高エネルギー重イオン衝突のための 相対論的tsallis...
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高エネルギー重イオン衝突のための
相対論的Tsallis流体模型の研究
武蔵工大 知識工学部 長田剛
Andrzej Sołtan Inst. G. Wilk
基礎物理学研究所 研究会 熱場の量子論とその応用京都大学 2007.09.06
基研研究会熱場の量子論とその応用9月6日
Scales in hydrodynamical model
macroscopic scalethe system’s size
L
microscopic scale(e.g. mean free path)
lideal hydrodynamics
- strongly coupled system -l L
‘cell’ of the hydrodynamics
基研研究会熱場の量子論とその応用9月6日
Standard hydrodynamics based on extensivityThe basic assumption of standard statistical mechanics is that the system under consideration can be subdivided into a set of non-overlapping subsystems.
B ln ,
( ) ( ) ( )
i ii
S k p p
S A B S A S B
= − ∑
⊕ = +
As a consequence the Boltzmann-Gibbs entropy is extensive in the sense that the total entropy of two independent subsystems is the sum of their entropies.
extensivity
基研研究会熱場の量子論とその応用9月6日
Assumptions in Boltzmann-Gibbs factor
some assumptions leading tothe Boltzmann-Gibbs (BG) factor may be too tight?
* absence of memory effects,* negligible local correlation* absence of long-range interaction
+ Boltzmann H-theorem (based on the extensive entropy)
extensivity:entropy, measure of information about the particle distribution in the states available to the system, is extensive in the sense that the total entropy of two independent subsystems is the sum of their entropies.
B ln ,
( ) ( ) ( )
i ii
S k p p
S A B S A S B
= − ∑
⊕ = +
基研研究会熱場の量子論とその応用9月6日
Correlations and non- extensivityIf memory effects and long-range forces are present, this property is no longer valid and the entropy, which is a measure of the information about the particle distribution in the states available to the system, is not an extensive quantity.
B ln ,
( ) ( ) ( )
i ii
S k p p
S A B S A S B
= − ∑
⊕ ≠ +non- extensivity
★ memory effect (temporal correlation) ★ long range forces (spatial correlation)
are present
基研研究会熱場の量子論とその応用9月6日
.
Non-extensive entropy
・In 1988 Tsallis proposed a generalization of the entropy of the BG entropy C. Tsallis, J.Stat.Phys.52(1988) 749.
1
B
1ln
1
l ,nq
q
q i iq
iq
ff
q
S k p p− −
≡−
= − ∑
( ) ( ) (1 ) ( ) )( ) (q qq q qS A B S A S B q S A S B+ −⊕ = +
Tsallis’s non-extensive entropy
基研研究会熱場の量子論とその応用9月6日
Introduction: Why Tsallis- hydro?
• the equilibrium in ‘standard’ thermodynamics relies on exponential particle spectra, while experiments definitely show a power-law tailin high energy transverse momenta.
⇒ hydro +(other dynamical origins…) ≟ Tsallis- hydro + (other dynamical origins…)
↑including (momentum) correlation
• ideal Tsallis (q- )hydro = (usual q=1) dissipative hydro ?
q: a parameterof non-extensivityintroduced by C. Tsallis in 1988.
q=1: usual the Boltzmann-Gibbs thermodynamics,statistics.
基研研究会熱場の量子論とその応用9月6日
Relativistic non-extensive kinetic theorynon-extensive version of Boltzmann equation:
A. Lavagno, Phys.Lett.A301(2002) 13.
33 31 11
0 0 01 11 1
1
1
( , ) ( , ),
( , | ,[ , ]
[ , ]
)1( , )( , | , )2
q q q
q q q
qq q
q
p f x p C x p
W p p p pd pd p h fd pC x pW p p p pp p h fp
ff
µµ
′
∂ =
′ ′⎧ ⎫′′ ⎪ ⎪= ⎨ ⎬′ ′′ ′ ⎪⎩
′
− ⎪⎭∫
1 1binary collisions: p p p p′ ′+ → +
1 1
1 1 1 Boltzmann Stosszahlansatz - (for =1)
[ , ] [ ( , ), ( , )]
[ , ] ( , ) ( , )q q
q q
h f f h f x p f x ph f f f x p f x p→ −
′ ′=
′ ′= ⋅
★ correlation function same space-time x but different p
1 1 1 1
transition rate between two particle state, assuming the detailed-balance: ( , | , ) ( , | , )W p p p p W p p p p•
′ ′ ′ ′=
基研研究会熱場の量子論とその応用9月6日
Non-extensive H theorem
q-generalized entropy current
[ ]1 1
1/(1 )
[ , ] exp [ln ln ]
exp ( ) 1 (1 )
q q
q
q
q
q qh
X q X
f f f f−= + −
= +
J. A. S. Lima, R. Silva and A.R. Plastino,Phys.Rev.Lett86(2001),2983
{ }3
B 3 0( ) ( , )ln ( , ) ( , )(2 )
qq q q q q
d p ps x k f x p f x p f x pp
µµ
π≡ − −∫
{ }3
B 3 0
by A .Lavagno,Phys.Lett.A301 (2002 )13 .
( ) ( , ) ln ( , )(
))
,2
(qq q q
qq qfd p ps x k f x p f
ppx p x
µµ
π≡ − −∫
3B
3 0
3B
3 0
( ) [ln ](2 )
[ln ] ( , ) 0 for all space-time point(2 )
qq q q q
q q q
k d ps x f p fp
k d p f C x pp
µ µµ µπ
π
∂ = − ∂
= − = ≥
∫
∫
revised by Osada and Wilktherm. dyn. rel. OK
q-generalized Boltzmann Stosszahlansatz
基研研究会熱場の量子論とその応用9月6日
q- equilibrium
3
3 0
setting ( ) 0, ( ) const.
1( ) ( , )(2 )
qq q
a x b x
d pT x p p f x pp
µ
µν µ ν
π= =
= ∫
q- energy-momentum tensor:
3
0 [ ] ( , ) ( , ) 0
collision invaria
if
nt:
( , ) ( ) ( )
qd pF x p C x pp
x p a x b x p µµ
ψ ψ
ψ
= ≡
= +
∫
B3
B
( )( ) [ln ] 0 setting ( ) 0, ( )
(2 ) ( )q q q
u xks x F f a x b xk T x
µµµ µπ
−∂ = = = = −
q- equilibrium distribution function:1
1
B
( )( , ) 1 (1 )
( )
q
q
p u xf x p q
k T x
µµ
−⎡ ⎤= − −⎢ ⎥
⎢ ⎥⎣ ⎦
基研研究会熱場の量子論とその応用9月6日
Thermodynamic relations
42 2 2 /(1 )
2
42 2 1/(1 )
2
32 2 /(1 ) 1/(1 )
2
[1 (1 ) ]2
[1 (1 ) ]2
[1 (1 ) ] [1 (1 ) ]2
q qq z
qq z
q q qq z
gT de e z e q e
gTP de e z e q e
gTs de e z e q e q e
επ
π
π
−
−
− −
= − − −
= − − −
⎡ ⎤= − − − + − −⎣ ⎦
∫
∫
∫
(Gibbs-Duhem)
q q q
Ts PdP
sdT
ε= +
=
/z m T=
hold also for the q- generalized quantities.
基研研究会熱場の量子論とその応用9月6日
q- hydrodynamical model
1+1 ( - ) relativistic -hydrodynamics:qτ η
1( ) 0
1( ) 0
q q q q qq q q
q q q q qq q q
vP v
v P PP v
ε ε α αε
τ τ η τ τ η
α αε
τ τ η τ τ η
∂ ∂ ∂ ∂⎧ ⎫+ + + + =⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭
∂ ∂ ∂ ∂⎧ ⎫+ + + + =⎨ ⎬∂ ∂ ∂ ∂⎩ ⎭
1( ) cosh( ), sinh( ) , ( ) tanh( )q q q q qu x v xµ α η α η α ητ
⎡ ⎤≡ − − ≡ −⎢ ⎥⎣ ⎦
2 22
1 1 metric =(1,- ), , ln2
t zg t zt z
µν τ ητ
+≡ − ≡
−
; ;( ) 0q q q q q qT P u u P gµν µ ν µν
µ µε⎡ ⎤= + − =⎣ ⎦
基研研究会熱場の量子論とその応用9月6日
Standard hydrodynamics vs. q- hydrodynamics
Tsallis hydrodynamics,locally conserve q- entropy current:
{ }3
B 3 0( ) ( , )ln ( , ) ( , )(2 )
qq q q q q
d p ps x k f x p f x p f x pp
µµ
π≡ − −∫
( ) ( ) (1 ) ( ) )( ) (q qq q qS A B S A S B q S A S B+ −⊕ = +
including correlations
{ }3
B 3 0( ) ( , )ln ( , ) ( , )(2 )d p ps x k f x p f x p f x p
p
µµ
π≡ − −∫
( ) ( ) ( )S A B S A S B⊕ = +
standard hydrodynamics, locally conserve entropy current:
without correlations between cells
much smaller than system size- strongly coupled system -
much smaller than system size- strongly coupled system -
基研研究会熱場の量子論とその応用9月6日
1E-4 1E-3 0.01 0.1 1 10
0.20
0.25
0.30
0.35P q /
ε q
εq [GeV/fm3]
q =1.0 q =1.1 q =1.2
q- EoS: non-extensive relativistic π gas
- 0/ , 0.14GeV, 3( for , , )z m T m g π π π+= = =
42 2 2 /(1 )
2
42 2 1/(1 )
2
[1 (1 ) ]2
[1 (1 ) ]2
q qq z
qq z
gT de e z e q e
gTP de e z e q e
επ
π
−
−
= − − −
= − − −
∫
∫
基研研究会熱場の量子論とその応用9月6日
q- Gaussian Initial condition
2(in)
0 2
0
( 1fm, ) exp ,2
( 1fm, )
q q
q
ηε τ η εσ
α τ η η
⎡ ⎤= = −⎢ ⎥
⎣ ⎦= =
BRAHMS Collab.Phys.Rev.Lett.93(2004),102301
2tot T 0 0 0 T
tot part part
( , ), 1.0fm, 6.5fm
, 357, 73 6GeV qE A d A
E N E N E
π τ η ε τ η τ= = =
= ∆ = ∆ = ±∫
RHIC Au + Au 200GeV/nucleon
基研研究会熱場の量子論とその応用9月6日
q- initial condition (for fixed σ )
0 1 2 3 40
5
10
15
20
25
30
q =1.00, ε(in)q =28.7 GeV/fm3
q =1.05, ε(in)q =25.9 GeV/fm3
q =1.10, ε(in)q =22.4 GeV/fm3
q =1.15, ε(in)q =16.8 GeV/fm3
Initial condition: σ = 1.25 fixed
η
ε q( τ 0, η
)
基研研究会熱場の量子論とその応用9月6日
Single particle spectra by q- hydro model
0 1 2 3 4 5 610-5
10-4
10-3
10-2
10-1
100
101
102
103
1/2π
1/p T
dN
/dp T
dy
pT [GeV/c]
q=1.08 q=1.09 q=1.10STAR Collab.
Au+Au 200GeV/nucleon
σq=1.30, ε(in)=20.5 [GeV/fm3]
TF=100 MeV
基研研究会熱場の量子論とその応用9月6日
Single particle spectra by q- hydro model
0 1 2 3 4 5 60
50
100
150
200
250
300
350
BRAHMS Au+Au 200GeV/nucleon
dN/d
y
y0 2 4 6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
q=1.08TF=100 MeV
STAR Au+Au 200GeV/nucleon
σq=1.26 ε(in)=22.6 [GeV/fm3]
σq=1.28 ε(in)=21.5 [GeV/fm3]
σq=1.30 ε(in)=20.5 [GeV/fm3]
σq=1.32 ε(in)=19.5 [GeV/fm3]
σq=1.34 ε(in)=18.5 [GeV/fm3]
σq=1.36 ε(in)=17.6 [GeV/fm3]
1/2π
1/p T
dN
/dp T
dy
pT [GeV/c]
基研研究会熱場の量子論とその応用9月6日
Perfect q-hydrodynamics = (q=1) dissipative hydrodynamics?
;
;
[ ]
[ ( ) ] 0q q q q q qT u u P
u u P W u W u
µν µ ν µνµ
µ ν µν µ ν ν µ µνµ
ε
ε π
= + ∆
= + + Π ∆ + + + =
rewusi ritng ,P and , ,
in a dissipative like form
e
:qu Tformallyµ µνε
q q
q q
q q
P P P P
u u u uµ µ µ µ
ε ε ε ε→ ≡ + ∆
→ ≡ + ∆
→ ≡ + ∆
non-extensivity q affects thermodynamical quantities and flow velocity
,
1 ( ) ,2( )
q q
q q q q
q q q q
P P P
W u P u
P u u
µ µ µ
µν µ ν
ε ε
π ε
Π ≡ ∆ = −
≡ ∆ + + ∆
≡ + ∆ ∆
基研研究会熱場の量子論とその応用9月6日
q≠1 near equilibrium picture
1 stationary state near ( 1)equilibriumq
q≠
=
eq eq 1,q qT T T T T
T W u W u
µν µν µν µν µν
µν µν µ ν ν µ µν
δ
δ π== + ≡
= Π∆ + + +
;;entropy production: usu T
Tµ µνν
µµδ⎡ ⎤ = −⎣ ⎦
Landau matching condition [U. Heinz, et.al., Phys.Rev.C73,034904(2006) ]
0u T uµνµ νδ = guarantee to use the same of ( ) as of ( ).
qT T T Tε ε
solution exsists only for 1 caseu qµ <
1 'ture' equilibriumq ≠
q- hydrodynamical model:
基研研究会熱場の量子論とその応用9月6日
Entropy production in the q≠1 near equilibrium picture
0 2 4 6 810-6
10-5
10-4
10-3
10-2
10-1
100
[suµ ]
;µ
σ=1.25 fixed
q=0.95
τ=5.0 fm τ=25.0 fm τ=75.0 fm
ηηη0 2 4 6 8
0 2 4 6 8
;; usu T
Tµ µνν
µµδ⎡ ⎤ = −⎣ ⎦
基研研究会熱場の量子論とその応用9月6日
Summary
Tsallis- (nonextensive-) hydrodynamical model is formulated by based on the relativistic nonextensive kinetic theory.perfect q- hydrodynamics may be connected with the dissipative ‘standard’( i.e., q=1) hydrodynamics.
What’s next ?2+1 or full dimension q-hydro model with QGP EoS.write down the fluctuation-dissipation theorem
for a nonextensive version and find transport-coefficients using q.
基研研究会熱場の量子論とその応用9月6日
Appendix:non-extensive and assumptions some assumptions leading to
the Boltzmann-Gibbs (BG) factor may be too tight?* absence of memory effects,
interactions can be described as a successions of simple binary collisionsalways possible determine a time interval does not change appreciably and its rate of change at time t depends only on its instantaneous value and not on its previous history.
* negligible local correlationassumptions for the space dependence of the distribution function its rate of change at a spatial point depends only on the neighborhood of that point.i.e., the range of the interactions is short with respect to the characteristic spatial dimension of the system.
* absence of long-range interactionthe momenta of two particles at the same spatial point are not correlated and the corresponding two-body correlation function can be factorized as a product of two single particle distributions.
+ Boltzmann H-theorem (based on the extensive entropy)
基研研究会熱場の量子論とその応用9月6日
q- hydrodynamical evolution of Temperature T
3in 20GeV/fmε =
0 2 4 6 80.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8σ=1.25 fixed
q = 1.00
η
0 2 4 6 80.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 τ =1fm τ =5fm τ =25fm τ =75fm
q = 1.10
T [G
eV]
η
基研研究会熱場の量子論とその応用9月6日
q- flow vs. q- energy density
10-2 10-1 100 1010.0
0.1
0.2
0.3
0.4
0.5σ=1.25 fixed
q = 1.00
αq-
η
10-2 10-1 100 101
0.1
0.2
0.3
0.4
0.5
εq [GeV/fm3] εq [GeV/fm3]
τ = 1.2 fm τ = 1.4 fm τ = 1.6 fm τ = 1.8 fm τ = 2.0 fm
q = 1.10
基研研究会熱場の量子論とその応用9月6日
Single particle spectra: freeze out surface
f
3
3 3 ( , )Cooper-Frye: (2 ) q
d N gE d fpp
pd
xµµσ
π Σ= ∫
0 2 4 60
50
100
150
200
250
300
TF=100 MeV
τ [f
m]
η
σ=1.25 fixed ε(in)=28.7 [GeV/fm3] fixed
0 2 4 60
50
100
150
200
250
300
q = 1.00 q = 1.05 q = 1.10 q = 1.15
η