Description of the nuclear stopping process within anisotropic thermo-hydrodynamics

Volume 240, number 3,4

PHYSICS LETTERS B

26 April 1990

D E S C R I P T I O N OF T H E NUCLEAR S T O P P I N G PROCESS WITHIN ANISOTROPIC THERMO-HYDRODYNAMICS B. KAMPFER a.~, B. LUKJ~CS b, G. WOLF b.2 and H.W. BARZ a a ZentralinstitutJ~r Kernforschung, Pf 19, DDR-8051 Rossendorf(Dresden). GDR h Central Research Institute for Physics. Pf 49, 11-1525 Budapest 114. tlungary Received 11 January 1990

A generalized hydrodynamics is presented, which takes into account the anisotropy of the nucleon momentum distribution. The dynamics of the nuclear stopping process is studied in this framework.

High energy nuclear collisions are always accompanied by nonequilibrium phenomena, particularly in early collision stages. The substantial mean free path leads to some initial partial transparency and anisotropic momentum distribution. Later the system evolves towards local equilibrium as shown by collective flow effects of heavy nuclei. Still, the observed differences in energy spectra in and perpendicular to the beam direction indicate anisotropy even at breakup. If one wants to include this effect of the mean free path and at the same time retain the practical advantages of fluid dynamics and the nuclear equation of state, one must go beyond standard dissipative thermo-hydrodynamics. Such attempts were made in refs. [ 1-3 ], however, only for stationary effects. Here we present a real dynamical calculation to describe nuclear stopping with anisotropic hydrodynamics. The essence of our approach is a genuinely anisotropic energy-momentum tensor for an anisotropic relativistic fluid. We show that then at least one extra vector field (in addition to the four-velocity), or a tensor (in addition to the metric tensor) must appear. The present paper is directly connected to a previous one [ 1 ] which studied the thermodynamics of anisotropic systems. Here the nuclear matter is regarded as a single and Present address: Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, SU-101 000 Moscow, USSR. 2 Present address: lnstitut f'tir Theoretische Physik, JustusLiebig-Universit~it Gieflen, Heinrich-Buff-Ring 16, D-6300 GieBen, FRG.

indivisible interacting unit. This is the decisive contrast to multi-fluid models, where the matter is described by a superposition of subsystems, each in local equilibrium. We include the anisotropy by a suitable parametrisation of the Lorentz-invariant distribution f(p', x*). The usual parametrisation f=f(urpr; cd) where p' is the particle momentum, u ' ( x k) is the four-velocity, and a~(k k) stands for a set of thermodynamic parameters) describes isotropic situations, since in the local rest frame, where u ' = ( 1,0),fdepends only on the square of the threemomentum. So even if the set a is wider than minimal, only local relaxation (temporal nonequilibrium) effects are included [4], but by no means any anisotropy. For the latter at least a second four-vector t' is needed to order to parametrize the distribution function by . f = f ( U r p r, trp ~, 0~') .

(1)

(In ref. [ 5 ] a second rank tensor is used for anisotropy via Q,.,p,pS, however, a vector is the simplest possibility.) Sincc here the phenomenology is considered, specification of the parameters may be postponed. Even then, the structure is fixed. Conserved baryon current n' and energy-momentum T 'k a r e well defined identifiable physical quantities. They are connected to the distribution function f v i a n'= f

'l'd3p

P J---E-:'

T'k= f d3p .1 p'pkf E

(2)

These quantities obey balance laws:

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

297

Volume 240, number 3.4 n':,=0,

T'~:,=0.

PHYSICS LETTERSB (3)

Now one can define the four-velocity by the Eckart gauge as (4)

n'-nu',

where u' is a unit vector and n is the baryon density (measured by the observer moving with u'). Using this field and the (still unspecified) t', T 'k can be decomposed as T~k=eu'uk + ~( U't~ + t'Uk) + q t - 2t'tk + p(g'k + uiuk) + ....

(5)

where the further terms would contain vectors orthogonal to both u ~and t', so they must be built up from gradients. (Two well-known examples from isotropic fluids are the heat conduction and shear viscosity.) Such gradients drive equilibration between different points via transport. We concentrate on local equilibration, so we neglect transport terms here. Eq. (5) is the general decomposition of a symmetric two-tensor with respect to two-vectors, if the remaining part is isotropic (see also ref. [6] ). In usual two-fluid models [7] the two-velocity fields possess six independent components. Here it is also the natural degree of freedom. Namely, t' stands for spatial anisotropy, so it is physical to choose it as a pure space vector in the comoving frame: u,t ~- O. Requiring reflection invariance for the change of definition t ~--,-t i one gets fl=0. We then have in the comoving frame

T,~=

p 0 0

0 p 0 p+q

/i °°

(6)

Structure (6) supports our choice of symbols in ( 5 ), namely e for energy density and p for isotropic thermodynamic pressure. The quantity q is just the difference between dynamic pressures in the beam direction and perpendicularly. Now remember that the vector t ~ is still unspecified, except for its direction, that it parametrizes the anisotropy of the one-particle distribution, and that a larger anisotropy results in a larger q as well. Without looking microscopically, any choice has the same rights, so we take the simplest one, q= nt= n x / ~ t ~ and in addition consider 298

26 April 1990

q as the new extensive density characterizing the anisotropic local thermodynamic state. Refs. [2,8] contain techniques how to select the particular form of the extra extensive parameter t.'fthe momentumdependence of the interactions is fully known, which is definitely not the case in the present situation. The entropy current vector can be decomposed as (7)

s ' = s u i + Tt' + ....

where again the neglected terms contain gradients (and automatically vanish for plane symmetry). Thermodynamic relations for the entropy s(e, n, q) as a potential can be found in ref. [ 1 ], and more details will be given elsewhere. Here we only recall e= T s - p + Itn+ vq , 7._~_ Os 0e'

-I~_ T

Os On '

-v Os T 0q'

(8)

thus defining the usual intensive parameters as well as v, the measure of the relative velocity of subsystems if one tries to find a link to the two-fluidpicture. The second law of thermodynamics requires sr:~>~0.

(9)

Using eqs. (3), (4), (6) and the balance equations the second law in (8) gives an inequality constraint for ~ (where the dot stands for the comoving derivatives u'0,). (The formula is rather long and will not be given here. ) The inequality must hold for any velocity field, anisotropy, and c. This prescribes definite counterterms in the evolution equation of q, and a structure as well. One gets 7=fl(Os/Oe)

(10)

(but in the present approach fl= 0 anyway ), and then also it=)t_qu" + q t - 2 OS(OS~ - ' Oe \OqJ t~t~u~'"'

( 1I )

by which eq. (8) reduces to 0s 0q2>~0. Therefore only the ). term is responsible for the entropy production, and the microscopic picture behind it may be the thermalisation of longitudinal mo-

Volume 240, number 3,4

PHYSICS LETTERS B

menta. Eq. ( 11 ) suggests the use of a relaxation time approach:

)..~ - q / z .

(12)

In ref. [ 5 ] the time scale of isotropisation of an initially ellipsoidal momentum distribution is evaluated by means of nucleon-nucleon cross sections, and in our language the result can be written as r = ( m / ) -t , r / ~ 2 f m 2 .

(13)

Now we have a complete system of evolution equations cast into the form

26 April 1990

we find v to agree with UCMS,indicating that v is half of the relative velocity of the counterstreaming subsystems (as expected from the last equation of (7) together with the energy considerations in ref. [ 1 ] ). Some numerical results are displayed in fig. l for Ebeam= l GeV/A, r/=2 fm 2 and thickness l0 fm. The evolution is followed in a special "comoving" system (~, t) defined by

x=x(t;~),

x(O,~)=~,

OxlO~

n--

Y

=const.,

where y stands for the Lorentz factor. Then ~ is a label

h + nO=O , b+(e+p+q)O=O, a ' + ( e+ p+ q ) - ' [ (p+ q ) ' u ' - (p+ q),rg 'r] = 0 ,

?1+ ( q+ q/v)O+ rlnq=O ,

2

3

(14)

where 0= Ur:r and a ' - - u':ru r. Note the formal similarity between the roles of q and the von NeumannRichtmyer artificial viscosity. Now, as a simplified but already nontrivial dynamical calculation, we consider the evolution of variables for a slab-geometry collision. The nuclear equation of state is as in refs. [1,9] with a quadratic compression ( K = 240 MeV) + thermal Boltzmann, pions are neglected. Our first observation is that at high r/values (large nuclear stopping power) the prediction of the shock front model is recovered; the same holds for unrealistically thick slabs. This is an interesting byproduct of our approximations. In particular, at large beam energies the numerical solutions of eq. (14) with a crudely centered staggered leapfrog scheme coincide better with the solutions of the shock-front model than those relying on the use of the artificial viscosity. I.e. the decaying anisotropy dissipates more efficiently the ambidirectional flow energy. The other limiting case is r/-,0, when there is no dissipation of the ambidirectional flow. Then, due to the assumed mirror symmetry fl=O the spatial component of the four-velocity u vanishes, as expected for two identical independent interpenetrating fluids in CMS. However, even in this case our model clearly differs from two-fluid models, in which there are 2 × 3 = 6 independent variables at each point (2 × (e, n, u) ), while we have four variables (e, n, u, v). Still,

.35I

0

~

5

10

15

t(fm/c) Fig. 1. The time evolution of density n, temperature T, spatial component of flow velocity u and measure of relative velocity v in comoving elements labelled by their initial locations ¢ for the right hand side of a symmetric collision. (Labels: 1:0.5 fm, 2:2.5 fm, 3:4.5 fm, 4:6.5 fm, 5:8.5 fm. ) More details can be found in the text.

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o f a fluid element and t is roughly y × ( C M S t i m e ) . Fig. ! shows that the compression proceeds in two steps: a relative steep density increase is followed by a slow further one. The transition stage coincides with the m a x i m u m o f v. During the first stage the flow velocity substantially decreases but at the transition its value is not yet 0. The later slow decrease goes parallel with the decays o f the anisotropy q (not disp l a y e d ) and relative velocity v. In this second stage a considerable part o f the b e a m energy is not yet dissipated, the present p a r a m e t e r s ( r / a n d the thickness) are not large enough to produce a picture resembling the shock model. One is left with rather inhomogeneous profiles. The density remains definitely below the one given by the shock model, while the central t e m p e r a t u r e is largely above it. In our picture stopping means two things, namely, that both the spatial c o m p o n e n t o f u ~ (in C M S ) and v (or q) vanish. In the central region both conditions are roughly fulfilled after 15 fm/c. After its maxim u m v decays exponentially (as q does, cf. ref. [ 6 ] ). However, the surface elements bounce at t > 12 f m / c, when a rarefaction wave travels inwards. In the rarefaction zone 0 > 0 and (as seen from eq. (14) and fig. I ) this drives a fast non-exponential decrease o f q. In this sense the matter (at least in our a p p r o a c h ) cannot be called completely stopped at any time, because u and v d o not vanish simultaneously, except at the very center, approximately. F o r smaller nuclei the inhomogeneous profiles are even more p r o n o u n c e d [ 9 ] and the stopping is even less complete. The value r/= 2 fm 2 is based on .free n u c l e o n - n u c l e o n cross sections. As shown in ref. [ 10 ], collective i n - m e d i u m effects may considerably d i m i n i s h the mean free path, e.g. enlarge the r/value.

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Such effects, increasing the stopping power, may give results much more resembling shock fronts. Therefore, for comparison with experimental d a t a and for d e t e r m i n i n g the nuclear equation o f state, more reliable d e t e r m i n a t i o n s o f r/will be needed as well as to extend the geometry o f the flow to 3D calculations. In s u m m a r y , here the simplest variant o f a fluid d y n a m i c a l model including anisotropy o f the nucleon m o m e n t u m distribution has been presented. The model visualizes the d y n a m i c s o f dissipation o f the interpenetrating flow and stopping. The usual onefluid picture is a clear limiting case, while some features o f two-fluid pictures are recovered at very high energies, but there the two descriptions are not identical with respect to the degrees o f freedom.

References [ 1] H.W. Barz, B. K~mpfer, B. Luk~ics,K. Martinzisand G. Wolf, Phys. Left. B 194 (1987) 15. [ 2 ] L. Neisse, H. St6cker and W. Greiner, J. Phys. G 13 ( 1987 ) LI81. [3] M. Cubero, M. Sch6nhofen, M. Gering, M. Sambataro, H. Feldmeier and W. N6renberg, Nucl. Phys. A 495 (1989) 347. [ 4 ] B. Luk~lcsand K. Marlin;is. Ann. Phys. 45 ( 1988 ) 102. [5] I. Lovas, G. Wolf and N.L. Bal~izs,Phys. Re,,'.C 35 (1987) 141. [6] B. K~impfer. B. Luk~ics,G. Wolf and H.W. Barz, preprint JINR-E2-88-588, Sov. J. Nucl. Phys., in print. [ 7 ] H.W. Barz and B. K~impfer,Phys. Lett. B 206 ( 1988 ) 399. [ 8 ] B. Kampfer, B. Luk~tcsand Gy. Wolf, Europh. Left. 8 ( 1989 ) 239. [ 91 B. K.~impfer,G. Wolf, B. Luk~icsand H.W. Bar-z,Contrib. paper NASI Conf. on The nuclear equation of state (Peniscola. Spain, 1989), ed. W. Greiner (Plenum. New York), in print. [ 10] M. Gyulassy and W. Greiner, Ann. Phys. 109 (1977) 485.