26 may – 1 june 2008, teipei-hualian li-zhi fang university of arizona
TRANSCRIPT
collaborators
supported by
US NSF Ast-0506734US NSF Ast-0507340
University of Arizona, Physics Li-Zhi Fang, Ji-Ren Liu Ping He Yi Lu
Brown University, Applied Mathematics Chi-Wang Shu Jing-Mei Qiu
Purple Mountain Observatory, Long-Long Feng, John Hopkins University Wei Zheng
High Order Accurate Weighted Essentially Non-Oscillatory (WENO) Algorithms with Applications to Cosmological Hydrodynamic Simulations
turbulence and large scale structure of the universe
structure formation of the universe, 2007
(1757)
(1923)
(1941)
minihalo model of quasar’s Ly-alpha forests
• self-gravitating objects:
Bachall & Salpeter 1965 Black 1981
• pressure-confined clouds
Sargent et al. 1980 Ostriker 1988
• low mass object Bond, Szalay & Silk 1988
• miniholes Rees 1986 Murakami & Ikeuchi 1993
A
B
Yi-Hu Fang et al 1996
Lyman-alpha absorption clouds are un-clustered, not virialized (1996)
low mass object,pressure-confined clouds,self-gravitating objects:minihalo model are incorrect
quasar HS 1700+64Ly-alpha transmittance flux
clusters
Jackhedkar, Zhan, Fang 2000
±Fr (x) = F (x +r) ¡ F (x)
Jackhedkar, Zhan, Fang 2000
PDF of
sigmaLyman-alpha transmitted flux fluctuations are highly non-Gaussian (2000)
±F = F (x + r) ¡ F (x)
intermittency is the alternation of phases of apparently periodic and chaotic dynamics. Consider a dynamical system. Let x be the observed variable. If x plotted as a function of time exhibits segments of relative constant values (laminar phase) interspersed by erratic bursts, the system dynamics is intermittent.
intermittent distribution random variables
random variable
»j can be 0 and 2 with probabilities 1=2
P =
8<
:
(2N ¡ 1)=2N »= 01=2N »= 2N
0 others
»= ¦ Nj =1»j = »1»2 ¢¢¢»N
h»i =0+0+¢¢¢+0+2N
2N =1
h»2i ¡ h»i2 =0+0+¢¢¢+22N
2N¡ 1=2N ¡ 1
ph»2i ¡ h»i2 À h»i
Probability Distribution Function (PDF)
P =
8<
:
(2N ¡ 1)=2N »= 01=2N »= 2N
0 othersP (») =1
¾p
2¼e¡ »2=2¾2
h»i =0+0+¢¢¢+0+2N
2N = 1
h»2i ¡ h»i2 =0+0+¢¢¢+22N
2N¡ 1= 2N ¡ 1
¾=p
h»2i ¡ h»i2 ' 2N =2
0
0.015
»= 8¾
long tail
N=6
Ut + f (U)X + g(U)Y +h(U)Z = F (t;U)
U = (½;½u;½v;½w;E )
f (U) = (½u;½u2 +P;½uv;½uw;u(E +P ))
g(U) = (½v;½uv;½v2 +P;½vw;v(E +P ))
h(U) = (½w;½uw;½vw;½w2 +P;w(E +P ))
E =P
° ¡ 1+
12½(u2 + v2 + w2)
F (t;U) =
µ0;¡
_aa
½V +½G;¡ 2_aa
E +½V ¢G ¡ ¤net
¶
G = ¡ r R © r 2©(x;t) = 4¼G[½tot(x;t) ¡ ½0(t)]=a
drDM
dt =1avDM ;
dvDM
dt =¡_aavDM +G
baryon fluid(Navier-Stokes)
gravity (Einstein)
dark matter (Newton)
a comsic factor
heating and cooling
peculiar velocity is irrotational, or potential.
structure formation: growth mode
the dynamical equation of baryon gas is
stochastic force driven Burgers' equation
or KPZ equation
v = ¡1a
r Á
Berera, Fang, PRL (1994), Jeans, ApJ (1999), Matarrese, Mohayaee, ApJ (2002)Feng, Pando, Fang, ApJ, (2003)
gravitational potential
@Á@t
¡1
2a2(r Á)2 +
ºa2
r 2Á = '
Jeans diffusion
Burgers’ turbulence
• Reynolds number
3/40 )/( arR c
Correlation length of random gravitational field Jeans smoothing length
Polyakov, PRE, (1995)Boldyrev, Linde, Polyakov, PRL, (2004)
scale free regime, hierarchical clustersfully developed turbulence
statistically quasi-equilibrium state
)()( xvrxvvr
prvS pr
pr
||
Kolmogorov (1941)
structure function
Gaussian field
S2pr = (2p¡ 1)!!(S2
r )p
S2pr =(S2
r )p = (2p¡ 1)!! r ¡ independent
Spr / r»(p)
»(p) = ®p
S2pr =(S2
r )p; r ¡ independent
self-similar field
Spr / r»(p)
S2pr =(S2
r )p / r»(2p)¡ p»(2); r ¡ dependent
»(p) is a nonlinear function of p:
intermittent field
models of hierarchical clusters
• beta model• linked-pair hierarchy• hierarchical Gaussian fluctuation (Press-Schechter
theory)• lognormal model• halo model
Soneira, Peebles 1977
beta modelradius R; R=̧ ; R=̧ 2;¢¢¢
N = 3
N(R=̧ )3 < R3
N¸¡ 3 = ¯ < 1
number of objects
test of beta model
p-dependence is linear. Not an intermittent field.
¹Spr / r»
»= ¡ (p¡ 1)(d¡ · ) < 0
N(R=̧ )3 < R3
N¸¡ 3 = ¯ < 1
N ?
linked-pair hierarchical clustering
Q_n are constant S. White 1979
Feng, Pando, Fang, 2001
d dimension
j / k
Q2n =1
2dj (n¡ 1)
¹S2nj
P n¡ 1j
hierarchical Gaussian fluctuation (Press-Schechter model)
Bond, Cole, Efsthathiou, Kaiser 1991
k-space
hierarchical models based on randomly additional process generally do notproduce intermittent field.
randomly multiplicational process
(central limitation theorem)
»= ¦ Nj =1»j = »1»2 ¢¢¢»j
ln»= ln»1 + ln»2 + ¢¢¢+ ln»N
»= »1 + »2 +¢¢+»N
hierarchical clusters
testing of hierarchical Gaussian fluctuation
Pando, Lipa, Greiner, Fang 1998
randomly multiplicational process
randomly additional process
halo modelmass fields are given by a superposition of the halos on various scales, and therefore, all non-Gaussian behaviors of the density field can be described by a universal density profile
The halo-halo correlation function
on scales larger than the halo size
is given by the two point correlation
function of the initially linear
Gaussian field.
log-Poisson hierarchical model
)()( xrxr
)1/()]/[ln(
!/)exp()(
)/(
)()(
21
21
12
21
2121
21
1212
rr
mmP
rrW
rWr
rr
mrrrr
mrr
rrrr
121
rrW
Liu, Fang, 2008, Lu, Fang, 2008
)]1/()1([
||
p
p
pr
pr
pp
rS p
Poisson random
m=1
m=2
m=N
r_1r_2
)1/()]/[ln(
!/)exp()(
)/(
)()(
21
21
12
21
2121
21
1212
rr
mmP
rrW
rWr
rr
mrrrr
mrr
rrrr
hierarchical clusteringrandomly multiplicative processscale invariance, self-similaritytranslational invarianceinfinite divisibility (the difference |r_1-r_2| can be finite or infinitesimal)
drr 21 lnln
tindependen- is )/(
spectrumpower ,||
)]1/()1([
1
22
222
rSS
rS
ppp
nr
nr
rr
pp
Gaussian field
S2nr = (2n ¡ 1)!!(S2
r )n
S2nr =(S2
r )n = (2n ¡ 1)!! r ¡ independent
¯ = 1 Gaussian ¯eld
¸ r1r2 ! 1 ; when ¯ ! 1
)1/()]/[ln(
!/)exp()(
)/(
)()(
21
21
12
21
2121
21
1212
rr
mmP
rrW
rWr
rr
mrrrr
mrr
rrrr
structures
rrFp
SSrF
rSp
r
pr
prp
ppr
r
)( ,
/)(
structures uppick
structuressinglar most by dominant is ,high for
structures
1
Fp(r) / r¡ ° (1¡ ¯ p )
log-Poisson hierarchical
. therefore,
tests of log-Poisson hierarchy
Intermittent exponent
beta-hierarchy
high order moment
scale-scale correlations
statistical properties
samplesmass density field of IGM, HI (simulation)
velocity field (simulation)
Lyman-alpha transmitted flux (simulation, observation)
scaling relations (simulation, observation)
He, Liu, Feng, Shu Fang, 2006Zhang, Liu, Feng, Fang, 2006Liu, Fang 2008Lu, Fang, 2008
beta- hierarchical
/123
/11
1
)](/)([)(/)(
........
)](/)([)(/)(
)(/)()(
rFrFrFrF
rFrFrFrF
rSrSrF
pp
ppp
p-invariance
)](/)(ln[)/1()(/)(ln
)](/)([)(/)(
231
/1231
rFrFrFrF
rFrFrFrF
pp
pp