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.