Free path lengths in quasi crystals

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by  Bernt Wennberg

FREE PATH LENGTHS IN QUASI CRYSTALS
BERNTWENNBERG
ABSTRACT. TheLorentzgasisamodelforacloudofpointparticles(electrons)
inadistributionofscatterersinspace. Thescatterersareoftenassumedtobe
spherical with a fixed diameter d, and the point particles move with constant
2 velocitybetweenthescatterers,andarespecularlyreflectedwhenhittingascat-
1 terer. There is no interaction between point particles. An interesting question
0 concernsthedistributionoffreepathlengths, i.e. thedistanceapointparticle
2
movesbetweenthescatteringevents,andhowthisdistributionscaleswithscat-

n tererdiameter,scattererdensityandthedistributionofthescatterers.Itisbynow
a wellknownthatintheso-calledBoltzmann-Gradlimit,aPoissondistributionof
J scattersleadstoanexponentialdistributionoffreepathlengths,whereasifthe

2 scattererdistributionisperiodic,thedistributionoffreepathbehavesasymptoti-
callylikeaCauchydistribution.

] This paper considers the case when the scatters are distributed on a quasi
h
crystal, i.e. non periodically, but with a long range order. Simulations of a
p
one-dimensional model are presented, showing that the quasi crystal behaves
-
h verymuchlikeaperiodiccrystal,andinparticular,thedistributionoffreepath
t lengthsisnotexponential.
a
m
[
1. INTRODUCTION
1
v The Lorentz gas is a mathematical model for the motion of (point) particles in e.g. a
0 crystal, consistingofspherical, elasticscatterersofradiusawithcentersatafixedsetof
5 pointsΓ ⊂ Rn. Apointparticlemovesinstraightlinesbetweentheobstacles,onwhich
4 ε
itisspecularlyreflected. Atleasttwoverydifferentscattererdistributions, Γ havebeen
0 ε
. studied thoroughly: the standard lattice L ⊂ Rn, with interstitial distance ε(n−1)/n, or a
1
randomdistribution,whereΓ isPoissondistributedwithintensityε−(n−1). Inthispaper
0 ε
wearemainlyconcernedwiththesocalledBoltzmann-Gradlimitofthissystem,andthat
2
1 correspondstosettingthescatterradiusa=εandlettingε→0.Verylooselyspeaking,in
: thisscalingthemeanfreepathofthepointparticlesremainoforderoneasε→0.Consider
v
i now a density fε(x,v,t] of point particles moving between the scatterers. Gallavotti [9]
X
(see also [10]) considered the random obstacle distribution, and proved that when ε →
r 0, f (x,v,t) converges to a function that solves a linear Boltzmann equation. On the
a ε
otherhand, Golse[11]proved, basedonresultsin[3]and[12], thatintheperiodiccase,
the limiting particle distribution does not satisfy a linear Boltzmann equation. Caglioti
andGolse[6,7,8]andMarklofandStrömbergsson[15,16,17,18]independentlyfound
sharpestimatesofthedistributionoffreepathlengthsandthattheBoltzmann-Gradlimit
correspondstoaBoltzmannlikeequationinanextendedphasesspacethatdoesdescribe
the evolution of a particle density in the limit of small ε; whereas the results in [6, 7, 8]
are restrictedto twospace dimensionsand relyon an independenceassumption, [15,16,
17,18]providesacompleteproofvalidforanyspacedimension. Relatedresultsvalidfor
inthetwo-dimensionalcase,canbefoundin[2,1],andin[4]. Situationswherescatterers
arerandomlyplaceonaperiodiclatticehavebeenconsideredin [5]and [20].
Here we are interested in the case where the obstacles are distributed as the atoms in
a quasi crystal. By definition a crystal is ”a solid with an essentially discrete diffraction
1
2 BERNTWENNBERG
pattern” [21]. It has been known for a long time that periodic crystals in three dimen-
sionsmustbelongtooneoffourteensymmetryclasses. Aquasicrystalisacrystalwhose
diffractionpatternexhibitsaforbiddensymmetry. Thefirstreportsonexperimentalresults
indicating that such solids exist were treated with suspicion by the scientific community,
butin2011theirdiscoverywasawardedtheNobelPrizeinChemistry[13].Therearealso
manymathematicalabstractconstructionsthatgivethesameresults,startinge.g. fromthe
Penrosetiling(seee.g.[21]). ItisthenanaturalquestiontoaskwhethertheBoltzmann-
Grad limit of a Lorentz gas in a quasi crystal behaves more like the random or periodic
case. Inthispaperwepresentsimulationresultsonaone-dimensionalmodel,whichgive
strongsupportforthelatter:thefreepathlengthdistributiondecayspolynomially,justasin
theperiodiccase,whereasintherandomcase,thepathlengthdistributionisexponentially
decaying.
ThenextsectiongivesprecisedefinitionsoftheLorentzmodel,andofitsBoltzmann-
Grad limit, and some results concerning the periodic and random cases are discussed.
Thereafter, in Section 3, a construction of quasi crystals is given; that section is mainly
basedon[21].
Section4presentsthesimulationmethodandtheresults,andthepaperendswithcon-
clusionsandsomeprospectsforfutureresearchinSection5.
2. FREEPATHLENGTHDISTRIBUTIONSINTHELORENTZMODEL
The discussion in this section is restricted to the Lorentz gas in two dimensions, and
hence we consider a point distribution Γ ⊂ R2, or more precisely, a family of point
ε
distributionsparametrizedbyε.Ateachpointp∈Γ weputacircularobstaclewithradius
ε
ε andcenterat p, andthenwe study themotionofapoint particlemovingwithconstant
speed, |v| = 1, alongstraightlinesbetweentheobstacles, andspecularlyreflectedwhen
hittinganobstacle. HencethephasespaceforonepointparticleisΩ ×S1,where
Γε
Ω =R2\ (cid:91) B¯ (ε)
Γε p
p∈Γ
andB¯ (ε)istheclosedballofradiusεcenteredatp ∈ R2. Foranyinitialpoint(x ,v )
p 0 0
wedefine
Tt (x ,v )=(x(t),v(t)),
Γε 0 0
the position of the point particle at time t, taking into account all reflections on the set
of obstacles. As long as the obstacles do not overlap, this is well defined for all t ≥ 0.
Moreover,thethefreepathlengthisdefinedas
(cid:110) (cid:12) (cid:111)
τ (x ,v )=inf t>0(cid:12)x +tv ∈/ Ω .
ε 0 0 (cid:12) 0 0 Γε
Wealsodefinethepathlengthdistribution
m({(x,v)∈(Ω ∩B (R))×S1|τ (x,v)∈]a,b[)}
(1) φ (]a,b[)= lim Γε 0 ε .
ε R→∞ m(ΩΓε ∩B0(R))
Herem(·)istheLebesguemeasurerestrictedtoΩ ×S1.Incaseswhere,atleastformally,
Γε
onecanprovethatforallintervals[a,b],φ (]a,b[)remainsboundedfromaboveandbelow
ε
whenε→0onespeaksofaBoltzmann-Gradlimit.
The two typical examples of point distributions Γ considered in connection with the
ε
LorentzgasaretheperiodicdistributionsΓ =a(ε)Z2,andaPoissoneanrandomdistribu-
ε
tionwithintensitya(ε).
For a periodic distribution it is more natural to define the free path length distribution
byrestrictingtoalatticeunitcellratherthantoaballofradiusRasin(1). Ofcourse,in
thelimitR→∞theresultisthesame. InthiscasetheBoltzmann-Gradlimitcorresponds
tochoosinga(ε)=ε1/2. Thepathlengthdistributionφ intheBoltzmann-Gradlimithas
ε
been studied in [3, 12], and then, with very sharp bounds in [6, 7, 8] using the theory of
FREEPATHLENGTHSINQUASICRYSTALS 3
θ
q q q q
0 1 2 3
FIGURE 1. ComputingthefreepathlengthinaperiodicLorentzgasis
equivalenttotoadiscretemap.
continued fractions, and [15, 16, 17, 18] using methods based on Ratner’s theorem. The
resultisthatφ ([T,∞[)→CT−1asymptoticallyforlargeT whenε→0.
ε
ThedistributionΓ isaPoissondistributionwithintensitya(ε)ifandonlyifforanyset
ε
A⊂R2,
(m(A)a(ε))k
Prob(noofpointsinA=k)= e−m(A)a(ε),
k!
and in also, for A ⊂ R2 and B ⊂ R2 with A(cid:83)B = ∅, the number of points in Γ (cid:83)A
ε
(cid:83)
andΓ Bareindependentrandomvariables. HeretheBoltzmann-Gradlimitisachieved
ε
bysettinga(ε) = ε−1,andlettingεgotozero. Thisdistributiondiffersfromtheperiodic
one in several fundamental aspects. First, obstacles may overlap, and hence the trajecto-
ries(x(t),v(t))cannotalwaysbecontinueduniquely. However,themeasureofsuch,bad
trajectories goes to zero when ε → 0. Secondly, before taking the limit R → ∞ in the
definition of the path length distribution, the expression in the right hand side of equa-
tion (1) is a random variable, but one that converges to a deterministic value both when
R → ∞, and when ε → 0. The path length distribution φ ([T,∞[) converges to a dis-
ε
tributionφ([T,∞[) ∼ exp(−cT). TheLorentzgaswithPoissondistributedobstacleshas
beenstudiedindetailbyGallavotti[9],whoprovedthatiff (x,v)∈L1(cid:0)Ω ×S1(cid:1)are
densities of initial points for point particles, and f (x,v,t)ε,=0 E(cid:2)f (T−t(xΓ,εv))(cid:3), then,
ε ε,0 Γε
assumingthatf → f ∈ L1(R2 ×S1), itfollowsthatf (x,v,t) → f(x,v,t), which
ε,0 0 ε
solvesalinearBoltzmannequation.
Adifferentclassofrandomdistributionsofscattererscanbeconstructedstartingfrom
aperiodicdistribution, byremovingobstaclesrandomly, independently, withsomeprob-
ability p(ε). Although for a given, positive ε, the behavior is rather different from the
Poissonean case, this difference disappears in the limit as ε → 0, and in particular, it is
possibletorigorouslyderivealinearBoltzmannequationstartingfromsuchdistributions
(see[5,20]).
Thefollowingsectiongivesexamplesofmathematicalconstructionsofquasicrystals,
and the corresponding Lorentz gas. However, computing long point particle trajectories
inaquasicrystalLorentzgasisverytimeconsuming,andthereforethesimulationresults
presented in this paper are performed on a one-dimensional discrete model. In the two-
dimensional periodic case, the path length distribution can be computed almost exactly
by analyzing the discrete map given by the consecutive intersections of a trajectory with
lines parallel to the lattice containing lattice points, as in Figure 1. We assume here that
a = 1. For an initial point (x ,v) with x = (q ,q˜ ) sitting on a horizontal line as in
0 0 0 0
the figure, and with v ∈ S2 having an angle θ to the vertical line, the distance between
twoconsecutivepointsistan(θ). Tofindthefreepathlengthofatrajectoryinaperiodic
Lorentzgasisthen(almost)equivalenttocomputing
√
(2) k =min(cid:8)j (cid:12)(cid:12)dist(qj,Z)≤ ε/2(cid:9).
√
Infact,wewillhaveτ (x ,v)= ε(k±1)/cosθ.
ε 0
4 BERNTWENNBERG
3. QUASICRYSTALS
The material in this section is mostly taken from M. Senechal’s book ”Quasi Crystals
andGeometry”[21],whichgivesagoodintroductiontothesubject.
ThereareseveralwaysofconstructionpointsetsinRn thatsatisfytherequirementfor
being a quasi crystal: ”an essentially discrete diffraction pattern exhibiting a forbidden
symmetry”. Oneoftheseisthesocalledprojectionmethod:
• LetLp beapointlatticeinRk (usuallythestandardlattice),andletK betheunit
cellinthelatticethatcontainstheorigin.
• LetE beann-dimensionalsubspaceofRk,suchthatE∩Lp ={0},andletE⊥be
theorthogonalcomplementofE.
• LetΠandΠ⊥betheorthogonalprojectionsonE andE⊥respectively.
• LetX = Lp ∩(cid:0)Π⊥(K)⊕E(cid:1), i.e. thesetoflatticepointscontainedinthestrip
(orcylinder)obtainedbytranslatingtheunitcellK alongE.
ThequasicrystalisΛ = Π(X),theorthogonalprojectionsofX onE. Thissetisdis-
(cid:8) (cid:12) (cid:9)
crete(inf |p1−p2|(cid:12)p1,p2 ∈Λ =r0 >0),nonperiodicandrelativelydense(meaning
thatthereisR >0suchthateverysphereinE ofradiusgreaterthanR containsatleast
0 0
onepointofΛ)(seeFigure2). ThatΛisdiscreteandrelativelydenseimpliesinparticular
thatitispossibletodefineascalingthatcorrespondstotheBoltzmann-Gradlimit.
Thetheoryofquasicrystalsstartedwiththe(experimental)discoveryofamaterialex-
hibitingafive-foldsymmetry. Mathematicallythiscanbeconstructedusingtheprojection
method,startingfromtheregularlatticeinR5. Aunitcellinthislatticeisthehypercube
Q with32vertices
5
5
1(cid:88)
α e , ,α ∈{−1,1}
2 j j j
j=1
wheree = (1,0,0,0,0),....,e = (0,0,0,0,1). Afivefoldsymmetryisgivenbycyclic
1 5
permutationoftheunitvectorse ,atransformationthatcanberepresentedbytherotation
j
matrix
 
0 0 0 0 1
 1 0 0 0 0 
 
A= 0 1 0 0 0 
 
 0 0 1 0 0 
0 0 0 1 0
whichbyanorthogonalchangeofvariablesbecomes
 cos(2π) −sin(2π) 0 0 0 
5 5
 sin(2π) cos(2π) 0 0 0 
 05 05 cos(4π) −sin(4π) 0 
 0 0 sin(45π) cos(4π5) 0 
5 5
0 0 0 0 1
Inthesenewcoordinates,Ewillbethesubspacespannedby(1,0,0,0,0)and(0,1,0,0,0).
Thequasicrystalwillthenbethetheset
Π(cid:0)Lp∪(cid:0)Π⊥(Q )⊕E(cid:1)(cid:1)⊂E.
5
A one-dimensional quasi crystal can be defined in the similarly as a projection on a
onedimensionalsubspaceofR2. Particularexamples,whichareconvenientfornumerical
computations,arethepointsequencesknownasFibonaccisequences. Theyaredefinedby
taking
E =(cid:8)x∈R2|x·ω =0(cid:9) ,
where √
1+ 5
ω =(−1,τ) with τ = .
2
FREEPATHLENGTHSINQUASICRYSTALS 5
E⊥
E
K
FIGURE 2. Atwodimensionalrepresentationoftheprojectionmethod
forconstructingquasicrystals
√
Letν =|ω|= 1+τ2.Thereisanexplicitformulaforcomputingthesequenceofpoints
thatconstitutetheone-dimensionalquasicrystalF obtainedbythisconstruction:
(cid:91) m 1 (cid:13)m(cid:13)
(3) F = {x } with x = + (cid:13) (cid:13) ,
m m ν τν (cid:13)τ (cid:13)
m∈Z
where(cid:107)·(cid:107)denotesthedistancetothenearestinteger. Thissequencehasmanyinteresting
properties,andinparticularitsatisfiesthecriteriathatdefinesaquasicrystal. Aparticular-
ityisthattheintervalbetweentwoconsecutivepointsareofexactlytwokinds,shortand
long:
1 1 1
x −x = or + .
m m−1 ν ν ντ
Aswehaveseenabove,thereisaverydirectconnectionbetweentheperiodicLorentz
gas in two dimensions and a discrete model, at least when it comes to computing the
freepath-lengthdistribution,illustratedinFigure1andEquation(2). Herewedefinethe
rescaledfreepathlength
(cid:8) (cid:12) (cid:9)
(4) τε(q0,v)=εmin j (cid:12)dist(q0+jv,F)≤ε/2 .
√
(NotethatcomparedtoEquation(2)thescalingfactor εisreplacedbyε,theonlyreason
beingtomakenotationmoreclean).
In the following section we simulate trajectories in order to estimate the path length
distributionandcomparewithsimulationresultswhenF isreplacedbyα−1Z,andwitha
Poissonstreamofpointsx withintensityα. Thefactorαischosensothatallthreepoint
j
distributionshavethesamedensity,asymptotically:
#([−R,R](cid:84)F) τ2
α= lim = .
R→∞ 2R ν
Thisisnotequivalenttothefreepathlengthdistributioninatwo-dimensionalLorentzgas
asitisintheperiodiccase,however.
4. SIMULATIONMETHODANDRESULTS
Herewestudythedistributionoffreepathlengthsofapointparticlejumpingonthereal
line, i.e. the number of jumps needed before the particle falls into an interval of width ε
withcenterateitherthequasicrystalF,asdefinedpreviously,theperiodiclattice ν Z,or
τ2
onaPoissondistributedsetofpoints,Γ withintensityτ2/ν. TheintensityofthePoisson
ε
6 BERNTWENNBERG
distributionandthescalingoftheperiodiclatticearechoseninorderthatallthreeobstacle
distributionshavethesamedensity.
Therandomstartingpointsq arechosenrandomly,uniformlyintheinterval
0
10000ν/τ2. Obviously,forcomputingthepathlengthdistributionintherandomobstacle
distributionthisisnotnecessary,q =0wouldgiveexactlythesameresult. Moreover,in
0
thiscasethedistributioncanbecomputedanalyticallyinthelimitofsmallε. Thatcaseis
includedinthesimulationonlyforcomparison. Intheperiodicdistributionofscatterersit
wouldbemorenaturaltochosearandominitialpointuniformlyintheinterval[0,ν/τ2],
butbydefinitionthequasicrystalisnotperiodic,andthereforeitisnaturaltochosealarger
interval.
Forthesimulationspresentedhere,vischosenuniformlyin[0,ν/τ2].Toobtainaresult
completely equivalent to the periodic Lorentz gas in two dimensions, one should have
takenv = ν tanθ withθ uniformlychosenintheinterval[0,π/4],butsimulationswith
τ2
differentjumplengthdistributionsshowgiveverysimilarresults,andthisisnotpresented
here. However,wehaveonlytrieddistributionswithboundeddensities.
Thepositionofthepointparticleafterj jumpsisdenotedq =q +vj,andthepoints
j 0
x ∈ F intheFibonaccisequencearecomputedwiththeformula(3). Thepointsinthe
m
Poissondistributionarecomputedindependentlyforeachtrajectoryq .
m
Thesimulationprocedureisthen
(1) Choseq andvrandomly
0
(cid:8) (cid:12) (cid:9)
(2) ComputekF,ε =min j (cid:12)dist(q0+jv,F)≤ε/2
(3) ComputekZ,ε =min(cid:8)(cid:8)j (cid:12)(cid:12)(cid:12)dist(q0+jv,τν2Z)≤ε(cid:9)/2(cid:9)
(4) ComputekΓ,ε =min j (cid:12)dist(q0+jv,Γ)≤ε/2 ,whereΓistherandomobsta-
cledistribution. Notethatforagiventrajectoryq ,thefreepathtimeisarandom
k
variable,dependingontherealizationoftheobstacledistribution. Onlyonereal-
izationoftherandomobstaclehasbeenchosen.
(5) Makearecordofeachofthesepathlengths,andrepeatalargenumberN times.
Forallsimulationresultspresentedbelow,N =2×107,andtheresultsresultshavebeen
usedtoestimateProb[k ≥K],where∗representsthedifferentobstacledistributions.In
∗,ε
connectionwiththeLorentzequation,therelevantquantityisthepathlengthdistribution
expressedintimeunits,Prob[εk ≥T],whichisestimatedas
∗,ε
Numberoftrajectorieswithk ≥T ε−1
(5) φ ([T,∞[)= ∗,ε
∗ε N
Thefirstexample,showninFigure3,comparesthefreepathlengthdistributionswith
thethreedifferentdistributionofscatterersthatweconsiderhereforε=10−5. Thegraph,
plotted in logarithmic scale, show the 1/T-behavior of φ ([T,∞[) both for the quasi
∗,ε
crystalF andfortheperiodicdistribution,whiletherandomdistributiongivesadifferent
result(anexponentialdecay,asexpected).
Figure4showsthesameasFigure3butwithε=10−4(left)andε=10−3(right).Note
thatthepathlengthdistributionfortherandomobstacledistributionisexponentiallyonly
inoftheexponentialdistributionforlargeT. Thiscorrespondsthefractionoftrajectories
withajumplengthv <εwhicharestoppedbythefirstobstacletheyreach.
Finally, Figure 5 shows the path length distribution φ ([T,∞[) for the quasi crystal
F,ε
with three different values of ε. The graph on the left shows distribution of the number
of free steps , k , and the graph on the right shows the three curves expressed as a
F,ε
function of T, as in Equation 5. That these curves almost coincide is a strong indication
that φ [T,∞[) converges to some distribution φ [T,∞[) as ε → ∞, just as in the
F,ε F
periodiccase.
FREEPATHLENGTHSINQUASICRYSTALS 7
FIGURE 3. The free path length distribution based on 107 trajectories
and ε = 10−5. The curves show the result for the Fibonacci distribu-
tion of scatters (green), the periodic distribution (blue) and the random
distribution(magenta).
FIGURE4. ThefreepathlengthdistributioncomputedasinFigure3but
withε=10−4(left)andε=10−3(right)
5. CONCLUSIONS
Thesimulationresultsshowthatatleasttheone-dimensionaldiscretetimeLorentzgas
inaquasicrystalbehavesessentiallyastheperiodicLorentzgas,andatleastthatthepath
length distribution is not exponentially decaying, as in the random case. The simulation
examplesallrefertothespecificcaseofaFibonaccisequencebasedquasicrystal. There
areotherconstructionsofonedimensionalquasicrystals,andinparticularonecanconstruct
awholefamilyofpointdistributionsconstructedasnon-peridicsequencesoftwodifferent
8 BERNTWENNBERG
FIGURE 5. The free path length distribution in the quasi crystal, for
ε = 10−5,10−4 and10−3. Totheleftthecurvesrepresentthenumber
of steps and on the right, the curves represent the scaled time before a
trajectoryhitsanobstacle.
intervals(thebookbySenechal[21]presentssome,andgivemanyreferences). Inpartic-
ularIwantedtoseewhetherthespecialchoiceofthegoldenratioasrationbetweenlong
andshortintervalswouldgiveaqualitativelydifferentresultfromother,saytrancendent,
ratios. Butthequalitativebehaviourseemstobethesame,andhencenosimulationresults
arepresentedhere.
However,thesimulationshavebeencarriedoutinamuchsimplifiedmodelofthegas,
andthereforeitisnotobviousthatasimulationofareal(two-dimensional)Lorentzgasin
aquasicrystalwouldgivethesameresult.
Because the quasi crystals considered here are constructed as projections of a regular
latticeinhigherdimension,itispossiblethatthemethodsofMarklofandStrömbergsson
referredtoabovewouldalsogiveresultsinthiscase. Thispossibilityismentionedin[14],
butasofnow,nofurtherresultshavebeenpublished.
Ontheotherhand, thereareratherexplicitresultsforadiscreteSchrödingerequation
inaone-dimensionalquasicrystalliketheonestudiednumericallyhere[19].
ACKNOWLEDGMENTS
I would like to thank Andreas Strömbergsson for kindly providing information on his
andMarklof’swork,andforpointingoutseveralrelevantreferences. Iwouldalsloliketo
thankEmanueleCagliotiandFrançoisGolseformanyinterstingdiscussions.Theresearch
hasbeenpartiallysupportedbytheSwedishResearchCouncil.
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DEPARTMENT OF MATHEMATICAL SCIENCES, CHALMERS UNIVERSITY OF TECHNOLOGY, SE41296
GOTHENBURG,SWEDEN
10 BERNTWENNBERG
DEPARTMENTOFMATHEMATICALSCIENCES,UNIVERSITYOFGOTHENBURG,SE41296GOTHENBURG,
SWEDEN
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