A potential including Heaviside function in 1+1 dimensional
Landau hydrodynamicsTakuya Mizoguchi*
*Toba National College of Maritime Technology, Japan
ContentsIntroductionLandau model and three solutionsData analyses of hadrons (RHIC , K)Explanation of net-protonSummary
H. Miyazawa**, M. Biyajima**, M. Ide** **Shinshu University, Japan
Introduction• 1+1 dimensional hydrodynamics proposed by
Landau– Landau’s solution (1953)– A boost non-invariant solution by Srivastava et al.
(1993)– Our solution including Heviside function (2008, 9)
• Three solutions cannot explain the net-proton at RHIC– New approach (2009)– Preliminary results on net-proton
Perfect fluid
(1+1) dimension
Landau's solutionL. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953)
Solution by Srivastava et al.D. K. Srivastava et al., Annals Phys. 228, 104 (1993)
Bessel function
ycs
y
c
s
ycs
Our analytical solution with the Heaviside functionMizoguchi, Miyazawa, Biyajima, Eur. Phys. J. A 40, 99 (2009)
(Cf. D. G. Duffy, “Green‘s functions with applications”, Ivar Stakgold, "Boundary Value Problems of Mathematical Physics“)
= 3 520
Competition!!
• T. Mizoguchi and M. Biyajima, Genshikaku Kenkyu (in Japanese), Vol. 52 Suppl. 3 (Feb. 2008) 61.
• Our talk in Annual Meeting for Physical Society of Japan (Mar. 2008, Kinki Univ. (Osaka))
The same expression as our solutionBeuf, Peschanski, Saridakis, Phys.Rev.C78 (2008) 064909
Comparison of three analytical solutions
Potencial (y, ) and Contour map
Trajectory of
Srivastava
Bjorken(x/t=const.)
t = x
= 1
Ours( = 2.5)
Bjorken
t = x
= 1
Bjorken: boost invariant solution.(Cf. J.D. Bjorken, Phys. Rev. D27 (1983) 140.)
Cooling law at y = 0
0: proper time at (y, ) = (0, 0).Bjorken: T3/T0
3 0= const. (cs2 = 1/3)
Rapidity distribution of hadrons
Data analyses• Parameter fitting by lea
st-squares (CERN MINUIT is used)
• Values to input : Tf, B
• Free parameters: f, cs
2(<=1/3), c,
Temperature and baryon chemical Potential (cf. Andronic et al., Nucl. Phys. A772 (2006) 167)
Analyses of charged and K data Mizoguchi, Miyazawa, Biyajima, Eur. Phys. J. A 40, 99 (2009)
RHIC K- (200 GeV) RHIC K+ (200 GeV)
LandauSrivastavaOurs
RHIC + (200 GeV)RHIC - (200 GeV)
Au+Au
Comparison of parameter f
f( Srivastava) < f ( Ours) < f ( Landau)
We cannot determine which temperature is right.
Attention!We cannot determine the value uniquely except for Landau’s solution.
• Solution of Slivastava et al. depends on y0
• Our solution doesn’t depends on y0
Contribution of the derivative term of H(Q)
RHIC - (200 GeV) RHIC + (200 GeV)
=1.7 =3.4
If is large, contributions of the derivative terms are small.
H-termH’-termH’’-term
Preliminary works
Analyses of net-proton data
• Our solution cannot explain the characteristic peak of RHIC net-proton data.
• Thus we consider another approach.
Parameter fitting by means of our solution
RHIC(62 GeV) RHIC(200 GeV)
Derivative terms of H
Remember the famous book!!Morse and Feshbach, ``Methods of Theoretical Physics'', C
hapter 7 (1953)
See also, Masoliver, Weiss, ``Finite-velocity diffusion '', Eur. J. Phys. 17 (1996)190
Analyses of net-proton data at AGS and SPS
GaussianI0-term
I1/p-term
AGS(5 GeV) SPS(17 GeV)
Analyses of net-proton data at RHIC
GaussianI0-term
I1/p-term
RHIC(62 GeV)RHIC(200 GeV)
Parameters and initial Temperature
upper limit Existence of missing proton !!
Estimation of initial Temperature
G. Wolschin, Phys. Lett. B 569 (2003) 67.
For fitting our solution to with this form, it is necessary to impose another conditions.
Summary• We consider the (1+1)-dimensional hydrodyna
mics, and derive the solution including the Heaviside function.
• Our solution explains the data and K distributions fairly well.
Preliminary work• Since our solution doesn't explain the character
istic peak at large y of net-proton distribution, we have considered a new analytic solution.
• The new solution explains the data of net-proton fairly well, except for data at 200 GeV.