Development of Fractal Geometry in a 1+1 Dimensional Universe

0.65 MB·English

by  Bruce N. Miller

Development of Fra
tal Geometry in a 1+1 Dimensional Universe
Bru
e N. Miller
Department of Physi
s and Astronomy,
Texas Christian University∗,
Fort Worth, Texas 76129
Jean-Louis Rouet
Institut des S
ien
es de la Terre d'Orléans (ISTO) - UMR 6113 CNRS/Université d'Orléans,
1A, rue de la Férollerie and Laboratoire de Mathématique,
Appli
ations et Physique Mathématique - UMR 6628 CNRS/Université d'Orléans,
UFR des S
ien
es, F-45067 Orléans Cedex 2, Fran
e
7 Emmanuel Le Guirre
0 Laboratoire de Mathématique,
0 Appli
ations et Physique Mathématique - UMR 6628 CNRS/Université d'Orléans,
2 UFR des S
ien
es, F-45067 Orléans Cedex 2, Fran
e

n
(Dated: February 6, 2008)
a
J
Observationsofgalaxiesoverlargedistan
esrevealthepossibilityofafra
taldistributionoftheir

3 positions. The sour
e of fra
tal behavior is thela
k of a length s
ale in the two body gravitational
intera
tion. However,evenwithnew,larger,samplesizesfromre
entsurveys,itisdi(cid:30)
ulttoextra
t
1 information
on
erning fra
tal properties with
on(cid:28)den
e. Similarly,simulationswith a billion par-
v
ti
lesonlyprovideathousandparti
lesperdimension,fartoosmallfora

urate
on
lusions. With
5
onedimensional models these limitations
an be over
omeby
arrying outsimulations with on the
0
orderofaquarterofamillionparti
leswithout
ompromisingthe
omputationofthegravitational
0
for
e. Herethemultifra
talpropertiesofagroupofthesemodelsthatin
orporatedi(cid:27)erentfeatures
1
of the dynami
al equations governing the evolution of a matter dominated universe are
ompared.
0
Theresultsshareimportantsimilaritieswithgalaxyobservations,su
hashierar
hi
al
lusteringand
7
apparent bifra
tal geometry. They also provide insights
on
erning possible
onstraints on length
0
andtimes
ales forfra
tal stru
ture. They
learly demonstratethatfra
tal geometryevolves inthe
/ µ
h (position, velo
ity) spa
e. The observed properties are simply a shadow (proje
tion) of higher
p dimensionalstru
ture.
-
h
PACSnumbers: 05.10.-a,05.45.-a,98.65.-r,98.80.-k
t
a
m
I. INTRODUCTION the intergala
ti
distan
e seemed to support the fra
tal
:
v
onje
ture. In fa
t, in the past it has been argued by
i Pietroneroand
olleaguesthat the universemay even be
X
Starting with the work of Vau
oulours,[1℄ questions fra
tal on all s
ales.[4℄[6℄ Of
ourse, if this were true it
r
a have been posed about the geometri
properties of the would wreak havo
with the
on
lusions of
osmologi
al
distribution of matter in the universe. Ne
essary for theory.
all modern
osmologies are the assumptions of homo-
M
Cauley haslookedat this issuefrom afew di(cid:27)erent
geneity and isotropy of the mass distribution on large
perspe
tives. He points out that sin
e the powerlawbe-
length s
ales.[2℄ However, observations have shown the
havior of the
orrelation fun
tion is only quantitatively
existen
e of very large stru
tures su
h as super-
lusters

orre
t over a (cid:28)nite length s
ale, the universe
ould not
and voids.[3℄ Moreover, as te
hnology has advan
ed, so
beasimple(mono)fra
tal.[7℄Thenthelogi
alnextques-
has the length s
ale of the largest observed stru
tures.
tion is whether or not the matter distribution is multi-
(Forare
entreviewseeJoneset. al.[4℄.) Itwasproposed
fra
tal. By
onsidering
ounts in
ells, Bailin and S
ha-
by Mandelbrot,[5℄ based on work of Peebles,[2℄ that the
e(cid:27)erhypothesizedsometimeagothatthedistributionof
matterdistributionintheuniverseisfra
tal. Supportfor
matter is approximately bifra
tal, i.e. a superposition of
this
onje
ture
ame primarily from the
omputation of
fra
tals with two distin
t s
aling laws.[8℄ M
Cauley has
thepair
orrelationfun
tionforthepositionsofgalaxies,
also addressed the possibility of inhomogeneity. Work-
aswellasfromdire
tobservation. Thefa
tthatthe
or-
ing with Martin Kers
her, he investigated whether the
relation fun
tion was well represented by a power law in
universehas multifra
tal geometry. His approa
hwasto
examine the point-wise dimension [9℄ by looking for lo-

al s
aling of the density around individual galaxies in
∗
Ele
troni
address: b.millert
u.edu; two
atalogues. Their
on
lusion was that, even if lo
al
URL:http://personal.t
u.edu/bmiller s
aling were present, there are large (cid:29)u
tuations in the
2
s
aling law. Moreover, the sample sizes were not su(cid:30)- tivity. [15℄ Then a question arises
on
erning the in
lu-

ient to be able to extra
t good lo
al s
aling exponents. sion of the Hubble (cid:29)ow into the dynami
al formulation.
Althoughlargergalaxy
atalogueshavebe
omeavailable This has been addressed in two ways: In the earliest,
sin
e their analysis, they do not meet the restri
tive
ri-
arriedoutby Rouetand Feix,thes
alefun
tionwasdi-
teria of su(cid:30)
ient size, as well as uniform extension in re
tlyinsertedintotheonedimensionaldynami
s.[16,17℄
all dire
tions, ne
essaryto measurelo
al dimensions. [4℄ Alternatively, starting with the usual three dimensional
Their(cid:28)nal
on
lusionwasthat,whilethegeometryofthe equationsofmotionandembeddingastrati(cid:28)edmassdis-
observeduniverseis
ertainlynotmonofra
tal,itwasnot tribution, Fanelli and Aurell obtained a similar set of
possibletoirrefutably
on
ludewhetheritismultifra
tal,
oupleddi(cid:27)erentialequationsfortheevolutionofthesys-
orwhethertheassumptionsofhomogeneityandisotropy tem in phase spa
e.[18℄ While the approa
hes are di(cid:27)er-
prevail at large length s
ales.[7℄ ent, from the standpoint of mathemati
s the twomodels
If
urrent observations aren't able to let us determine are very similar and di(cid:27)er only in the values of a single
the geometry of the universe, then we need to turn to (cid:28)xed parameter, the e(cid:27)e
tive fri
tion
onstant. Follow-
theory and simulation. For the most part, we are
on- ing Fanelli,[18℄ we refer to the former as the RF model

erned with the evolution of matter after the period of and the latter as the Quinti
, or Q, model.
re
ombination. Consequently the Hubble expansion has By introdu
ing s
aling in both position and time, in
sloweddownsu(cid:30)
iently thatNewtoniandynami
sisap- ea
hmodelautonomousequationsofmotionareobtained
pli
able, at least within a (cid:28)nite region of spa
e.[10℄ A in the
omoving frame. In addition to the
ontribution
number of theoreti
al approa
hes to the
omputation of for the gravitational (cid:28)eld, there is now a ba
kground
fra
tal dimensions have been investigated. Ea
h of them term,
orresponding to a
onstant negative mass distri-
is predi
ated on some external assumption whi
h is not bution, and a linear fri
tion term. By eliminating the
yet veri(cid:28)ed. For example, de Vega et. al. assume that fri
tion term a Hamiltonian version
an also be
on-
the universe is
lose to a thermodynami

riti
al point stru
ted. At least three other one dimensional mod-
[11℄ , while Gruji
explores a (cid:28)eld theory where va
uum els have also been investigated, one
onsisting of New-
energy predominates over inert matter, and the latter is tonian mass sheets whi
h sti
k together whenever they
assumed to have a fra
tal stru
turing [12℄. An exam-
ross,[19℄ one evolved by dire
tly integrating the Zel-
ination of the re
ent literature reveals that theory has dovi
h equations,[20, 21℄ and the
ontinuoussystem sat-
not
onverged on a
ompelling and uniformly a

epted isfyingBurger'sequation[22℄. Inaddition,fra
talbehav-
theory of fra
tal stru
ture in the universe. ior has been studied in the autonomous one dimensional
In the last few de
ades dynami
al N-body simula- systemwherethereisnoba
kgroundHubble(cid:29)ow.[23,24℄
tion of
old dark matter CDM has experien
ed rapid In dynami
al simulations, both the RF and Quinti
advan
es due to improvements in both algorithms and model
learly manifest the development of hierar
hi
al
µ
te
hnology.[13, 14℄ It is now possible to
arry 1o0u9t grav-
lusteringin both
on(cid:28)gurationand spa
e(the proje
-
itational N-body simulations with upwards of point tionofthephasespa
eontheposition-velo
ityplane). In
mass parti
les. However, in order to employ simulation
ommonwiththeobservationofgalaxypositions,astime
methods for systems evolving over
osmologi
al time, evolves both dense
lusters and relatively empty regions
it is ne
essary to
ompromise the representation of the (voids)develop. Intheirseminalwork,by
omputingthe
gravitationalintera
tionoverboth longand shortlength box
ounting dimension for the RF model, Rouet and
s
ales. For example, tree methods are frequently em- Feix were able to dire
tly demonstrate the formation of
ployed for large separations, typi
ally the gravitational fra
talstru
ture.[16, 17℄Theyfound avalueofabout0.6
(cid:28)eldis
omputedfromagrid,andashortrange
ut-o(cid:27)is for the box
ounting dimension of the well evolved mass
employedto
ontrolthesingularityintheNewtonianpair points in the
on(cid:28)guration spa
e, indi
ating the forma-
potential.[13, 14℄ Unfortunatel1y0,9even if the simulations tion of a robust fra
tal geometry. In a later work,Miller
w10e3reperfe
t,asystemofeven parti
lesprovidesonly and Rouet investigated the generalized dimension of the
parti
lesperdimensionandwouldthusbeinsu(cid:30)
ient RFmodel.[25℄In
ommonwiththeanalysisofgalaxyob-
to investigate the fra
tal geometry with
on(cid:28)den
e. servationsbyBailinandS
hae(cid:27)er[8℄theyfoundeviden
e
As a logi
al
onsequen
e of these di(cid:30)
ulties it was for bi-fra
tal geometry. Although they did not
ompute
natural that physi
ists would look to lower dimensional a
tual dimensions, later Fanelli et. al. also found a sug-
models for insight. Although this sa
ri(cid:28)
es the
orre
t gestion of bi-fra
tal behavior in the model without fri
-
dynami
s, it provides an arena where a

urate
ompu- tion [26℄. It is then not surprising that the autonomous,
tations with large numbers of parti
les
an be
arried isolated,gravitationalsystemwhi
hdoesnotin
orporate
out for signi(cid:28)
ant
osmologi
al time. It is hoped that the Hubble (cid:29)ow alsomanifests fra
tal behaviorfor short
insights gainedfrom makingthis trade-o(cid:27) arebene(cid:28)
ial. times as long as the virial ratio realized by the initial
In one dimension, Newtonian gravity
orresponds to a
onditions is very small.[23, 24℄ In addition, in the one
systemofin(cid:28)nitesimallythin, parallel,masssheetsofin- dimensional model of turbulen
e governed by Burger's
(cid:28)nite spatial extent. Sin
e there is no
urvature in a equation, the formation of sho
ks has the appearan
e of
1+1dimensional gravitationalsystem, we
annot expe
t densitysingularitiesthataresimilartothe
lustersfound
to obtain equationsof motion dire
tly from general rela- in theRF and Quinti
model. Atypeofbi-fra
talgeom-
3
etry has also been demonstrated for this system.[22℄ du
tion, we are interested in the development of den-
Belowwepresenttheresultsofourre
entinvestigation sity (cid:29)u
tuations following the time of re
ombination so
of multifra
tal properties of the Quinti
model and the that ele
tromagneti
for
es
an be ignored. For that,
modelwithoutfri
tion. Inse
tionIIwewill(cid:28)rstdes
ribe and later, epo
hs, the Hubble expansion has slowed suf-
the systems. Then we will give a straightforwardderiva- (cid:28)
iently that Newtonian dynami
s provides an adequate
tion of the equations of motion and explain how they representationof the motion in a (cid:28)nite region.[10℄ Then,
3+1
di(cid:27)erfromtheothermodelsmentionedabove. Inse
tion ina dimensionaluniverse,theNewtonianequations
III we will explain how the simulations were
arried out governinga mass point are simply
and des
ribe their qualitative features. In se
tion IV we
de(cid:28)ne the generalized dimension and other fra
tal mea- dr dv
=v, =E (r,t)
sures and present our approa
h for
omputing them. In dt dt g (2)
se
tion V we will present the results of the multifra
tal
analysis. Finally
on
lusions will be presented in se
tion Eg(r,t)
where,here, isthegravitational(cid:28)eld. Wewishto
VI.
followthe motionin aframeofreferen
ewhere theaver-
age density remains
onstant, i.e. the
o-moving frame.
II. DESCRIPTION OF THE SYSTEM Thereforewetransformto anewAs(pta)
e
oordinrat=ewAh(ti)
xh
s
ales the distan
e a

ordingto . Writing
we obtain
We
onsider a one dimensional gravitational system
N
(OGS) of parallel, in(cid:28)nitesmimally thin, mass sheets d2x 2 dAdx 1 d2A 1
ea
h with mass per unit area . The sheets are
on- dt2 + A dt dt + A dt2 x=A3Eg(x,t) (3)
strained to move perpendi
ular to their surfa
e. There-
x
fore we
an
onstru
t a Cartesian axis perpendi
ular
N
to the sheets. We imagine that the sheets are labeled where, in the above we have taken advantage of the in-
j = 1,...,N x
beay
h sheet
an.beFraosmsigintsedinttheerse

otoiordninwaittehxtjheand-axthise, Evegr(sxe,stq)u=areAd12eEpegn(rd,etn)
wehofertehethgreavfuitna
ttiioonnaall(cid:28)deeldpetnodwerni
tee
v

orrespondingvelo
ity j. FromGauss'law,thefor
eon is preserved. In a matter dominated (Einstein deSitter)
ea
hsheetisa
onstantproportionaltothenetdi(cid:27)eren
e universe we (cid:28)nd that
between the total "mass" (really surfa
e density) to its
2
right and left. The equations of motion are A(t)= t 3 , ρ (t)= 6πGt2 −1
b
(cid:18)t0(cid:19) (4)
dx dv (cid:0) (cid:1)
i i
=v , =E =2πmG[N N ]
i i R,i L,i
dt dt − (1) t0
where issomearbitraryinitialtime
orresponding,say,
G
wthhere Ei is theNgRra,ivitNatLio,inal (cid:28)eld experien
ed by the ttiootnhael
eopnos
thanotf,raen
odmρbb(int)atisiotnh,eaviesrtahgee,uunnivifeorrsmal,gdreanvsititay-
i parti
le and ( ) is the number of parti
les
frequently referred to as the ba
kground density. These
(sheets) to its right (left) . Be
ause the a

eleration of
results
anbeobtaineddire
tlyfromEq.3bynotingthat
ea
hparti
leis
onstantbetween
rossings,theequations
if the density is uniform, so that all matter is moving
of motion
an be integrated exa
tly. Therefore it is not
with the Hubble (cid:29)ow, the (cid:28)rst two terms in Eq.3 van-
ne
essary to use numeri
al integration to follow the tra- A
ishwhereasthethird term(times ) mustbeequatedto
je
tory in phase spa
e. As a
onsequen
e it was possible
to study the system dynami
s on the earliest
omputers the gravitational (cid:28)eld resulting from the uniformlyAdxis-
tributed mass
ontained within a sphere of radius | |.
and it
an be
onsidered the gravitational analogue of
Then the third term of Eq.3 is simply the
ontribution
the Fermi-Pasta-Ulam system.[27℄ It was (cid:28)rst employed
bNy Le
ar and Cohen to investigate the relaxation of an dareinssinitgybfryomsubthtreas
ptihnegret.[h2e℄ N(cid:28)eoltdindgutehattoAth3eρbb(ta)
k=grρobu(tn0d)
body gravitational system.[28℄ Although it was (cid:28)rst
N for
es the result. Alternatively, also for the
ase of uni-
thought that the body system would reNa
2h equilib- form density, taking the divergen
e of ea
h side of Eq. 3
riumwith arelaxationtimeproportionalto ,[29℄ this
andassertingthePoissonequationfor
esthesameresult.
was not born out by simulations.[30℄ Partially be
ause
ThustheFriedmans
alingis
onsistentwiththe
oupling
of its relu
tan
e to rea
h equilibrium, both single and
of a uniform Hubble (cid:29)ow with Newtonian dynami
s.[2℄
two
omponentversionsof the system havebeen studied
For
omputational purposes it is useful to obtain au-
exhaustively in re
ent years.[31, 32℄[33℄[34℄[35℄[36℄ Most
tonomous equations of motion whi
h do not depend ex-
re
ently it has been demonstrated that, for short times
pli
itlyonthetime. This
anbee(cid:27)e
tivelya

omplished
and spe
ial initial
onditions, the system evolution
an
[16,17℄bytransformingthetime
oordinatea

ordingto
be modeled by an exa
tly integrable system.[37℄
To
onstru
t and explorea
osmologi
alversionofthe
A(t) t
OGSweintrodu
ethes
alefa
tor foramatterdom- dt=B(t)dτ, B(t)=
inated universe.[2℄ Moreover, as dis
ussed in the intro- t0 (5)
4
σ2
yielding the autonomous equations where v isthevarian
eofthevelo
ityattheinitialtime,
v = σ = a/√3 a
T v
is the thermal velo
ity, and is the
d2x + 1 dx 2 x=E (x). maximalabsolutevelo
ityvalue giventoaparti
le[ wah,aen
dτ2 3t0 dτ − 9t20 g (6) the initial distribution of velo
ities is unifLor=mnoλn,− 0<℄.
j
Then, with the further requirement that
n<N,
in these units our equations are
For the Newtonian dynami
s
onsidered here it is ne
-
essary to
on(cid:28)ne our attention to a bounded region of
Ω Ω d2x 1 dx 1 n
spa
e, . It is
ustomary to
hoose a
ube for and as- i i
+ x = [N N ].
sumeperiodi
boundary
onditions. Thusourequations dτ2 √6 dτ − 3 i (cid:16)N(cid:17) R,i− L,i (13)

orrespond to a dissipative dynami
al system in the
o-
1/3t0
movingframewithfri
tion
onstant andwithfor
es
Thereisstilla(cid:28)nalissuethatwehavetoaddress. Ifwe
arisingfrom(cid:29)u
tuationsinthelo
aldensitywithrespe
t
assumeforthemomentthatthesheetsareuniformlydis-
to a uniform ba
kground.
tributedand movingwith theHubble(cid:29)ow, thenthe(cid:28)rst
For the spe
ial
ase of a strati(cid:28)ed mass distribution
t two terms above are zero. Unfortunately, if we take the
the lo
al density at time is given by
averageoverea
h remaining term, we (cid:28)nd that they dif-
fer by a fa
tor of three. This seeming dis
repan
y arises
ρ(x,t)= mj(t)δ(x xj) be
ausewestartedwithspheri
alsymmetryaboutanar-
− (7)
X bitrary point for the Hubble (cid:29)ow and are now imposing
mj(t) jth axial symmetry. If we imagine that we are in a lo
al re-
where is the mass per unit area of the sheet
gionwithastrati(cid:28)edgeometry,thesymmetryisdi(cid:27)erent
and, from symmetry, the gravitational (cid:28)eld only has a
x andthishastobere(cid:29)e
tedintheba
kgroundterm. This

omponent in the dire
tion. Then, for the spe
ial
ase
of equal masses mj(t) = m(t), the
orre
t form of the apparentxidis
repan
y
anbe re
ti(cid:28)ed by multiplying the
term in by three. Our (cid:28)nal equations of evolution are
gravitational(cid:28)eldo

urringon therighthandside ofEq
i then
6 at the lo
ation of parti
le is then
d2x 1 dx n
i i
Eg(xi)=2πm(t0)G[NR,i−NL,i] (8) dτ2 + √6 dτ −xi=(cid:16)N(cid:17)[NR,i−NL,i]. (14)
sminj
(et)w=e malr(et0a)d/yAi2mpli
itly a

ounted for the fa
t that The des
ription is
ompleted by assuming that the sys-
in Eq.6. The equations are further
m(t0) temsatis(cid:28)esperiodi
boundary
onditionsontheinterval
simpli(cid:28)ed by establishing the
onne
tion between 2L
ρb(t0). , i.e. when a parti
le leaves the primitive
ell de(cid:28)ned
andtheba
kgrounddensityattheinitialtime, Let 2L < x < 2L
2N by − on the right, it re-enters at the left
us assume that we have parti
les (sheets)
on(cid:28)ned
2L 2L<x<2L
hand boundary with the identi
al velo
ity. Note that, in
within a slab with width , i.e. − . Then

omputing the (cid:28)eld, we do not attempt to in
lude
on-
x
i
tributions from the images of outside of the primitive
ρb(to)= 6πGt2o −1 =(cid:18)NL(cid:19)m(t0), (9)
ell.
(cid:0) (cid:1) Finally, we mention that the RF model is obtained
fromthereversesequen
ewhereone(cid:28)rstrestri
tsthege-
we may express the (cid:28)eld by ometryto1+1dimensionsandthenintrodu
esthetrans-
formation to the
omoving frame. In this approa
h it is
2 L
Eg(xi)=2πm(t0)G[NR,i−NL,i]= 3t20 (cid:18)N(cid:19)[NR,i−NL,i], onfotxinea
sewsseardyidtohemxrea.kTehthiseisadqjuuis
tkmlyenseteinnbtyheno
otien(cid:30)g
tihenatt
thedivergen
eof isthreetimes greaterthan thediver-
(10) xxˆ
gen
e of whi
h one would obtain by dire
tly starting
and the equations of motion for a parti
le in the system
with the one dimensional model. However, in the RF
now read
model, the
oe(cid:30)
ient of the (cid:28)rst derivative term (the
1/√2 1/√6
d2x 1 dx 2 2 L fri
tion
onstant) is instead of . This simply
i i
dτ2 + 3t0 dτ − 9t20xi=3t20 (cid:18)N(cid:19)[NR,i−NL,i]. (11) milleunstsrioanteasl tuhnaivt,erssine,
ethtehreereisisandoeg
ruerevaotfuarerbiintraar1in+e1ssdiin-

hoosing the (cid:28)nal model. It
annot be obtained solely
Itis
onvenienttorefertoJeanstheoryforthe(cid:28)nal
hoi
e from General Relativity. For a dis
ussion of this point
of units of time and length [38℄ see Mann et. al. . [15℄ A Hamiltonian version
an also
3 be
onsidered by setting the fri
tion
onstant to zero.
T =ω−1 =(4πGρ)−1/2 = t
j j r2 o Intheirearlierwork,usingthe linearizedVlasov-Poisson
equations, Rouet and Feix
arried out a stability analy-
3
λj =λj =vT/ωjq3σv2/ωj2 = √2σvto, (12) sis of the model without fri
tion. They determined that
thesystemfollowedtheexpe
ted behavior,i.e. when the
5
system size is greater than the Jean's length, instability
o

urs and
lustering be
omes possible.[16, 17℄ We men- tan(2θ)=(1+ρ+ρ )/γ,
b
tionthat, whenthefri
tiontermisnotpresent,both the (18)
Q and RF models are identi
al so, with the assumption θ
where de(cid:28)nesthe anglewith the
oordinateaxisin the
that the fri
tion term won't have a large in(cid:29)uen
e on µ
spa
e. Thus,inregionsoflowdensity, weexpe
ttosee
short time, linear, stability, the analysis of the Hamilto-
linesofmassbeingstret
hedwith
onstantpositiveslope
nian version applies equally to ea
h version.
inthephaseplane. Wewillseebelowthatthispredi
tion
The Vlasov-Poisson limit for the system is obtained
N m 0
by letting → ∞ and → while
onstraining the is a

urately born out by simulations.
density and energy (at a given time). Then, in the
o-
movingframe, thesystemisrepresentedbya(cid:29)uidinthe
µ f(x,v;t)
III. SIMULATIONS
spa
e. Let representthe normalized distribu-
(x,v) t
tion of mass in the phase plane at time . From
f
mass
onservation, satis(cid:28)es the
ontinuity equation An attra
tion of these one-dimensional gravitational
systems is their ease of simulation. In both the au-
∂f ∂f ∂af ∂φT(x) tonomous(purelyNewtonianwithoutexpansion)andRF
+v + =0, a= γv
∂t ∂x ∂v − − ∂x (15) models it is possible to integrate the motion of the in-
dividual parti
les between
rossings analyti
ally. Then
whi
h
an also be derived using the identi
al s
aling as the temporalevolution ofthe system
anbe obtained by
a(x,v)
aboveS. In Eq. (15), is the lo
al a

eleration and followingthe su

essive
rossingsofthe individual, adja-
φ
T
isthepotentialfun
tionindu
edbythetotaldensity,
ent, parti
le traje
tories. This is true as well for the Q
ρb yi = xi+1 xi
in
ludingthee(cid:27)e
tivenegativeba
kgrounddensity,− . model. If we let − , where we have ordered
a(x,v) φ
T
Note that both and are linear fun
tionals of the parti
le labels from left to right, then we (cid:28)nd that
f(x,v;t) y
i
. Coupling Eq. (15) with the Poisson equation the di(cid:27)erential equation for ea
h is the same, namely
yieldsthe
ompleteVlasov-Poissonsystemgoverningthe
f(x,v;t)
evolution of . Depending on the (cid:28)nal
hoi
e of d2y 1 dy 2n
γ i + i y = .
thefri
tion
onstant, ,the
ontinuumlimitofeitherthe dτ2 √6 dτ − i −N (19)
Quinti
orRF model
an be representedby Eq.15. Note
that an exa
t solution is
The general solution of the homogenious version of
ρ v2 E. 19 is a sum of exponentials. By in
luding the par-
b
f(x,v,t)= √|2πσ| 2 exp−2σ2, σ(t)=σ0exp−γt ti
ular solution of the inhomogenius equation (simply a

onstant) we obtain a (cid:28)fth order algebrai
equation in
u = exp(τ/√6)
(16) . Hen
e the name Q, or Quinti
, model.
Using dynami
al simulation, we will see that it is ex- These
an be solved numeri
ally in terms of the initial
tremelyunstablewhenthesystemsizeex
eedstheJeans'
onditions by analyti
ally bounding the roots and em-
length at the initial time. ployingthe Newton-Raphson method.[39℄ (Note that for
Useful information
an be had without
onstru
ting the RF model a
ubi
equation is obtained.) A sophis-
an expli
it solution. For example, we qui
kly (cid:28)nd that ti
ated, event driven, algorithm was designed to exe
ute
thesystemenergyde
reasesatarateproportionaltothe thesimulations. Twoimportantfeaturesofthealgorithm
kineti
energy. The Tsallis entropy is de(cid:28)ned by are that it only updates the positions and velo
ities of a
pair of parti
les when they a
tually
ross, and it main-
1 fqdxdv
tains the
orre
t ordering of ea
h parti
le's position on
S = − , fdxdv =1
q Rq 1 Z Z (17) the line. Using this algorithm we were able to
arry out
−
runsforsigni(cid:28)
ant
osmologi
altimewithlargenumbers
q 1 S
q
Inthelimit → , redu
estotheusualGibbsentropy, of parti
les.
S1 = flnfdxdv
−R R . By asserting the Vlasov-PoissSo1n Typi
alnumeNri
a=ls2i1m8ulationswere
arriedoutforsys-
evolution,we(cid:28)ndthattheBolztmann-Gibbsentropy, , temswithupto parti
les. Initial
onditionswere
2γ q > 1 S
q
de
reases at the
onstant rate − while, for ,
hosen by equally spa
ing the parti
les on the line and
de
reases exponentially in time. This tells us that the randomly
hoosing their velo
ities from a uniform dis-
mass is being
on
entrated in regions of de
reasing area tribution within a (cid:28)xed interval. For histori
al reasons
of the phase plane, suggesting the development of stru
- we
all this a waterbag. Other initial
onditions, su
h
ture. These properties are immediately evident for the as Normally distributed velo
ities, as well as a Brown-
trivial solution given above. By imposing a Eu
lidean ian motion representation, whi
h more
losely imitates
metri
in the phase plane, we
an also investigate lo
al a
osmologi
al setting, were also investigated. The sim-
properties su
h as the dire
tions of maximum stret
hing ulations were
arried forward for approximately (cid:28)fteen
and
ompression, as well as the lo
al vorti
ity. We eas- dimensionless time units.
ily (cid:28)nd that the rate of separation between two nearby In F21i7g. 1 we present a visualization of a typi
al run
points is a maximum in the dire
tion given by with parti
les. The system
onsists of the Q model
6
µ
with an initial waterbag distribution in the spa
e. Ini- the multifra
tal analysisdes
ribed table I below in some
tially the velo
ity spread is (-12.5, 12.5) in the dimen- detail.
sionlessunitsemployedhereandthesystem
ontainsap-
proximately 16,000 Jeans' lengths. In the left
olumn
we present a histogram of the parti
les positions at in- IV. FRACTAL MEASURES

reasing time frames, while on the right we display the
µ

orresponding parti
le lo
ations in (position, velo
ity) Itisnatural to assumethatthe apparentlyself similar
spa
e. It is
lear from the panels that hierar
hi
al
lus- stru
turewhi
hdevelopsinthephaseplane(seeFigs.1,2)
teringiso

urring,i.e. small
lustersarejoiningtogether astimeevolveshasfra
talgeometry,but wewillseethat
to form largerones, sothe
lustering me
hanism is (cid:16)bot- things aren't so simple. In their earlier study of the RF
toms up(cid:17)[2℄. The (cid:28)rst
lusters are roughly thτe=siz6e of a model, Rouet and Feix found a box
ounting dimension
Jean's length and seem toτ a=pp1e4ar at about and for the parti
le positions of about 0.6 for an initial wa-
there are many while bµy there are on the order terbag distribution (uniform on a re
tangle in the phase
of 30
lusters. In the spa
e we observe that between plane-see above) and a fra
tal dimension of about 0.8 in
the
lusters matter is distributed along linear paths. As the
on(cid:28)gurationspa
e(i.e. oftheproje
tionofthesetof
µ
time progresses the size of the linear segments in
reases. pointsin spa
eonthepositionaxis). [17℄. Asfaraswe
Thebehaviorofthµeseunder-denseregionsisgovernedby know, Bailin and Shae(cid:27)er were the (cid:28)rst to suggest that
the stret
hing in spa
e predi
ted by Vlasov theory ex- the distribution ofgalaxypositions are
onsistentwith a
plainedabove(seebeforeEq.(18)). Theslopesoftheseg- bifra
tal geometry [40℄ . Their idea was that the geome-
ments are the same and in quantitative agreement with tryofthegalaxydistributionwasdi(cid:27)erentinthe
lusters
Eq. (18). Qualitatively similar histories are obtained andvoidsand,asa(cid:28)rstapproximation,this
ouldberep-
for the RF model and the model without fri
tion (see resented as a superposition of two independent fra
tals.
below), as well as for the di(cid:27)erent boundary
onditions Of
ourse, their analysis was restri
ted solely to galaxy
mentioned above. However, there are some subtle dif- positions. Sin
e the stru
tureswhi
h evolveare strongly
feren
es. In Fig. 2 we zoomµin and show a sequen
e of inhomogeneous, to gain further insight we have
arried
magni(cid:28)ed inserts from the spa
e at time T=14. The out a multi-fra
tal analysis [41℄ in both the phase plane
hierar
hi
al stru
ture observed in these models suggests and the position
oordinate.
theexisten
eofafra
talstru
ture,but
arefulanalysisis Themultifra
talformalismsharesanumberoffeatures
requiredto determine if this is
orre
t. Simulationshave with thermodynami
s. To implement it we partitioned
µ
alsobeenperformedwherethesystemsizeislessthanthe ea
h spa
e
on(cid:28)guration spa
e and spa
e, into
ells of
l
Jean'slength. Theresultssupportthestandardstability length . Atea
htimeofobservationinthesimulation,a
µ =N (t)/N i N (t)
analysisinthathierar
hi
al
lusteringisnotobservedfor measure i i wasassignedto
ell ,where i
i t N
these initial
onditions. is the population of
ell at time and is the total
Histori
ally,powerlawbehaviorin the density-density number of parti
les in the simulation. The generalized
q

orrelationfuntionhasbeentakenasthemostimportant dimension of order is de(cid:28)ned by [41℄
signature of self similar behavior of the distribution of
galaxypositions.[4℄ In Figure 3 weprovidea log-logplot 1 lnC
C(r) D = lim q, C =Σµq.
of the
orrelation fun
tion at T=14 de(cid:28)ned by q q 1l→0 lnl q i (20)
−
1 r+∆ wheNre(NCq(l)1)is the e(cid:27)e
tive partition fun
tion [9℄, D0 is
C(r)=<δρ(y+r)δρ(y)>∼ ∆Zr i,Xj,i6=jδ(r′−(yi−yj))t−he boqxL
2−ou1ntindgrd′imension, D1, obtained by takinDg2the
 limit → ,is the information dimqension, and 0is
the
orrelationdimension [41℄[9℄. As in
reasesabove ,
i j L/4<x <3L/4 D
i,j q
whereparti
les and aresu
hthat the provideinformationonthegeometryof
ellswith
∆
toavoidboundarye(cid:27)e
tsand isthebinsize. Notethe higher population.
0.1 < ln(l) < l 0
existen
e of a s
aling region from about In pra
ti
e, it is not possible to take the limit →
30 l
, a range of about 2.5 de
ades in . Note also the witha(cid:28)nitesample. Instead,onelooksforas
alingrela-
lnl
noise present at larger s
ales. Sin
e the (cid:29)u
tuations are tion overa substantial rangeof with the expe
tation
lnC lnl
q
vanishingly small at s
ales on the order of the system that a linear relation between and o

urs, sug-
C l
q
size, this is most likely attributable to the presen
e of gesting power law dependen
e of on . Then, in the
shot noise. This
ouldberedu
edbytakingan ensemble mostfavorable
ase,the slopeofthelinear regionshould
γ q 1
average over many runs. By
omputing the slope, say provide the
orre
t power and, after dividing by − ,
D
q
, of the log-log plot in Fig. 3 we are able to obtain an thegeneralizeddimension . Asarule,orguide,ifs
al-
D2 =1 γ
estimateofthe
orrelationdimension − foraone ing
anbefound eitherfromobservationor
omputation
γ = .42 l
dimensional system. We (cid:28)nd that for a s
aling overthreede
adesof ,thenwetypi
allyinfer thatthere
l
region of about two de
ades in . This suggests that is good eviden
e of fra
talτqstru
tureC.[q7℄ Alτlqso of interelst
the
orrelation dimension is approximately 0.6, whi
h is is the global s
alingindex , where ∼ for small .
τ D
q q
in agreement within the standard numeri
al error with It
anbe shownthat and arerelatedto ea
hother
7
µ 217
Figure 1: Evolution in
on(cid:28)guration and spa
e for the quinti
model with parti
les from T=0 to T=14. The initial
distribution is a waterbag with velo
ities in therange (-12.5, 12.5) and a size of about16,000 Jeans' lengths.
D (1 q)=τ
q q
throughaLegendretransformationby − for dominant, we (cid:28)nd a simple relation between the global
q =1 τ α
q
6 [9℄. Here wepresentthe resultsofourfra
tal anal- index, , and ,
ysis of the parti
le positions on the line (position only)
µ
and the plane (position, velo
ity olr spa
e). τq =αq f(α).
If it exists, a s
aling range of is de(cid:28)ned as the in- − (22)
lnC lnl
q
terval on whi
h plots of qv=ers1us are lΣinµealrn.(µO)f µ

ourslne,(lf)or the spe
ial
ase of we plot − i i Re
all that a Eu
lidean metri
ωjis =imp1o,sed on the -
vs. to obtain the infoDrmation dimension. [9℄ If a spa
e. Be
ause, in ∆ouxr =uni∆tsv, weLdivide it
s
alingrange
anbe found, q isobtained by taking the inLto
e1l0ls,0s0u0
h that . Then, as is large
appropriatederivative. Itiswellestablishedbyproofand ( ≈ ), so is the extent of the partition in the
example that, for a normal, homogeneous, fra
tal, all of velo
ity spa
e. On this s
ale the initial distribution of
the generalized dimensions are equal, while for an inho- parti
ules appears as a line so that the initial dimen-
mogeneous fra
tal, e.g. the Henon attra
tor,Dq+1 ≤Dq sion is also about unity. In fa
t, initially the velo
ity of
l C (l)
[41℄. Inthelimitofsmall , thepartitionfun
tion, q , the parti
les is a perturbation. It is not large enough

analsobede
omposedintoasumof
ontributionsfrom to allow parti
les to
ross the entire system in a unit of
regions of the inhomogeneous fra
tal sharing the same time. During the initial period the virial ratio rapidly
α
point-wise dimension, , os
illates. Afterawhilethegranularityofthesysµtemde-
stroys the approximate symmetry of the initial -spa
e
C (l)= dαlαqρ(α)exp[ f(α)] distribution. Breaking the symmetry leads to the short
q
Z − (21) time dissipative mixing that results in the separation of
thesysteminto
lusters. Althoughtheembeddingdimen-
f(α)
where is the fra
tal dimension of its support sionisdi(cid:27)erent,thebehaviorofthedistributionofpoints
q
[41℄[42℄[9℄. Then if, for a range of , a single region is in
on(cid:28)gurationspa
eissimilar. Theinitialdimensionis
8
5000
As time progresses, however, for the initial
onditions
q >0
dis
ussedabove,for typi
allytwoindependents
al-
4000
ing regimes developed. Of
ourse this is in addition to
l
the trivial s
alingregionsobtained forvery small ,
or-
l
3000 responding to isolated points, and to large on the or-
der of the system size, for whi
h the matter distribution
v 2000 lookssmooth. Theobservedsizeofea
hs
alingrangede-
pendedonboth theelapsedtimeintothesimulationand
q
1000 the value of . While the length of ea
h s
aling regime
q
varied with both and time, in some instan
es it was
l
0 possibleto(cid:28)nd goods
alingoverup1 tlonfCourde
adelsniln !
In Fig. 4 we provide plots of q−1 q versus in
µ q
−1000 -spa
e for four di(cid:27)erent values of
overing most of
56000 57000 58000 59000 60000 61000 62000 63000 64000 5 < q < 10
(a) x the range we investigated (− ). To guarantee
that the fra
tal stru
ture was fully developed, we
hose
T = 14
for the time of observation and the initial
on-
q = 5 q = 0
2500 ditions are those given above. For − and
we
learly observe a single, large, s
aling range where
0.5 < lnl < 8
,
orresponding to about three de
ades in
2000 l q
. In the remaining panels (
, d), where we in
rease
5 10
to , and then , we see a dramati

hange. The large
v
s
alingrangehassplitintotwosmallerregionsseparated
1500 l =1.5
at about , and the slope of the region with larger
l
has de
reased
ompared with the s
aling region on its
1.5 < lnl < 1.5 l
left (− ). The s
aling range with larger ,
1000 1.5 < lnl < 8.5,

orresponds to just over three de
ades
l
in . Note that in panels
and d of Fig. 4, the s
aling
l
range with larger is more robust.
58450 58500 58550 58600 58650 58700 58750 58800 58850
(b) x Now that we have identi(cid:28)ed the important s
ales, we
D
q
are able to
ompute the generalized dimension, , and
τ D q
q q
the global index, . In Fig. 5(a) we plot vs
al-
Figure2: Conse
utiveexpansions(zooms)onsu

essivesmall µ T = 14
µ

ulated in the -spa
e for the quinti
model at
squaressele
tedinthe spa
epanelsatT=14forthequinti
q > 0
model. They have theappearan
e of a randomfra
tal whi
h for for the
ase of the larger s
aling range. As
suggests self-similarity. expe
ted, it is a de
reasing fun
tion and µ
learly demon-
strates multifra
tal behavior. Although spa
e is two
D0 D0 > D1
dimensional, is about 0.9. While is typ-
i
al for multifra
tals (information dimension
overs the
D
q
importantregion), the fa
tthat is stri
tlyde
reasing
suggests greater"fra
tality"from in
reasingly overdense
regions, i.e. the system is strongly inhomogeneous. In
τ q
q
Fig. 5b we plot versus in mu spa
e. Two linear re-
0 < q < 1 1 < q < 10
gions ( and )
an be distinguished
suggestingbifra
talgeometry,i.e. asuperpositionoftwo
α f(α)
fra
tals with unique values of and .
We have seen that the one dimensional gravitational
systemrevealsfra
talgeometryinthehigherdimensional
µ
spa
e (frequently referred to as phase spa
e in the as-
tronomi
al literature). Histori
ally self similar behavior
was (cid:28)rst inferred in gravitational systems by studying
the distribution of galaxy lo
ations, i.e in the system's

on(cid:28)guration spa
e, and sear
hing for powerlaw behav-
Figure 3: Figure 3. The
orrelation fun
tion at T=14 whi
h iorinthe
orrelationfun
tion(seeabove).[40℄In
ommon
shows as
alingregion fromabout0.3toabout30,arangeof withthisapproa
h,weproje
tthemuspa
edistribution
γ =.42
about 2 de
ades with a s
aling exponent . on the (position) line to study the geometry of the
on-
(cid:28)guration spa
e. In Fig. 6 a,b,
,d, we provide plots of
1/(q 1)ln(I ) ln(l) x
q
− vs in (
on(cid:28)guration)spa
eforthe
nearly one until
lustering
ommen
es. At this time the quinti
model prepared as above at T=14. In
ommon
µ µ q = 5 q =0
dimensions in -spa
e and
on(cid:28)guration spa
e separate. withthe -spa
edistribution,for − and ,there
9
−9 1
−9.5
0.95
q,l)) −10
C( 0.9
n(
−1)) l −10.5 q,l) 0.85
1/(q −11 D(
(
−11.5 0.8
−12 0.75
−4 −2 0 2 4 6 8 10
(a) ln(l)
0.7
−2 0 2 4 6 8 10
(a) q
−4
7
q,l))
C( −6 6
n(
−1)) l −8 5
q
1/( 4
(
−10 q,l) 3
(
t
−12 2
−4 −2 0 2 4 6 8 10
(b) ln(l) 1
−2
0
−4 −1
C(q,l)) −6 (b) 0 2 4 q 6 8 10
n(
1/(q−1)) l −8 sF
iagluinreg5in:dGexenτeqrvasl.izqed(pdaimneelnbs)ioinnµD-qspvas
.eqfo(pratnheelqau)inatnid
mgloobdaell
( q>0
−10 at T=14 with .
−12
−4 −2 0 2 4 6 8 10 q = 10
(d) the existen
e of two s
aling ranges be
omes
(c) ln(l) 4 < ln(l) < 0 0 < ln(l) < 8 q = 5
apparent, − and for
−2 3<ln(l)<1 1<ln(l)<8
(about3.3de
ades),− and for
q = 10
(about 3 de
ades). Note that the s
aling ranges
−4 µ
are similar for both the and
on(cid:28)guration spa
e, but
q,l)) they are not identi
al.
C( −6
−1)) ln( −8 dimInenFsiigounreD7qwanedilluglsotbraatleitnhdeeTxbe=τhqa1vf4oiorrtohfeth
eogne(cid:28)ngeurraaltizioend
1/(q spa
eof the quinti
modedl a=t 1 D0 . Althoug0h.7now the
( embedding dimension is , is about so the
−10
distributionisde(cid:28)nitely fra
tal. As anti
ipated from the
µ D q
q
-spa
e distribution, is a de
reasing fun
tion of so
−12
−4 −2 0 2 4 6 8 10 it is also inhomogeneousand multifra
tal. In Fig. 7b we
τ q
(d) ln(l) plot q vs in
on(cid:28)guration spa
e, for the same system.
µ
In
ontrast with the -spa
e index, here we observe a
Figure 4: S
aling behaviorin µ-spa
e. Plots of q−11lnCq ver- nearly linear fun
tion withqslight
onvexity (de
reasing
lnl
derivative with respe
t to ). Perhaps this results from
sus are providedforfour valuesof q: a) q=-5,b) q=0.
) α
q=5, d) q=10. the sfu(pαe)rposition of0t<hrqee<li1n.e5ar1r.5eg<ionqs<w7ith7d<isqtin<
t10
and , say with ; ; ,
butthis
annotbeinferredfromtheplotwithoutfurther
analysis.
3<ln(l)<3 q = 5
isasingle,larges
alingregion(− for − For
ompleteness, in Fig. 8 we examine the behavior
2<ln(l)<6 q =0 q =5 D τ q < 0 q
q q
and− for ). Similarly,at (
)and of and for . In general, negative is im-
10
−9 0.8
−9.5 0.75
q,l)) −10 0.7
C(
n(
−1)) l −10.5 q,l) 0.65
1/(q −11 D( 0.6
(
−11.5
0.55
−12
−4 −2 0 2 4 6 8 10 0.5
(a) ln(l)
0.45
−2 0 2 4 6 8 10
(a) q
−4
6
q,l))
n(C( −6 5
−1)) l −8 4
q
1/(
(
−10 q,l) 3
(
t 2
−12
−4 −2 0 2 4 6 8 10
1
(b) ln(l)
−2
0
−4 −1
C(q,l)) −6 (b) 0 2 4 q 6 8 10
n(
1/(q−1)) l −8 sF
iagluinreg7i:ndGeexneτrqavlsi.zeqdd(pimaneenlsibo)niDnq
ovns.(cid:28)gqu(rpaatnioenla()xa)nspdag
leobfoarl
( q>0.
−10 thequinti
model at T=14 with .
−12
−4 −2 0 2 4 6 8 10
an
e - it is in
reasing. The sour
e of the problem
an be
(c) ln(l) τq
determined by examiningthe behaviorof . It is nearly
−2 10 < q < 0

onstant over the range − with a value of
τ 1
q
approximately ≃ − , implying that the underdense
−4
regions are dominated by a single lo
al dimension.
q,l)) Part of our goal is to
ompare how fra
tal geometry
C( −6
n( arises in a family of related models. So far we have pre-
−1)) l −8 sented results for the quinti
, or Q, model. However we
1/(q have also
arried out similar studies of the Rouet-Feix
( (RF) model, the model without fri
tion (whi
h
an be
−10
obtained from either of the former by nullifying the (cid:28)rst
derivative
ontributioninEq(14), andsimplyanisolated
−12
−4 −2 0 2 4 6 8 10 system without fri
tion or ba
kground. The latter is
(d) ln(l) a purely Newtonian model without a
osmologi
al
on-
ne
tion. In Table I we list the important
hara
teris-
1/(q−1)ln(Cq) ln(l) x ti
dimensions and exponents for the the quinti
model
Figure6: Plotsof vs in
on(cid:28)guration(
) spa
e for thequinti
modeql prepqa=red−a5s aboqve=a0t T=q14=ar5e and the model without fri
tion for
omparison. While
provided for four values of : a) , b) ,
) , there are similarities in the fra
tal stru
ture of ea
h of
q=10
d) . these models, they are not the same. For example in
D
q
the quinti
model the generalized fra
tal dimension, ,
is
onsistently smaller in ea
h manifold than the
orre-
portantforrevealingthegeometryoflowdensityregions sponding dimension in the model without fri
tion. It is
D
q
(voids). We see that has a very unphysi
al appear- noteworththatthebox
ountingdimensioninthe
on(cid:28)g-