Table Of ContentNormaldiffusionincrystal structures andhigher-dimensional billiardmodels withgaps
David P. Sanders∗
DepartamentodeF´ısica,FacultaddeCiencias,UniversidadNacionalAuto´nomadeMe´xico,Me´xicoD.F.,04510Mexico
(Dated:January26,2009)
Weshow,bothheuristicallyandnumerically,thatthree-dimensionalperiodicLorentzgases—cloudsofparti-
clesscatteringoffcrystallinearraysofhardspheres—oftenexhibitnormaldiffusion,evenwhentherearegaps
throughwhichparticlescantravelwithoutevercolliding,i.e.,whenthesystemhasaninfinitehorizon. Thisis
9 thecaseprovidedthatthesegapsarenot“toobig”,asmeasuredbytheirdimension. Theresultsareillustrated
0 withsimulationsofasimplethree-dimensionalmodelhavingdifferenttypesofdiffusiveregime,andarethen
0 extendedtohigher-dimensionalbilliardmodels,whichincludehard-spherefluids.
2
PACSnumbers:05.60.Cd,05.45.Jn,05.45.Pq,66.10.cg
n
a
J
TheLorentzgasisaclassicalmodeloftransportprocesses, to an apparent general belief that the diffusive properties of
6
inwhichacloudofnon-interactingpointparticles(modelling higher-dimensionalsystems should be analogousto those in
2
electrons)undergofreemotionbetweenelasticcollisionswith the2Dcase. HypercubicLorentzgases(withinfinitehorizon)
] fixedhardspheres(atoms)[1]. Ithasbeenmuchstudiedasa ind≤7dimensionswerestudiedin[16],butnostrongcon-
h
modelsystemforwhichtheprogrammeofstatisticalphysics clusionsaboutdiffusivepropertiescouldbedrawn.
c
e canbe carriedoutin detail: to relatethe knownmicroscopic Inparticular,itwasbelievedthatafinitehorizonwasnec-
m dynamics to the macroscopic behavior of the system, which essary fora system to show normaldiffusion,with weak su-
- inthiscaseisdiffusive[2–4]. perdiffusionoccurringforaninfinitehorizon[12,17]. While
t
a Whenthescatterersarearrangedinaperiodiccrystalstruc- periodicLorentzgaseswithfinitehorizonanddisjointobsta-
t ture, the dynamicsofthisbilliardmodelcan bereducedtoa cles have been proved to exist in any dimension [18], con-
s
. singleunitcell[2]. Thecurvedshapeofthescatterersimplies structingsuchamodelturnsouttobeadifficulttask—weare
t
a thatnearbytrajectoriesseparateexponentiallyfast,sothatthe not aware of any known explicit examples, even in three di-
m
systemishyperbolic(chaotic)andergodic[5]. mensions. Furthermore, crystals of spheres arranged in any
- Intwodimensions,ithasbeenshownthatthecloudofpar- Bravais lattice (and in many other crystal structures) always
d
n ticlesintheperiodicLorentzgasundergoesnormaldiffusion, havesmallgapswhichpreventafinitehorizon[18–20].
o provided that the geometrical finite horizon condition is sat- InthisLetter,weshow,usingheuristicargumentsandcare-
c isfied: particlescannottravelarbitrarilyfarwithoutcolliding fulnumericalsimulations,thatinfactperiodicLorentzgases
[
with a scatterer [5, 6]. By normal diffusion, we mean that inthreeandhigherdimensionswithinfinitehorizon—thatis,
2 thedistributionofparticlepositionsbehaveslikesolutionsof withgaps,orholes,inthestructure—canexhibitnormaldif-
v the diffusion equation; in particular, the mean-squared dis- fusion. The keyobservationis thatthegapsinconfiguration
5
placement (variance) grows asymptotically linearly in time: space,whicharehigher-dimensionalanalogsofthecorridors
3
hr(t)2i∼2dDt when t →¥ , where r(t) is the displacement in2D,canbeofdifferentdimensions.Structureswithgapsof
2
2 ofaparticleattimet fromitsinitialposition,d isthespatial the highest possible dimension exhibit weak superdiffusion,
. dimension,h·idenotesameanoverinitialconditions,andthe asinthe2Dinfinite-horizoncase,whereaslower-dimensional
8
0 diffusioncoefficientDgivestheasymptoticgrowthrate. gapsgivenormaldiffusion. Nonetheless,highermomentsof
8 Whenthehorizonisinfinite,however,particlescanundergo thedisplacementdistributionareaffectedbythesmallpropor-
0 arbitrarily long free flights along corridors in the structure. tionofarbitarilylongtrajectoriesinthestructure.
:
v It was long argued[7–9] and has recently been proved[10], To test the analytical arguments, we perform careful nu-
i thatthereisthenweaksuperdiffusivebehavior,withhr(t)2i∼ mericalsimulationsofa3DperiodicLorentzgasmodelwith
X
tlogt,sothatthediffusioncoefficientnolongerexists. spheres of two radii, which can be varied to obtain differ-
r
a Forhigher-dimensionalperiodicLorentzgases,rigorousre- enttypesofdiffusiveregime. Inparticular,aafinite-horizon
sults on ergodicproperties[11] and diffusiveproperties[12] regime may be obtained by allowing the spheres to overlap;
havebeenobtained;recentprogressintheiranalysishasbeen otherwise, gapsof differentdimensionscan be found. Here,
made[13, 14],includinginthelimitofsmallscatterers[15]. resultswillbegivenforrepresentativecasesineachregime;a
In particular, higher-dimensionalLorentz gases are believed detailedanalysisofthemodelwillbegivenelsewhere.
toexhibitnormaldiffusionwhenthehorizonisfinite[12]. Finally,weextendtheargumentstohigher-dimensionalbil-
Nonetheless,thestudyofbilliardmodelsinhigherdimen- liards,includingtheclassofhard-spherefluids[21],thuspro-
sions, especially three dimensions, has received surprisingly vidinganapproachtothediffusivebehaviorofsuchsystems
little attention from the physics community,despite their in- intermsofthegeometryoftheirconfigurationspace.
terestassimplemodelsoftransportinthree-dimensionalcrys- Modelandgapsinconfigurationspace:- Webeginbyin-
tals. Thiscanbeattributedtoincreasedsimulationtimesand troducing a simple two-parameter 3D periodic Lorentz gas
the difficulty of visualisation in higher dimensions, but also model,withwhichthedifferenttypesofdiffusiveregimecan
2
(a) (b) (c)
FIG.1: (Coloronline)Sphericalscatterers(lightcolor;purpleonline)andgaps(darkcolor;greenonline)inthe3DperiodicLorentzmodel
discussedinthetext:(a)verticalplanargapsfora=0.25andb=0.15;and(b)verticalcylindricalgapsfora=0.4andb=0.4(abody-centred
cubicstructure). Thegapsareshowninasingleunitcell,butinfactextendthroughoutthewholeofspace. (c)Whena=0.55andb=0.4,
thescatterersoverlap,leavinganinfinite,connectedavailablespacefortheparticles,whichisdepicted;forclarity,thespheresareomitted.In
thiscase,thehorizonisfinite—therearenogapsinthestructure.
beexplored. Themodelconsistsofacubiclatticeofspheres tions in a unit cell, which have a free path length T before
ofradiusa,withanadditionalsphericalscattererofradiusbat collidingwhichisgreaterthant [7,24,25].
thecentreofeachcubicunitcell,themselvesforminganother
Consider straight trajectories which emanate in all direc-
(interpenetrating)cubic lattice. The side length of the cubic
tionsvfromagiveninitialconditionx lyinginsideagapG.
0
unitcell is taken equalto 1. By varyingthe radii a and b of
Since energy is conserved at collisions, all particles can be
the spheres, a range of models with different properties can
takentohavespeed1. Thepossiblepositionsx ofthetrajec-
t
beobtained;a“phasediagram”showingthepossibilitiesand
toriesattimet thenlieonasphereS ofradiust andsurface
t
adetailedstudyofitspropertieswillbepresentedelsewhere. areaS(t)=4p t2, centredonx . TheproportionP(T >t)of
0
Thisisa3Dversionofthe2Dmodelstudiedin[22,23].
trajectorieswhichhavenotcollidedduringtimet isgivenby
When b=0, we obtaina simple cubiclattice of spherical theratioP(T >t):=A(t)/S(t),whereA(t)istheareaofthe
scatterers. Inthiscase,wecaninsertplanesparalleltothelat- intersectionI :=G∩S ofthegapGwiththesphereS.
t t t
tice directionswhichdonotintersectanyobstacles—wecall
IfGisaplanargap,thentheintersectionI isapproximately
theseplanargaps. Thisremainsthecaseforsmallenoughb, t
theproductofacircleofradiust withanintervalofthesame
asshowninfig.1(a). Forb≥ 1−a,however,allofthepla-
2 widthwasthegap. ThusA(t)≃2p wt,givingtheasymptotic
nargapsareblocked. Therearestillgapsofinfiniteextentin
behaviorP(T >t)∼C/t whent→¥ ,whereCisaconstant.
the structure,buttheyare nowcylindricalgaps, as shownin
This result was previously found for a simple cubic lattice
fig.1(b).Theseareinfinitelylongtubeswhichdonotintersect
[7, 12]; a detailed calculation is given in [25]. When G is
anyscatterer,givenbytheproductofalinewithanarea;the
a cylindricalgap, however,its intersectionI with the sphere
latteristheprojectionofthegapalongtheaxisofthecylinder. t
S isasymptoticallythecross-sectionalareaAofthecylinder,
By tuning a and b appropriately, it is also possible to ob- t
givingtheasymptoticsP(T >t)∼C/t2.
tain an explicit 3D periodic Lorentz gas with finite horizon.
To do so, we allow the scatterers to overlap,since otherwise ThetailP(T >t)ofthefree-pathdistributionisstronglyre-
constructingsuchamodelisverydifficult. Alladjacentpairs latedtothesystem’sdiffusiveproperties. Friedman&Martin
ofa-spheresoverlapwhena> 1;choosingtheradiusbofthe [7]proposedthattheasymptoticdecayrateofthevelocityau-
2
central sphere large enough then allows us to block all gaps tocorrelationfunctionC(t):=hv(0)·v(t)iisthesameasthat
inthestructure,givingafinite-horizonmodel,asshownelse- ofP(T >t),sinceC(t)isdominatedbytrajectorieswhichdo
where.Unlikeinthe2Dcase,in3Dthefreespacebetweenthe notcollideup to timet. The finite-timediffusioncoefficient
overlapping scatterers forms an infinite connected network. D(t):= ddthr(t)2iisgivenbyD(t)= d1R0tC(s)ds,sothatD(t)
Physically,thiscancorrespondtoasphereofnon-zeroradius converges,to the diffusioncoefficientD, only if the velocity
collidingwithdisjointscatterers.Notethatrigorousresultson autocorrelationC(t)decaysfasterthan1/t[2].
higher-dimensional Lorentz gases assume disjoint scatterers Thuswe expectthata 3D periodicLorentzgasshouldex-
[14],andthusdonotdirectlyapplytoourmodel. hibitnormaldiffusionwhenP(T >t)decaysfasterthan1/t,
Distributionoffreepaths:- Severalapproachestothedif- asisthecasewithcylindricalgaps(andwhenthehorizonisfi-
fusive properties of infinite horizon systems involve the tail nite),butweaksuperdiffusionwhenitdecayslike1/t.Thisis
of the free-path length distribution, that is, the proportion alsoinagreementwithanequivalentconditiononthemoment
P(T >t) of trajectories, starting from random initial condi- ofthefreepathdistributionbetweencollisions[9].
3
7.0 sitionsforwhichthese non-collidingtrajectoriespointin the
a=0.4;b=0.0
a=0.4;b=0.21 samedirection(s)agapinconfigurationspace. Notethatitis
6.0
a=0.4;b=0.4
possibleforagivensetofinitialconditionstohavesuchtra-
a=0.55;b=0.4
5.0 jectories pointing in different, unconnected directions—this
t/ 4.0 is the case, for example, in fig. 1(b), where there are also
(cid:11) cylindricalgapsinahorizontaldirection(notshown).Insuch
2
)
t( 3.0 cases, we consider each such set of different directions as a
r
(cid:10)
distinct gap. For a discussion of higher-dimensionalgaps in
2.0
thecontextofspherepackings,seeref.[20].
1.0 Asshownaboveforthe3Dcase,thekeygeometricalprop-
ertydeterminingthediffusivebehaviorofa systemisthedi-
0.0
102 103 104 105 106 mensionofitsgaps.WedefinethedimensionofagapGtobe
thedimensiongofthelargestaffinesubspacewhichliescom-
t
pletelywithinthegap,i.e.,whichdoesnotintersectanyscat-
FIG.2: (Coloronline)Linear–logplotofhr(t)2i/t vs.t indifferent terer. In a system with a d-dimensionalconfigurationspace,
diffusiveregimes:finitehorizon(a=0.55;b=0.4);cylindricalgaps therecanbegapswithanydimensionbetween1andd−1,or
inabody-centredcubiclattice(a=b=0.4);singlelargecylindrical nogapsatall(finitehorizon).
gap(a=0.4;b=0.21);andsimplecubiclattice(a=0.4;b=0.0) To calculate the tail P(T >t) of the free-pathdistribution
withplanargaps. Meansaretakenoverupto4×107 initialcondi-
duetosuchgaps,wetakecoordinatesx:=(x ,...,x )inthe
tions; errorbarsareoftheorderofthesymbolsize. Lineargrowth 1 d
(weaksuperdiffusion)occursonlywhenthereareplanargaps. d-dimensionalconfigurationspace,withtheinitialpositionat
theorigin. ThesphereS isthengivenby(cid:229) d x2=t2. Con-
t i=1 i
sideragapG,ofdimensiong.Insidethegap,thereisalargest
Numericalresults:- Totesttheabovehypotheses,weper- subspace,alsoofdimensiong,i.e.,ithasgfreely-varyingco-
formcarefulnumericalsimulationsofourmodeltocalculate ordinates. By a rotation of the coordinate system, this sub-
the time-evolutionof the mean-squareddisplacementhr(t)2i space can thus be written as x =x =···=x =0, where
1 2 c
ineachregime. We useastringenttesttodistinguishnormal c:=d−g is the codimension of the gap, giving the number
from weakly anomalous diffusion: hr(t)2i/t is plotted as a ofcoordinatesinthesubspacewhicharefixed. Theintersec-
function of logt [22, 26, 27]. Normal diffusioncorresponds tion I =G∩S of the gap with the sphere is thus given by
t t
to an asymptoticallyflat graph,since the logarithmiccorrec- (cid:229) d v2 =t2. Thisis a g-dimensionalsphere, with surface
i=c+1 i
tion is absent, and the diffusion coefficient is then propor- area K tg−1, where K is a dimension-dependent constant.
g g
tionalto the asymptoticheightof the graph. Weak superdif- The tail of the free-path distribution is given by the ratio of
fusivetlogt behaviorforthemean-squareddisplacement,on the area of intersectionI to the area of the sphereS, giving
t t
theotherhand,givesasymptoticlineargrowth[26]. theasymptotics
Numericalresultsareshowninfig.2. Weseethattheargu-
mentsgiven in the previoussection are confirmed: diffusion P(T >t)∼Z Kgtg−1 =Kt−(d−g)=Kt−c, (1)
isnormal,with hr(t)2i∼t, whenthehorizonisfinite, andis cK td−1
d
weakly superdiffusive, with hr(t)2i∼tlogt, when there is a
planargap. Furthermore,thenumericsclearlyshow thatdif- whereZc isthec-dimensionalcross-sectionalareaofthegap
fusionis normalalso in thecase thatthereare onlycylindri- inthedirectionsorthogonaltotheaffinesubspace,andKisan
calgaps. Thisisthecaseevenwhenthecylindricalgapsare overallconstant.
“large”, for example when a=0.4 and b=0.21, when the We thus see that the decay is faster for gaps of smaller
gapsdepictedin1(b)mergetoformasinglecylindricalgap, dimension (larger codimension), but it is always eventually
still without any planar gaps in the structure. Thus we con- dominatedby the contributionof trajectorieslying along the
cludethattheheuristicargumentscorrectlypredictthetypeof gaps. The dominant contribution to the tail of the free-path
diffusionwhichoccursinthesesystems. distribution,andhencetothediffusiveproperties,thuscomes
Gaps in higher-dimensional billiards:- Fluids of hard fromthegapoflargestdimension.
spheres are isomorphic to higher-dimensional chaotic bil- Wethusconjecturethatd-dimensionalchaotic,periodicbil-
liard models, although with cylindrical instead of spherical liard models can generically be expected to exhibit normal
scatterers [21]. By extending the above arguments, we can diffusion,atthelevelofthemean-squareddisplacement,pro-
hopetoobtaininformationoncorrelationdecayanddiffusive vided that the largest-dimensional gap is of dimension less
propertiesforgeneralhigher-dimensionalchaoticbilliardsby thand−1,thatis,ifitscodimensionislargerthan1.
analysingthegapsintheirconfigurationspace. Higher moments:- A more sensitive probe of diffusive
To define these higher-dimensionalgaps, we consider ini- properties is given by the growth rates g (q) of the qth mo-
tial positions in a configuration space of dimension d, from ments of the displacement distribution, hrq(t)i∼tg(q), as a
whichinfinitelylongnon-collidingtrajectoriesemanatealong functionoftherealparameterq[24,28,29]. IfP(T >t)de-
certaindirections. Wecallaconnectedsetofsuchinitialpo- caysliket−c,thenlongtrajectoriesdominatehrq(t)iforlarge
4
8 a=0.4;b=0.0 manuscript critically. Supercomputing facilities were pro-
a=0.4;b=0.21
vided by DGSCA-UNAM, and financial support from the
a=0.55;b=0.4
DGAPA-UNAM PROFIP programmeis also acknowledged.
6
Theauthorisgratefultotheanonymousrefereesforinterest-
ingcomments.
)
q
( 4
g
2
∗ Electronicaddress:[email protected]
[1] H.A.Lorentz,Proc.Roy.Acad.Amst.7,438,585,684(1905).
0
[2] P.Gaspard,Chaos,ScatteringandStatisticalMechanics(Cam-
0 2 4 6 8
bridgeUniversityPress,Cambridge,1998).
q [3] D.Sza´sz,ed.,HardBallSystemsandtheLorentzGas(Springer-
Verlag,Berlin,2000).
FIG.3: Growthrateg(q)ofhighermomentshrq(t)iasafunctionof [4] R. Klages, Microscopic Chaos, Fractals and Transport in
q; geometriesareasinfig.2. Fora=0.4andb=0.21,themeans NonequilibriumStatisticalMechanics(WorldScientific,Singa-
were calculated over 2.4×108 initial conditions, up to a timet = pore,2007).
10000,tocapturetheweakeffectofthecylindricalgaps.Thestraight [5] N. Chernov and R. Markarian, Chaotic Billiards (American
lines show the expected Gaussian behavior (q/2) and behavior for MathematicalSociety,Providence,RI,2006).
largeqwithplanar(q−1)andcylindrical(q−2)gaps. [6] L.A.BunimovichandY.G.Sinai,Comm.Math.Phys.78,479
(1981).
[7] B.FriedmanandR.F.Martin,Phys.Lett.A105,23(1984).
q, giving g (q)=q−c, while the low moments show diffu- [8] A. Zacherl, T. Geisel, J. Nierwetberg, and G. Radons,
sive Gaussian behavior, with g (q)=q/2 [24]. A crossover Phys.Lett.A114,317(1986).
[9] P.M.Bleher,J.Stat.Phys.66,315(1992).
betweenthetwobehaviorsthusoccursatq=2c. Ifthehori-
[10] D.Sza´szandT.Varju´,J.Stat.Phys.129,59(2007).
zonisfinite,thentherearenolongfreeflights,andGaussian
[11] Y.G.SinaiandN.I.Chernov,Russ.Math.Surv.42,181(1987).
behaviorisexpectedforallq. Highermomentsforthefinite- [12] N.I.Chernov,J.Stat.Phys.74,11(1994).
horizonLorentzgaswerestudiedin[30]. [13] P. Ba´lint, N. Chernov, D. Sza´sz, and I. P. To´th, Ann. Henri
The numerical calculation of higher moments is difficult, Poincare´3,451(2002).
due to the weak effect of free flights [26]. Nonetheless, by [14] P. Ba´lint and I. P. To´th, Exponential decay of correlations in
multi-dimensionaldispersingbilliards,submitted,URLhttp:
taking means over a very large number of initial conditions,
//www.math.bme.hu/∼pet/pub.html.
it is possible to see the effect of the different types of gaps
[15] J.MarklofandA.Stro¨mbergsson,Thedistributionoffreepath
forour3DLorentzgasmodel: asshowninfig.3,theyarein
lengths in the periodic Lorentz gas and related lattice point
agreement with the above argument. Thus, higher moments problems(2007),arXiv:0706.4395.ToappearinAnn.Math.
candistinguishthesubtleeffectsofdifferenttypesofgaps. [16] J.-P.BouchaudandP.LeDoussal,J.Stat.Phys.41,225(1985).
Inconclusion,wehaveshownthatthediffusiveproperties [17] C. P. Dettmann, in Hard Ball Systems and the Lorentz Gas,
ofperiodicthree-dimensionalLorentzgases,andbyextension editedbyD.Sza´sz(Springer,2000),pp.315–365.
[18] M.HenkandC.Zong,Mathematika47,31(2000).
ofhigher-dimensionalperiodicbilliardmodels,dependonthe
[19] A.Heppes,Ann.Univ.Sci.Budapest.Eo¨tvo¨sSect.Math.3–4,
highest dimension of gap in the configurationspace. By in-
89(1960/1961).
troducingasimple3Dmodelinwhicheachtypeofdiffusive
[20] C.Zong,Bull.Amer.Math.Soc.39,533(2002).
regime occurs, we showed that if there is a finite horizon or [21] D.Sza´sz,PhysicaA194,86(1993).
cylindricalgaps,thenthediffusionisnormal,whereasplanar [22] P.L.GarridoandG.Gallavotti,J.Stat.Phys.76,549(1994).
gapsgiveweaksuperdiffusion.Nonetheless,highermoments [23] D.P.Sanders,Phys.Rev.E71,016220(2005).
distinguishbetweendifferenttypesofgaps.Theconceptofin- [24] D. N. Armstead, B. R. Hunt, and E. Ott, Phys. Rev. E 67,
021110(2003).
finitehorizonisthusnolongersufficientlypreciseforhigher-
[25] F.GolseandB.Wennberg,M2ANMath.Model.Numer.Anal.
dimensionalsystems, and must be replaced by maximal gap
34,1151(2000).
dimension. In future work we will extend our numericalin-
[26] D.P.SandersandH.Larralde,Phys.Rev.E73,026205(2006).
vestigationstohigher-dimensionalmodels. [27] T.GeiselandS.Thomae,Phys.Rev.Lett.52,1936(1984).
This work was initiated in the author’s Ph.D. thesis [31]. [28] P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and
He thanks P. Gaspard and R. MacKay for helpful com- A.Vulpiani,PhysicaD134,75(1999).
ments and M. Henk for usefulcorrespondence,and is grate- [29] M.Courbage,M.Edelman,S.M.S.Fathi,andG.M.Zaslavsky,
Phys.Rev.E77,036203(2008).
ful to the Erwin Schro¨dingerInstitute and the Universite´ Li-
[30] N.ChernovandC.P.Dettmann,PhysicaA279,37(2000).
bredeBruxellesforfinancialsupport,whichenableddiscus-
[31] D. P.Sanders, Ph.D.thesis, Mathematics Institute, University
sions with N. Chernov, I. Melbourne, D. Sza´sz, I.P. To´th
ofWarwick(2005),arXiv:0808.2252.
and T. Varju´, and especially T. Gilbert, who also read the
