the drift velocity of the flow through the relation (2). Indeed for ∆R 1 we expect l ∆R that gives ≪ ∼ v ∆R2/3. This dependence is close to the numer- f ∝ ical data shown in Fig. 4 (inset) even if the numerical [1] A. Lichtenberg and M. Lieberman, Regular and chaotic value of the exponent is approximated better by 0.5. In dynamics, Springer, Berlin (1992). the other limit R 1, we have l R2/a that gives [2] E. Ott, Chaos in dynamical systems, Cambridge Univ. ≫ ∼ v R4/3. This power dependence is in a satisfactory Press (1993). f agr∝eement with our numerical data (inset in Fig. 4) al- [3] F. Galton, Natural inherritance, London (1889). though the numerical value of the exponent is closer to [4] I. P. Kornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic theory, Springer, Berlin (1982). 1. We attribute these small deviations in the exponent [5] P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65, 1693 values to a restrictedintervalofvariationinR. Actually (1990); H. van Beijeren and J. R. Dorfman, Phys. Rev. we can not use very large/small values of ∆R since in Lett. 74, 4412 (1995). these limits the value of γ becomes comparable with γ c [6] D. Weiss, M. L. Roukes, A. Mesching, P. Grambow, K. andchaoticattractordisappears. Wenotethataccording von Klitzing, and G. Weiman, Phys. Rev. Lett. 66, 2790 to our data (see Fig. 4) the strange attractor exists even (1991). in the case R > Rc = 4/√3 when at f = 0,γ = 0 there [7] R.Fleischmann,T.Geisel,andR.Ketzmerick,Phys.Rev. arestraighttrajectoriescrossingthewholeplanewithout Lett. 68, 1367 (1992). collision. Apparently the contribution of these orbits is [8] J.M.Ziman,Principlesofthetheoryofsolids,Cambridge not significant if γ >0 and f is not directed along these Univ.Press (1972). lines. [9] L.D.Landau and E.M. Lifshitz, Hydrodynamics, Nauka, Moscow (1986). Let us now discuss a few consequences of the obtained results. The introduction of scatters qualitatively changes the flow properties: instead of a laminar flow we obtain a turbulent/chaotic one. The flow rate v de- f pends on the applied force f and γ in a nonlinear way given by equation (2). Contrary to the laminar case where v = f/γ in presence of scatters the flow veloc- f ity depends alsoonthe mass ofparticlesm. Infact light particles stream with higher velocity. This property of chaotic flow can be used for mass separation of particles suspended in viscous flows, for example isotopes. It is alsointerestingtonotethatinthederivationofequations (1), (2), (3) we didn’t use any specific properties of the disc distributioninthe plane. Thereforethese resultsre- mainalsovalidfor arandomdistributionofdiscscatters in the plane. We wonder if some rigorous mathematical results can be obtained in the limit of weak friction to provethe existence ofa chaoticattractorforaregularor random distribution of discs. Wethinkthattheobtainedresultsrepresentcertainin- terest for transport properties of particles suspended in a viscous flow streaming through a system of large scatters. Indeed, when a liquid streams laminarly through scatters with the velocity v the flow of suspended par- s ticles is characterized by the relations (1), (2), (3) with f = v γ where γ is an effective friction created eff s eff eff by the viscosityof the liquidand f is the resultingef- eff fective force acting on the suspended particles. Then for weak friction the drift velocity of particles v is smaller f thanthevelocityofthe liquidv . Suchkindoftransport s 4