Transport coefficients in ultra-relativistic heavy-ion collisions

Nuclear Physics A435 (1985) 826-843 @ North-Holland Publishing Company

TRANSPORT

COEFFICIENTS IN ULTRA-RELATIVISTIC HEAVY-ION COLLISIONS SEAN GAVIN

Department

of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA

I1 10 W. Green Street, Urbana,

Received 1 August 1984 Abstract: In the expansion of matter following an ultra-relativistic heavy-ion collision, entropy is produced

through viscous dissipation and thermal conduction. We derive expressions, in terms of scattering rates, for the thermal conductivities and coefficients of viscosity pertaining to matter in both the deconfined and hadronic phases, in the collision-time approximation. Variational calculations of the collision times are then performed for the hadronic (pion) phase. Dissipative effects in the pion fluid are then estimated.

1. Introduction The initial entropy produced in an ultra-relativistic central heavy-ion collision (&> 30 GeV/A) should provide a clear signature of the deconfinement phase transition le3). In order to read this signature, we must understand the mechanisms by which entropy is produced irreversibly in the expansion of the central rapidity region as well as the nuclear fragmentation regions. The matter in these various regions is likely to begin as a nearly thermalized quark-gluon plasma 4-6). As the system expands hydrodynamically, hadronization occurs, and in the central rapidity region the matter becomes essentially a pion fluid. Entropy production enters the hydrodynamic description of the matter through viscosity and thermal conduction. In the first part of this paper we derive expressions for the first and second coefficients of viscosity as well as the thermal conductivity, in the collision-time approximation for these systems. The second part contains a discussion of dissipation in the central rapidity region pion fluid. In particular, we present a variational calculation of collision times for viscosity and thermal conduction in terms of low-energy pion-pion scattering amplitudes. Finally, we discuss the implications of these calculations.

2. Transport coefficients in the collision-time approximation

Hydrodynamics is useful in describing small departures of a system from a local equilibrium state. Transport processes, which arise as the result of collisions between particles in a system, tend to resist these excursions. The interplay between microscopic (collisions) and macroscopic (hydrodynamic) phenomena is described by a 826

827

S. Gavin / Transport coeficients

kinetic equation. We start with the Boltzmann kinetic equation, which in the local rest frame (where the momentum density vanishes) is

(1) where u,, = at+/ap= P/.F~ is the single-particle velocity and I{f,} is the rate of change of the distribution function f, due to collisions. The stress-energy tensor Tp”, from which the viscosities are extracted, is determined from f, by T”’=

drp'v',f,

,

(2)

where dT = g d3p/(27r)3 and g denotes the number of internal degrees of freedom. Near local equilibrium the system can be described locally by the temperature T, the velocity u, and the baryon chemical potential ,_&b,which vary slowly in space and time. Pion and gluon systems are Bose fluids with no net baryon density (so that & = 0). For such systems near equilibrium, the distribution function assumes the local equilibrium form 1 fpO=e(C~-p.“)/‘_l

(3)

near the local rest frame, plus small corrections. Since I{f,“} vanishes, transport arises only when f, is different from f,“. In order to extend the definitions of T and u to non-equilibrium systems+, we demand that

(5) where Too = e is the energy density (note that a system with non-zero baryon density requires a similar condition on the local-equilibrium baryon current to fix the chemical potential in addition to T and u.) The right side of (4) is a function of T alone (in the local rest frame) and defines the temperature near equilibrium. Similarly, eq: (5) defines the non-equilibrium (energy) velocity++. In the collision-time approximation to (l), the non-zero eigenvalues of the linearized collision term I{f,} are taken to have a common value l/7, where T is the collision time. We write I{f,>=-(f,-f,“)/T ’ Strictly speaking,

(6)

they are well-defined only in global equilibrium. ” We follow, for the sake of clarity, the Landau-Lifshitz ‘) formulation of relativistic hydrodynamics in which the fluid velocity is defined in terms of the energy flow. The results are, of course, independent of this choice.

828

S. Gavin j Transport coe~cie~~s

where f,” is given by (3) for a Bose system. Moreover, we ignore possible energy dependence of 7. Multiplying eq. ( 1) by p“ = (Ed,p) and integrating over momenta we find that, with f,” defined by (4) and (5), the conservation laws apTI”” = 0 are satisfied, as required for a fluid in the absence of sources. Let us first consider the simpler case of bosonic matter with no net baryon content; later in this section, we study transport in a quark-anti-quark fluid, which in general contributes net baryon number to the system. The only transport coefficients relevant to a system with baryon number zero are the two viscosities; thermal conduction, which involves the relative flow of energy and baryon number, does not occur+. The first viscosity 7, and the second viscosity 5, enter the fluid equations through the dissipative part of the stress-energy tensor, TgL, = THv- Tg”, where To”” is the stress-energy tensor for the system in local equilibrium. In a local Lorentz frame in which the momentum density pi is small, the spatial part of the stress tensor Tf(i,, depends linearly upon the gradients of the local three-velocity in the form (7) plus terms of order (fi~/L)~, where BG 1 is the mean particle velocity and L-lPT/TI. The procedure for solving the Boltzmann equation for the transport coefficients is standard. From (1) and (6), one finds& =fi+Sfb where

>

(

Sf*=-7 ~f;+u,.vf,o

plus terms of order (t%/L)‘. Hence (2) implies that, to lowest order, T’j = Tg+

dl"p'v;lif

,

(9)

where (3) gives T{ = PSv and P is the local pressure. (Recall that, in an arbitrary Lorentz frame, rg” = eu”u y + P( gPLy+ U’U“)

(10)

for the metric with go’= -1, uPu” = -1, and e the energy density.) To calculate the first viscosity consider a steady flow of the form ui = (u,(v), 0,O) with T everywhere constant. Then (7) reduces to TXY= -v au,/ay. The time derivative in (8) vanishes for this flow; thus using (9) we find that

(11) ’ This is analogous absence of Umklapp

to the disappearance of the phonon thermal-conduction processes [see e.g. ref. *)I and in superfluid 4He.

mode in insulators

in the

829

S. Gavin / Transport coeficients

so that

(12) This result simplifies if we take the particle mass m = 0; then integration by parts yields

q(m=O)=$e(T),

(13)

where e(T) = $g1r2 p is the familiar black-body result for the energy-density of a massless Bose fluid. For a gas of nearly massless particles, we find the expansion ,=,(m=O)(l-$($+$(;T+.

. .).

(14)

On the other hand, in the nonrelativistic limit, n is most simply expressed in terms of the density of excitations n = I dTfj’ as T1=nrT(l-;+...). A simple numerical interpolation 17 n(m=O)=e

(15)

between these two limits is

_m,T l+m/T+1.817(m/T)Z+1.447(m/T)3’2+1.265(m/T)5’2 1+ 2.0( m/ T)Z {

1

which agrees well with a numerical integration of eq. (12) throughout the range of m/T.

Dissipation through second viscosity arises from the departure of a system from thermodynamic equilibrium as it is uniformly compressed; from the trace of (7), (Tdiss)i = -35(V * U) e

(16)

In terms off,“, (8) and (9) give (T~i~,)~=-~~d~(~~~+~p.~~~)~. Observe that in the local rest frame, d,T’” conservation laws to order (or/L)‘: ae -$=-wv.u, w-+-VP,

= 0

(17)

implies the energy and momentum

(18) (19)

where w = e + P is the enthalpy density. We see that (18) demands that a flow with

830

S. Gavin / Transport coeficients

non-vanishing divergence cannot be steady (in contrast to the pure shear flow considered above). In order that Tii,, have the form (7), the time derivatives of u and T arising from af,“/at in (17) must be eliminated using the conservation laws (18) and (19). Noting that C,= de/dT is the specific heat, and that dP = sdT = w dT/ T,and calculating the derivatives in (8) using (3), we find

6&=-2g ep&v.u--(q.V)(p.u) , I P1 ”

where the dependence on VT has disappeared judicious use of (2) and (4) we obtain

as expected. With eq. (18) and

w-4 For MP T, l-0.1627;+**.

, >

while in the non-relativistic

regime (T < m) .

(22)

Note that, in contrast to the case of a gas of conserved particles, l/v does not vanish in the non-relativistic limit, since a system of non-conserved particles can dissipate energy as it is compressed uniformly by creating particles out of the vacuum. As before, the results (21) and (22) can be combined in an interpolation formula which closely reproduces the result of numerical integration of (1) for all values of m: [=~m3Te-m~T 1 - 0.1627( m/ T)+ 0.00574( m/ T)5'2 1 + 0.0006( m/ T)4 The viscosities of a gluon liquid (g = 16 for SU(3) color) are described by the m = 0 results: ?7o=&7?7+, &SO.

(23) (24)

After hadronization occurs, entropy in the central rapidity region is produced mainly by collisions between pions. The pion fluid viscosities, v,, and {,,, which must be found numerically, are shown (relative to the collision time T) as a function of temperature in fig. 1. Since the second viscosity is -0.1% of the first viscosity, we expect that, for most purposes, second viscosity effects will be negligible+. ’ A pion gas will also sustain thermal conduction if phase space blocks reactions which change the number of pions. This point is addressed in the next section.

S. Gavin

1 Transport

coeflcients

831

Temperature(MeV) Fig. 1. The first and second viscosities 7 and 5, and the thermal conductivity K are shown, relative to the collision time, for a pion gas (g = 3, m, = 140 MeV) as a function of temperature. The thermal conductivity is relevant only to a gas of conserved pions and is discussed in sect. 3. Note that the second viscosity is multiplied by a factor of 103.

Let us now consider a quark-anti-quark gas with arbitrary pt,. The particle (anti-particle) distribution function f,+ takes on the familiar Fermi-Dirac form in local equilibrium, 1 (25) j-;* = e(Cp-PW+LbW-+1 ’ assuming local chemical equilibrium between quarks and anti-quarks (the flavor index on ,_&.b will be understood). The stress-energy tensor is found simply by replacing fP in eqs. (2) by the sum fP+ +f,-; the lowest-order stress-energy tensor is again given by (lo), so that (18) and (19) are again the lowest-order conservation equations. New to our problem is baryon conservation, $$,+~-jb=O, where Pb=

Wf,+-_&-I, I

jb=

dK$+-f,-)u, I

are the baryon density and three-current.

(26)

832

S. Gauin / Transport coeficients

A system with pb # 0 can produce entropy irreversibly through thermal conduction as well as viscous dissipation. We first calculate the heat current to lowest order in &-/I, and extract the thermal ~nductivity, and then find the two viscosities from T&s, following the Bose-case. Thermal conduction arises when energy flows relative to the baryonic enthalpy. The energy flux is T” so that the heat current is given by Ii = TOi_Ef

.

(27)

Pb

In a static situation with u = 0, the heat flux depends linearly upon VT: I’ = where

K

-KV’T,

(28)

is the thermal conductivity. To lowest order, (2), (26) and (27) then give

The Boltzmann equation implies that S$, = -r( p/e,,) Vf&, depending, when u = 0, on position only through &&b and T. Momentum conservation (cf. eq. (19)) shows that VP = 0 in the steady state; thus the Gibbs-Duhem relation dP = 0= w dT/ T-l-pbT d(pb/ T), relates the gradients of T and pti From (25), we obtain l

By comparing (28) with (29) we then find

The calculation of r] is identical to the & = 0 case as expected. The result is (31) We derive the second viscosity 5 from (16) and (17) as in the ,.&= 0 case, although a few words are in order as the conservation laws for finite &, are somewhat more involved. In the energy conservation law (cf. eq. (18)), e now depends on &, as well as T. Using the baryon number conservation equation, which is @b/at f pbV * II = 0 to lowest order in @k/L, we fmd a convenient form of ( 18):

We now proceed as in the pi, = 0 case to find Jo = lo+ i- Jo_, where (32)

833

S. Guuin / Transport coeficients

and we have used the thermodynamic (@l&L

identity

= Pb(@blQb)T + (aPlae),,T*(a(&

T)/aT),,

(33)

.

Note that as j,&btends to zero in (33), (aP/C?&)e vanishes, (aP/ae),+ w/C,T the structure of eq. (20) is recovered. Eqs. (30)-(32) can be evaluated numerically for arbitrary mass, temperature chemical potential. To lowest order in the particle mass, exact expressions for and 5 can be obtained in closed form for arbitrary CL,,.We spare the reader details and simply state the rather intricate results: no(m=O)=&re=&w,

and and 7, K the (34)

(354 the heat capacity is given by

where w and pb are w = ;g?r* P{& + 2y2 + y”} , Pb

= %grT3{Y3

(35c) (354

+ Y> ,

with y = pb/TT. In terms of y, we easily find

We) =&T*{ 1 + 3y*} .

(35f)

The expression for 5 is somewhat more complicated:

(364 where 1 +gy’+gy4+$jy6+gy* H(Y)

= 1 +yy’+2?Jy4+y.$3y6+2.gy*

*

Wb)

The results (34)-(36) are applicable to systems with m =GT, so they should be adequate to describe transport by light (u, d) quarks. As in the Bose case, we see that Jo is negligible compared with no (co vanishes as m4 in the relativistic limit). For strange quarks (m, - 250 MeV) no and KQ are well approximated by their m = 0 values; although co is still small compared with no and KQ, higher-order corrections in m/T are important in the second viscosity and to evaluate it we must resort to numerical integration of (32). Fig. 2 shows no/r, KQ/~ and ~Q/T for typical values of ,&, as a function of T (in these figures, we take g = 6 = 3(color) X 2(spin)).

834

S. Gavin / Transport coeflcients

71

1600

800

Temperature

(MeV)

‘r

-180

Temperature

(MeV

1

Fig. 2. The first and second viscosities and thermal conductivity n, 6 and K relative to the collision time for a massive quark-anti-quark fluid (g = 6, m = 250 MeV) are given. Results are shown as a function of temperature for CL,,=100 MeV (fig. 2a) and pb=300 MeV (fig. 2b). The first viscosity and thermal conductivity shown are about 10% less than their m = 0 values and the second viscosity, which vanishes when m = 0, is now non-zero (note that the second viscosity is multiplied by 103!). We see, by comparison of figs. 2a and 2b, that K/T varies strongly as a function of pb whereas q/r and l/ 7 are roughly independent of pb in this temperature range.

835

S. Gavin / Transport coejicients

The ,.$ = 0 limits of the transport coefficients (34)-(36) are important in the central rapidity region where few baryons are expected. In this limit, the first and second viscosities are To(m=O,pb=O)=-

Tr2g rT4 450 ’

(374

Wb) +O thermal conduction ceases so the fact that We have noted earlier that as ,.&b K~(??I = 0) (cf. eq. (35a)) diverges as pa2 is, at first glance, disturbing. This effect, which is absent (see fig.. 1) in the familiar case in which the total number of particles is conserved (this case is treated in sect 3), accounts for the increase in the magnitude of KQ shown in figs. 2a and 2b. This divergence is inconsequential, however, as only the factor KQ/~: enters the equations of motion; for example the entropy produced by thermal conduction [see e.g. eq. (2.22) of ref. ‘)I in the local rest frame is = KQ(PbT/d2(V(pb/ ($PP)tbermal

T))‘,

(38)

where uP is the entropy flux. Since KQ& remains finite as ,.&+ 0, tranSpOI? due to thermal conduction becomes irrelevant as V(&,/ T) + 0. Only momentum diffusion through viscous stresses remains as transport processes; The results (23), (24) and (34)-(36) describe the gluon system and the quark-antiquark system individually and, strictly speaking, apply to the quark-gluon plasma only when quark-gluon collision frequencies are small compared with characteristic dynamical frequencies. In general a calculation of transport coefficients for a realistic quark-gluon-anti-quark system will have to take quark-gluon interactions into account. More precisely, let loo be the collision time for the scattering of quarks (or anti-quarks) from gluons and let roe be the collision time for the scattering of gluons from quarks. If L is the characteristic length scale of inhomogeneities in the system and and oo (=S1) respectively are the mean velocities of quarks and ghOIIS in the system, then 06’ = L/fiQ and w;’ = L/tiG are the characteristic time scales over which the quark and gluon components of the system evolve. When ~o~oo s 1 and ~o~co > 1 while ~o~oo < 1 and oGTGG< 1 (where loo and TGG denote the quark-quark and gluon-gluon collision times, respectively) then quark-gluon collisions are irrelevant to the macroscopic behavior of the system and (23), (24) and (34)-(36) apply. In general, this is not the case and all the collision times may be comparable. The calculation of these collision times from first principles is complicated as the details of screening effects in many-body QCD systems are not well understood. However, Hosoya and Kajantie ‘“, have pointed out that T& should scale as af In (l/a,)T where a, is the running QCD coupling constant -1n (T/A)-’ with A - 200 MeV. The In (as) arises in their result as a consequence of screening effects. Recently, Danielewicz and Gyulassy “) have suggested that it is not unreasonable t?Q

836

S. Gavin / Transport coeficients

to assume that the collision times are limited by the interparticle spacing. In this, work we shall take these collision times as parameters and leave their precise c~culation as an interesting future problem. For arbitrary collision times, transport coefficients for the present three-component system are obtained by solving three coupled Boltzmann equations. This task is straightforward but the results are untransparent. In the limit where ~o~oo, ~o~oo. and taboo are all
TG=(T&+T&)-‘, and r (formerly

TQQ)

in

(3

1)

(394

is given by TQ=(&+&)-’

;

(39b)

more explicitly,

where wo=4eo=$Jzr2P.

(40b)

The results for the thermal conductivity and second viscosity are modified because e, P and w in the conservation equations (18) and (19) now refer to the combined gluon-quark-anti-quark system. For the thermal conductivity we find K=KG+KQ,

(414

where

and wr = WQ + WG. Observe that in a mixture with baryons the gluon fluid will contribute to thermal conduction (driven by the baryon chemical potential gradient due to the quarks and anti-quarks), this follows as the gluon fluid carries energy relative to the enthalpy carried by the baryons (note that this contribution vanishes when & = 0).

S. Gavin / Transport coe~~ents

837

The second viscosity can also be written as the sum of gluon, quark and anti-quark contributions: L=!CG+YQ++LQ-

(424

with

(42b)

The gluons contribute to the second viscosity of the system (despite the fact that they are massless) since compressing the gluon component of the fluid drives the (possibly massive) quark component out of local equilibrium. The appearance of second viscosity in a system of massless radiation quanta (e.g. gluons) coupled to matter has been discussed earlier by Weinberg ‘) in a different context; studying a photon gas interacting with a perfect fluid of massive particles in the absence of photon-photon interactions (the limit corresponding to OUT TGG + ~0 and roe + 0), he derives expressions formally analogous to KG and Jo. We have evaluated (40), (41) and (42) numerically for values of ~o/ro close to unity and find that they differ little (relative to the collision time) from those in figs. 2a and 2b. (Analytic expressions for the present case similar to (34)-(36) have been obtained when m ( T, but they are so unwieldly as to be of little use!) 3. Variational calculation of collision times The details of the specific scattering mechanisms that determine the transport coefficients (for an arbitrary system) are contained, within our treatment, in the collision (or relaxation) times for the two viscosities and thermal conduction (all of which are generally different). We turn in this section to a calculation of these quantities for the pion system in the central rapidity region where low-energy pion-pion scattering is predominant. The pion system can sustain heat conduction despite the fact that the pions themselves carry zero baryon number. This follows since the total number of pions is itself essentially conserved, as we can see by the following argument. In a pure pion system, the dominant reaction that changes the number of pions is 4~ --, 27~ This reaction, which is favored thermodynamically 3), is slow in sufficiently rarefied systems?. Consequently, as the post-hadronization pion cloud expands and comes out of global equilibrium a small positive pion chemical potential p,( r, t), develops in order to keep the pion number fixed. ’ The inverse process, 2~ + 4-75is also effectively absent. At temperatures below the deconfinement temperature (-200MeV), collisions between thermal pions cannot produce enough energy for this process to become appreciable; low-energy pion-pi& scattering is essentially elastic [see e.g. ref. ‘*)I.

838

S. Gavin / Tmnsport coeflcients

The thermal conductivity is easily derived following the methods of the previous section and one finds (43) where w, is the enthalpy and n,, is the total number density of pions+. We can evaluate this expression by taking CL,,= 0 (although V(y,,/ T) # 0) and the result is shown in fig. 1. Note that this result is finite in this limit (as opposed to the conserved buryon number case) since nsr f 0 due to the presence of pion pairs produced as the system hadronized. Also since K,,T/~= is of order unity, thermal conduction by this mechanism cannot be ignored. Let us now calculate the collision times for thermal conductivity and first viscosity. We leave the calculation of the relaxation time for second viscosity for a future work as we expect second viscosity effects to be small (see above). The variational method for calculating transport coefficients 13) allows for a simple accounting of the energy dependence of the various scattering mechanisms and gives lower bounds on these collision times. We use this method here as a starting point for an approximate calculation of these parameters. We begin by writing the distribution function in the form

where QjP,assumed small, will ultimately be our variational trial function. The rate of change of fP due to collisions I{&}, for the process pp +pg *p: +pg is specified in terms of QP by the differential cross section cr( S, cos 8) ; here, s = (pi + p2)2 and 0 is the scattering angle in the center-of-mass frame. The irreversible entropy production is found (to lowest order in @,,) to be

(454

where AA(f) = A(&) + A(p$) - A(pf) - A(pt) for any function A@‘“). If @z is the exact solution to the transport equation (1) for a shear viscous flow u,(y) then the first viscosity is given by (from here on the subscript “T” will be implicit) rl -’ = T$/ ( TX,)* ,

(46)

’ Recently, Hosoya and Kajantie lo) have calculated transport coefficients for pure gluonic matter and find results similar to (12) and (20). Their expression for K however, differs from (43) in that it does not contain the w/n term: this term arises when the calculation is made consistent with eq. (5).

839

S. Gawk / Transport eoe~cients

where Ts&Sis found from (9) with Sf, now given by (44) with Q, = @pp”. Similarly, if @p”is the exact solution for pure thermal conduction, then (47)

K -‘=S/(I/T)*,

where I is the heat current (27) in terms of @s, The variational principle rests on the fact that the right sides of (46) and (47) are minimized by the exact solutions. The viscosity and heat conduction relaxation times rT and rK can be defined using TJ/T from (12) and ~/r from (43) as (48) the subscript “var”, which hereafter will be implicit, indicates that eqs. (48) are obtained from variational solutions. We choose the trial functions 13) @;pp”=-

@5=-

(> $

(

g

(491

PrPyt

{t;,-w/n)p,. >

(Note that expressions (46) and (47) are independent of the overall normalizations of the trial functions.) The evaluation of these expressions is now straightfo~ard. For example, to find TV,we first calculate T& from (49):

Then (43, (46) and (48) give

c’= B I

so ( S, cos ~)8”(Ap”)(A(p”p’)}*,

where B(T)=

15(~,‘~) 2T C

J

d~(~~~/~~~)~4/~~

>

(5la)

-2

.

Wb)

Eq. (51) is easily evaluated if we write approximately: f,&( l-t&)( 1i-f,) = e-%+ep2)‘T (a good approximation even for Ta m,). An expression analogous to (51) for T, is easily found:

where C(T)=3(47T)

(I 2

> -2

dT (af,/ae,){&,-w/n}‘pu,

.

S. Gavin / Transport coeficients

840

Temperature

(MeV)

Fig. 3. The relaxation times for viscosity and thermal conduction r,, and rk are shown as a function of temperature for the pion fluid expected in the central rapidity region. lhe transport coefficients q and K, multiplied

by the temperature,

are also given (left-hand

scale).

In evaluating TVand T,, explicitly for a pion gas we have used a simple parameterization of the low energy scattering amplitudes containing the effects of the P-wave p-resonance (&= 770 MeV) and the D-wavef’ resonance (& = 1273 MeV) [ref. ‘*)I to obtain ci(s, cos 0). The numerical results, essentially insensitive to the inclusion of the f” resonance, are shown in fig. 3 along with the transport coefficients r] and KT. 4. Discussion We may now estimate the importance of entropy production in the expansion of the central rapidity region following hadronization. In the longitudinal expansion, which is approximately described by the Bjorken scaling solution 5), we may compare the magnitude of the entropy generation term with that of the longitudinal cooling term (from now on, T without a subscript denotes proper time!),

(52)

S. Gaoin / Transport coeficients

The entropy generation term in longitudinal

841

expansion is (53)

where we have neglected second viscosity. (Note that the entropy generation due to thermal conduction vanishes identically in the scaling limit because the gradient in (38) vanishes.) Following Baym ‘), we interpret magnitude of the ratio of these two terms as an effective Reynolds number Re: (54) where in the latter equation we have written n =&Ts~,, (cf. eq. (13)). Taking (for the sake of illustration) T - 4 fm and T - 200 MeV (so that T,,- 3 fm), then Re - 5. Since (53) is comparable in magnitude to (52), we conclude that the entropy density can be increased appreciably relative to the scaling solution and that realistic hydrodynamic models should take viscosity into account. The problem of estimating the total entropy produced is substantially more difficult. Though hydrodynamics provides a good starting point for this calculation, as the system expands it must eventually freeze out of local equilibrium into freely streaming fragments. Under these circumstances hydrodynamics must break down since, in freeze-out, the pion mean free path exceeds length scales of the system. It is possible, however, to gain some understanding of the freeze-out process in heavy-ion collisions from the following simple argument. Suppose that the system is initially in local equilibrium following hadronization. From (52) we see that the time scale of the longitudinal expansion is essentially T. If we neglect the pion mass and take T,, - (naJ’ where a, - 2 fm2 is an effective (roughly COnStant) mr cross section, then TV Will grow in proportion to T (since n scales approximately as s). This behavior persists for finite m, as can be seen from fig. 3, which shows that T,, increases as the temperature of the system drops. From the solution of (52) for finite m, we obtain T(T) which, we find, behaves similarly so that TV/ T - constant. The fact that T,, grows linearly with T implies that as long as the transverse motion of the system can be neglected the system, if in equilibrium after hadronization, will remain in local equilibrium. Eventually (T b RJ c, 2 fm +A”3) the rarefaction wave reaches the center and the motion of the system is effectively three-dimensional 6). The density in this regime will diminish as T-~ so that the scattering time will increase -TV; freeze-out will then occur. It is possible to get a handle on the effects of dissipation in the hadronic regime by neglecting the contribution from transverse rarefaction and solving for the energy as a function of proper time assuming solutions of the scaling form+. The entropy ’ This has been done by Hosoya and Kajantie lo) in the zero flavor limit (where CT- Te2). Their calculation also differs from ours in that gluons are not conserved, while here pions effectively are.

842

production

S. Gavin / Transport coeflicients

and number conservation equations can then be written (55)

Eq. (56) is readily integrated to give n = nO.rO/~where no is the total pion density (n++ n,+ n-) at the initial instant r. (hadronization~. We can rewrite (55) as an equation for the energy density using Ts = w - pn along with the first law of thermodynamics T ds = de - p dn and eq. (56): (57) If we take M, = 0 then 7 = fwr, and w = $e. Using T,,- (nu,,)’ = ~~(7~) + T/T,,, eq. (57) can be easily solved: e(r)=eO

0

-4{1-RR-‘}/3

_T

(58)

I

70

where we note that the Reynolds number as given by (54) is independent of 7 in this approximation. The average energy per pion E, emerging from the central rapidity region is then e(7f)

&3-_=r-n(q)

eO

7r

no 0 70

-ti-4Re-9/3

,

(59)

where TV is the time that entropy production effectively ceases. We expect that this should happen when the transverse expansion becomes important (so that TV2 fm +A”3 - 13 fm for 238U); taking T,- 4 fm and Re (ro) - 5 we see that in the central rapidity regime E, will be enhanced by dissipation by -35%. In this calculation we have assumed that the system emerges from the phase transition as a pion gas in local equilibrium. This need not be the case and it is indeed possible that the pions may form as freely-streaming fragments. Here, finite pion-mass effects may slow the expansion sufficiently as to allow the system to return to equilib~um. In either case, entropy production due to the eventual freezeout must itself be understood and incorporated into hydrodynamic descriptions of the (longitudinal and transverse) expansion before the total entropy production can be calculated with any confidence. Work on this problem is currently in progress. The author is grateful to Gordon Baym for his insightful guidance through the course of this work. This work has been’supported in part by a US National Science Foundation grant DMR 81-17182.

S. Gavin / Transport coeficients

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

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