The Boltzmann-Langevin model for nuclear collisions

Nuclear Physics A545 (1992) 35c-~6c North-Holland, Amslcrdam

NUCLEAR PHYSICS A

THE BOLTZMANN-LANGEVIN MODEL FOR NUCLEAR COLLISIONS S. Ayika, E. Suraudb, M. Belkacemb, and D . Boilleyb aTennessee Technological University, Cookeville, TN 38505, USA and Joint Institute for HeavyIon Research, Oak Ridge, TN 37831, USA bGANIL, B. P . 5027, F-14021, Caen, Cedex, France Abstract An extension of the one-body transport models is developed by incorporating correlations into the equation of motion in a stochastic approximation . In the semiclassical limit, this yields the Boltzmann-Langevin Equation for the fluctuating singleparticle density in the phase-space . In order to investigate the gross-properties of density fluctuations in heavyion collisions, a number of calculations have been carried out . Thos,; calculations reveal that large dynamcal fluctuations a.re introduced into the momentum space during the early stages of the collision, which cause the nuclear system to decay into a great variety of final channels . The effects of the fluctuations on the kaon production in heavyion collisions at sub-threshold energies are also investigated . 1. INTRODUCTION The dynamcal descriptions of energetic heavyion collisions are generally based on two different approaches, molecular dynamics and oue-body transport rrcodels . In the molecular dynamics approach, one tries to solve the full many-body problem under certain approximations . On the other hand, in transport models the system at any time is characterized by its single-particle density rather than by the full many body information . These one-body transport models, in the semi~lassical limit with a Boltzma~u-Uehling-Uhlenbeck (BUU) form of a collision term, have been extensively applied to describe heavyion collisions at intermediate energies [1]. Although these transport models provide a good description for the average properties of the one-body The observables, they cannot describe the dynamics of density fluctuations . fluctn.ation processes such as ch~ster production, dispersion of observables, multiplicity distributions of the produced particles cannot be investigated on the basis of these models. In order to describe the fluctuation processes, we need to improve the onebody transport models beyond the mean-field approximation by including the effects of correlations . Recently, Ayik and Gregoire have proposed an extension of onebody transport models by incorporating correlations into the equation of motion in a stochastic approximation and in a consistent manner with the fluctuation-dissipation theorem of non-equilibrium statistical mechanics [2] . This gives rise to stochastic transport equations for the fluctuating single-particle density and, in the semi-classical limit, the resultant equation is referred to as the Boltzmann-Langevin Equation (BLE) . A similar approach was later developed by Randrup and Remaud [3] . In Section 2, we briefly describe the BLE approach . In Section 3, we discuss a projection method for obtaining approximate numerical solutions of the BLE. Then in Section 4, we present some applications to nuclear collisions in which we investigate 0375-9474/92/$05.00 ©1992 - Elscvicr Scicncc Publishers B.V . All rights rescrvcd .

.S. Ayi~. et al. / Tlae Boltzmctnn-Langevin model the dynamics of density fluctuations and dhe influence of these fluctuations on the kaon production crosssection at sub-threshold energies. Finally in Section 5, we make some remarks on the further developments of the model and give some conclusions . 2.

T

N- A

E

N EQITATI®N

In energetic heavy-ion collisions, the nuclear system decays into a great variety of final states . In contrast to the experimental situation, the BUU model yields a unique (deterministic) trajectory for a given initial condition, which may be considered as an average ûvcl aii possible final states (ensenz6le averaging), and the dynamcal branching is not allowed. In particular the BUU description becomes worse when the spreading of the trajectories of the single-particle densities associated with the final states is large. This severe limitation follows from the independent collision approximation employed in the derivation of the BUU model. The derivation of the BUU model involves two levels of approximations . The first one is the t ur~ca,tion of dynamics at a two-body level, which is a good approximation for a sufficiently dilute system. IIowEVer, this i~ not sufficient . A more drastic approximation must be introduced by neglecting the correlations between subsequent binary collisions, which is usually referred to as the rrtolecular chaos assur>zption. As a result, the fluctuations are not propagated and the independent binary collisions always drive the system to the maximum entropy . In order to describe the fluctuations, i .e., the spreading of trajectories of the singleparticle densities, the effects of correlations must be restored into the equation of motion in some ways . The residual interactions, in general, play two different roles: (i) producing dissipation by randomizing the momentum distribution via binary collisions and (ü) inducing fluctuations by propagating correlations in the phase-space . By incorporating these two effects of the residual interactions into the equation of motion, one obtains stochastic transport equations, which can be studied in various representations [2,4] . In the semi-classical limit it takes the form of a BLE for the fluctuating single-particle density f(r,p,t) in the phase-space,

ere the left handside describes the Vlasov propagation determined in terms of the nuclear mean-field U( f) . The collision term K( f) has the usual BUU form but is expressed in terms of the fluctuating density f(r,p,t), K( f ) =

dpa

dp3

-

dp4 vV(12 ;34) [(1-f 1)(1-f z) f 3 f 4

f i f a (1-f 3) (1-f 4)]

(2.2)

where f - = f(r,pj ,t) and ~V(12 ;34) denotes tl°le transition rates, which can be given in terms o the in-medium nucleon-nucleon scattering cross-section [1-2] . On the righthandside of eq. (2.1), in addition to the usual collision term, it appears an additional Term SK(r,p,t), which is called the fluctuating collision terrrt . The fluctuating collision term arises from correlations not accounted for by the collision term . As a matter of fact, such an addaional term always arises in transport theory whenever we deal with a reduced description and it describes the coupling to the degrees of freedom, which are

,S . Ayik et al. . The Boltzmann-Lanycvin model

3 7c

not explicitly considered [5-61. The fluctuating collision term has many properties which are similar to the random force in a typical Langevin equation : (i) It varies rapidly in time with a characteristic time in the order of the duration time of a twobody collision. (ü) It is nearly impossible to calculate the fluctuating collision term explicitly because ~t is equivalent for exactly solving the manybody problem. (iii) It vanishes on the average with the molecular chaos assumption, hence, does not appear in the average description of the BUU model. The BLE contains full information about dissipation and fluctuation properties of the single-particle density. However, in order to have a tractable model, we need to ir~troduce approximations . In analogy with the Brownian motion, it is assumed that eq . (2 .1} describes a stochastic process in which the whole density is a stochastic variable and the fluctuating collision term acts like a random force [7l. In such a stochastic description the fluctuating collision term is characterized by a correlation function, <ôK(r,p,t)(2 bK(r',p',t' )> = C(p,p' } b(r-r' ) b(tt' )

.3)

which is assumed to be local in space and time without the rrcernory effects. With a specified correlation function, the BLE becomes a well-defined stochastic transport equation for the fluctuating singe-particle density. It provides a probabilistic description in contrast to the deterministic description of the BUU model. The BLE has many solutions with a given initial condition . Each solution produces an event and many solutions are needed for describing a collision process. The BLE was first proposed by Bixon and Zwanzig in order to describe the hydrodynamic fluctuations [8] . They evaluate the correlation function C(p,p' ) in equilibrium using the fluctuation-dissipation theorem as an input. As a result, their model is valid only for classical systems near equilibrium. In order to describe nonequilibrium fluctuations in quantal systems, Ayik and Gregoire calculate the correlation function directly in non-equilibrium within a weak~oupling approximation [2]. In the semi-classical limit, the correlation function is given by

C(P~P') -2

=f

J dPa dP4

dP3 dP4 W(11' ;34) [f l fi' (lß) (1 f4) + (1 f~) (1 fl~) ß f4l

W(12;1'4)

+ b(P-P')~dP2 dP3 dP4

[fl fa (1 fl') (1 f4) + (1 fl) (1 f2) fl' f4l

W(12 ;34)

[fl f2 (1ß) (1f4) + (1fl) ( 1f2) ß fol

(2 .4}

where W(12~34) is the same transition rate which enters into the collision term and The f~ = f(r,p~,t with f(r,p,t) as the locally averaged single-particle density. correlation unction is entirely determined by the one-body properties and is closely related to the collision term . Aside from the mean-field and the nucleonnucleon crosssection, no other information is needed for describing the fluctuations . The fluctuation and dissipation properties of density, which are described by the collision term and the correlation function, are not independent properties, but must be related to each other (as in any relaxation process} through a fluctuation-dissipation theorem. Therefore, the close relationship between the correlation function and the collision term can be regarded as a fluctuation~üssipation theorem associated with the stochastic evolution of the single-particle density. The BLE satisfies the conservation laws of total energy, total momentum, and total particle number . In contrast to the Brownian motion ~n which the energy conservation is satisfied on the average, each event of the BLE respects the conservation laws . This property follows from the fact that the fluctuating collision term in the BLE is an internal noise and the correlation function

,S. Ayik et al. l The Boltzmann-Langevin model satisfies certain sum rules. t he correlation function C(p,p' ) is valid for large uct ations in non-equilibrium and for small fluctuations around equilibrium ; it reproduces the known result of Bixon and Zwanzig .

he L describes a diffusion process in an abstract space of single-particle densities in which the correlation function C(p,p') acts as a driving force for the ctuations. The numerical solutions of the BLE can be obtained b em to in standard methods for solving a typical stochastic differential equation . Stârtingwith â definite density a s given time, the equation of motion generates a set of densities during the time step. For the next time step, we choose one such density as the new initi state and repeat this procedure. A direct simulation of this manner is not very practical and propagates too much detailed information, which is not needed for describing gross properties of the density fluctuations . n order to describe the gross properties of density fluctuations, it is sufficient to propagate the fluctuations associated with a few low order multipole moments of the momentum distribution . As can be seen from the fluid dynamical description, the evolution of density is co'apled to the fluctuation~issipation mechanism through the momentum flow tensor, which is nothing but the local quadrupole moment of the o ent m distribution . Therefore, we devP,lop a method for obtaining approximate solutions of the L y projecting fluctuations on the local multipole momentç of the momentum distribution 2, 11],

ere L( ) is the multipole moment operator of order L in the momentum space which is a L-i-1 dimensional vector with components QL ~(pj . The fluctuations of the multipole oments are characterized by a diffusion mdtri~, which can be deduced from the icroscopic correlation matrix C(p,p' ) as L(p')

C(p~p')

dp2 dpa dp4 ®Q L ~Q L , W(12;34) fl f2 (1-ß) (1-f4)

(3 .2)

with ®Q~ = L( ~) + ~,(pa) - QL(p3) - QL (p4) . This quantity determines the early growth rate of the fluctuations of QL's, and it can be easily computed at each time step with the pseudo particle simulation . The idea is now to simulate the evolution of Qt's y performing a mufti-dimensional random walk in accordance with the time and position dependent diffusion matrix given above . Then, a single dynamical trajectory can be determined according to the following algorithm : (i) Starting with a definite density f (r,p,t) at time t, its average evolution and the elements of the diffusion matrix are calculated during the time step ®t with the particle simulation, yielding f(r, ,t + ®t) and CAL , (r,t) . (ü) In the second step, the fluctuations of the multipole o ants are determined according to a mufti-dimensional Langevin equation,

S. Ayik et al . l The Boltzmann-Langevin model

QL(r,t+ot) = QL(r,t+ot) + E C L

t

~ r,t

~ LL' W L ° °

39~

(3 .3)

Here QL(r,t+Ot) is the multipole moment associated with the locally averaged density f(r,p,t+Ot), and the quantity in the second term is the square-root of the diffusion w matrix multiplied by the independent Gaussian random numbers W with unit L variance and zero mean for each multipole moment . The square-root of the diffusion matrix is defined in a standard way in terms of the orthogonal transformation which diagonalizes CLL , (r,t) [7] . (iii) Finally, the fluctuations are inserted into the phase-space by scaling the local momentum distribution to the new values of A A QL's, f (r,p,t+Ot) -+ f(r,p,t+fit) . This procedure is repeated at each time step. In the practical applications of this method, the multipole space must be truncated to a reasonable size. We expect that the propagation of fluctuations by the quadrupole scaling above should provide a good approximation for the gross properties of density fluctuations . Other simulation methods of the BLE-type equations have been proposed also in realistic situations by Bauer, et al. [9] and more recently in an infinite matter model by R,a,ndrup, et al . [10] . 4. APPLIC `l`Ii~NS TO HEA`T~YIOI~ COLLISIONS 4.1. Density Fluctuations The BLE approach provides a useful basis for studying dynamics of density fluctuations in nuclear collisions at intermediate energies and for investigating possible connections between the reaction mechanism of the nzultifragmentation process and the nuclear matter equation of state . The multifr~,g:nPntation process is usually associated with the instabilities in the spinodal region. In the spinodal region the nuclear system becomes unstable with respect to density fluctuations, and that causes tr~e system to break up into clusters. In order to study the density fluctuations in nuclear collisions we carry out a number of calculations [11] . In most of these calculations the BLE events are determined in the lowest order approximation by propagating only the fluctuations associated with the z~omponent of the quadrupole moment of the momentum distribution, Qzo(p) ~ Qa(p) = py. In this case, eq. (3 .3) 2p Z - pX reduces to a one-dimensional Langevin equation involving only a single diffusion coefficient C22(t) -_- C2(t). In the calculations, we employ a simplified three parameters (t o, t3, ~y) Skyrme interaction, which gives an incompressibility modules of 200 MeV, and use an energy dependent nucleon-nucleon cross~ection together with phenomenological medïum effects [a 2] . The computations are performed with 20 pseudo particles per physical nucleon and the collision integral is evaluated by means of, so-called, the full ensemble technique. First, we consiûef~ tl~e fluctuations associated with the momentum space and calculate the mean value <~~Z> and the variance QZ = 2 - z of the total quadrupole moment, which provide measures for the energy dissipation and the fluctuations in the momentum space, respectively . As an example, Figure 1 shows the time evolution of the mean value , the diffusion coefficient C2(t), the variance Q2(t) of the total quadrupole moment of the momentum distribution and the collision rate in 12C + 12C collisions at various energies. The mean value of the quadrupole moment describes the da znping of the relative motion

.S.

=~0~

Ayik ct al . I

Thc Boltzmann-Langcvin model

' 2C+' ZC(b=Ofm) "100 Events"

"100 Events°' , ~ , E/A = 40 MeV/n

0.1 -~ 0.08

- - - E/A = b0 MeV/n

r O.Ofi

----- E/A= 100MeV/n

n

E/A = 40 MeV/n - - - E/A = 60 MeV/n E/A = 100 MeV/n

0,04

0v

v 0.02 0

° (b)

60 50 ...E 40 30 N

10 0

0

30

60 90 TIME (fm/c)

120

150 0

30

60 90 TIIViE (fm/c)

120

Figure 1. Time evolution of the collision rate (a), the diffusion coefficient (b), the mean value (c) and the variance (d) associated with the total quadrupole moment of the momentum distribution in 12C + 12C collisions at bombarding energies of 40 (solid lines), 60 (dashed lines) and 100 (dotted lines) MeV per nucleon. and indicates that a large portion of the relative kinetic energy is converted into the internal excitations during a time interval of 20-40 fm/c. The diffusion coefficient is concentrated at the early stages and its peak value is at least an order of magnitude larger than the thermal background. As a result of this behavior, the variance of the quadrupole moment also goes through a maximum during thv early stages. There are strong correlations between the Dissipation rate, the collision rate and the magnitude of fluctuations . The dissipation rate and the magnitude of fluctuations are large when the collision rate is high, VVe also notice that the magnitude of fluctuations increases for increasing energy . This follows from the fact that with increasing bombarding energy, the Pauli blociCing becomes less effective and, consequently, the available p ase~space for decay becomes larger, which increases the dissipation rate and the magnitude of fluctuations . We also carry out calculations in which the BLE events are determined by the quadrupole plus octupole scaling . The results of these calculations for the quadrupole moment are similar to those obtained by the quadrupole scaling alone. I3owever, the situation is different for the octupole moment . The scaling of the

S. Ayik et al. / The Boltzmann-Langevin model

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quadrupole moment above does not generate fluctuations in the octupole mode, and as a result its variance remains small. In the case of the quadrupole plus octupole scaling, the fluctuations in the octupole mode is also propagated and its variance exhibits the characteristic bump. These calculations indicate that large dynamcal fluctuations are introduced into the momentum space during early non~quilibrium stages of the collision and the amplitude of fluctuations increases with increasing energy . These fluctuations are subsequently transmitted into density and depending on the size of the fluctuations, the system breaks up into many different final states . 4.2. K ~ Production at Sub-Threshold Energies On the basis of the BLE approach we can study the rare processes occurring in heavyion collisions and investigate the effects of correlations on the sub-threshold particle production and the hard photon emission . Here we discuss the calculations carried out for the K+ production in heavyion collisions at the bombarding energies below and far below free nucleonnucleon threshold. By investigating the particle production mechanism in heavy ion collisions we hope to learn information about the reaction dynamics and to understand whether the production mechanism is an incoherent or a collective process [13] . For this purpose kaons provide a good probe because they are not disturbed very much by the nuclear medium. It is very difficult to explain the particle production at low bombarding energies in the BUU approach in terms of a superposition of independent binary collisions. In order to explain the production mechanism, the effects of the correlations must be :ncorporated into the calculations . The correlations give rise to large fluctuations in the local momentum distribution . In particular, the fluctuations associated with the high energy tails of the momentum distribution can have a large effect on the particle production crosssections at sub-threshold energies. In addition, the multiparticle interactions involving more than two nucleons provide an efficient mechanism to produce sub-threshold particles [14] . Both effects should be incorporated into the cross-section calculations, In the present investigations, we calculate the production cross~ections by incorporating the effects of fluctuations in the BLE approach and compare the results with those obtained in the average description of the BUU model [15] . At energies below 1 .0 GeV/n, kaons are produced mainly m the elementary baryon-baryon collisions and the invariant K+ production cross-section is evaluated by folding the elementary cross-section with the momentum distribution of the colliding baryons, r E ~ ~QK~ = E J 2~rb db P c

f

dt ~dr dp dp' ~ v-v' ~ E

â-3 ~~~ P

~ F.

(4.1)

Here d3Q~/dpi denotes the elementary cross-section in channels "c", B+B-~B+K~+Y, with B as either a nucleon or a delta and Y representing a A- or a ~hyperon. The pion channels are also included in the calculations . In the BUU approach the factor F is given in terms of the ensemble averaged single-particle density by F ~ Fßuu ~- f(r~P~t) f(r~P' ~t) [1-f(r~P" ~t)]~

(4 .2)

In the BLE approach, the cross-section is evaluated in the same way, but the factor F is determined as an average over an ensemble of BLE events, (4 .3)

S. Ayik et al . I The Boitzmann-Langevin model

4~2c

1 2 10°

~. 1 2~ ______~

Numerical simulations with Fluctuations (BLE)

1Q 1

1~ 2

103

10~®0

600

800

1000 ~L~b

1200

1400

(~~e~~11}

Figure 2. The I{ ~ production cross~ection in 12C + 12C collisions as a function of the bombarding energy . Shown are the BLE simulations (solid line), the BUU simulations (dashed line) and the Gaussian model calculations (dots). The calculations are performed with a soft mean-field. i Z

10 -6

10 ® f;

l

v

200

+ i z~

C

Fluctuations (Bl,E) _ - - - No Fluctuations (BUU) 400

v

G0~1

800 Lab

v

I

1000 1200 1400 ( e~/ )

Figure 3. The energy dependence of the I{ + production cross-section in i~C + t2C collisions including the energies far below the free N-N threshold. Shown are the Gaussian model calculations with fluctuations (solid line) and the BUU simulations (dashed line) . The calculations ärt performed with a soft mean-field .

S. Ayik et al. I The Boltzmann-Langevin model

43c

We determine an ensemble of BLE e~,Aents using the projection method in the lowest order approximation with quadrupole scaling as discussed in Section 3, and calculate the K+ production cross~ection in both the BUU and the BLE approaches with the input data of ref. [16] . In order to improve the statistics of the numerical simulations at low bombarding energies, we also perform a model calculation by parameterizing the fluctuating momentum distribution in terms of the z-component of the total quadrupole moment Q of the momentum distribution f(r,p,t) ~ fQ(r,p,t), assuming a Gaussian form for the distribution function P(Q) of the quadrupole moments . In this case, the factor F in eq. (4.1) becomPç F -~ F HLE

- ,!

dQ P(Q) fQ(r,p,t) fQ(r,p',t) [1fQ(r,p",t) ] .

(4.4)

In this Gaussian model, the mean value and the variance Q of the distribution Q function P(Q) as well as the momentum distribution fQ (r,p,t) for each Qbin, are extracted from a large number of numerical simulations of the BLE, and the crosssection eq. (4.1), is evaluated by numerical integration. Figure 2 shows the total K+ production cross-section in i2C + 12C collisions as a function of bombarading energy . The solid line and the dashed line in Figure 2 represents the result of calculations in the BLE and the BUU approaches, respectively. These calculations are performed with a soft meanWield and by taking only the nucleon-nucleon channels into account . The BLE model with fluctuations, in particular at low energies, gives much larger cross-sections than those obtained in the BUU description . The dots in Figure 2 represent the calculations performed within the Gaussian model, which agree well with the numerical simulations . The nmerical simulations for the kaon production at energies far below the threshold are not reliable. They require a huge number of the BLE events . However, we can estimate the order of magnitude of the production crosssection at far below the threshold energies using the Gaussian model. In Figure 3, the total K + production cross-section in i2C + 12C collisions calculated with the Gaussian model is plotted as a function of bombarding energy, including the energies far below the free nucleon-nucleon threshold. The dashed line indicates the result of the BUU simulations . With this model we can also estimate K * production cross-section in 42C a + 42Ca collisions at a bombarding energy of 90 MeV/n. The estimated cross-section, which is shown in Table 1, is the same order of magnitude as the recent experimental data obtained by Julien, et al . [17] . However, more work is needed for a better understanding of the production mechanism at far below threshold energies . Table 1 The K+ production corss-section in 42Ca + 42Ca collisions at a bombarding energy of 90 MeV/n . The range of values of the calculations reflects the range of error of the Monte Carlo integrations . Gaussian Model

Experimental data

.S . Ayi~: et al . I The Boltzrnann-Land>evin model

~4~ . .

t

tic

cripti®a of Fission

The BLE approach can also be applied to study the gross properties of collective nuclear motion including inelastic collisions and induced fission . These studies can be carried out either at a microscopic level by a numerical simulation of the BLE or at a macroscopic level in terms of collective transport models. Within a fluid dynamcal treatment, we can deduce from the BLE collective transport models, extract transport coefficients, and compare the results with the phenomenological models [18] . Although the microscopic BLE is a arkovian equation, the deduced collective transport equations for collective variables q(t), in general, exhibit memory effects [19,20], +

~ qa + ~ _ -

c

dt ° ~y(t-t ° ) q(t ° ) + SF(t).

(4.5)

ere M(q) and V(q) are the irrotational mass parameters and the potential energy associated with the collective motion . On the right-hand-side, the retarded friction kernel is given by ere, T and Q are the relaxation time of the deformed momentum distribution and the reduced friction coefficients, respectively . The quantity bF(t) represents the random force on the collective motion and its correlation function is determined in terms of the friction kernel as with T as the nuclear temperature . We apply this model to study dynamics of the induced fission and investigate the influence of the memory effects on the transient times and the fission decay rates [21] . Recent experiments indicate that more prescission neutrons are emitted during fission than those predicted by the standard statistical model of Bohr and Wheeler. These observations may be useful to extract information about the magnitude of nuclear dissipation . About fifty years ago Kramers suggested that the induced fission can be described as a diffusion process over the fission barrier . Recently, using phenomenological transport models without memory effects, a number of calculations for fission were carried out by Grangé, et al . [22] and Abe, et al. [23], and an upper limit fnr the reduced friction coefficient is extracted from the prescission neutron data in fission of 15sEr nucleus . On the basis of the BLE, we explicitly calculate the transport coefficients associated with fission by assuming a one-dimensional quadrupolar shape evolution and retaining only the two-body dissipation mechanism . The calculated reduced friction coefficient is strongly temperature dependent and its magnitude is much larger than the estimates based on the prescission neutron data, even though the memory effects tend to reduce its magnitude. This result is in agreement with the conclusion reached in a recent work of Bush, et al . [24] . The large effective friction gives rise to smaller stationary fission rates and longer transient times, hence allows for more neutrons to be emitted .

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Ayik et al. l The Boltzmann-Langevin rnodel

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5. CONCLUSIONS The semi-classical BLE approach provides a good starting point in order to study the dynamics of density fluctuations in the heavy ion c~ilisions and to investigate the gross properties of reaction mechanisms at intermediate energies. However, the model needs improvements in several directions. We need to carry out better and more systematic numerical calculations, to develop semi-quantal transport treatments of the collision process and to extend the model to the relativistic collisions. Heavyion collisions at bombarding energies of a few GeV/n give rise to hi hly compressed and excited nuclear matter during the initial stages of the collision. ~t these energies the baryonic excitations and the mesonic degrees of freedom become important . In the framework of Waleckestype field theory, the BLE approach can be extended to the relativistic collisions. This _yields a ss~t of relativistic BLE's for the interacti~nE baryons and mesons in which the fluctuating collision terms are characterized by a correlation matrix [25~ . It will be interesting to investigate the influence of correlations on the particle production mechanism and the collective flow properties at high energies. Authors gratefully acknowledge Ch. Grégoire, G. F. Bertsch and P. G . Reinhard for many fruitful discussions . One of us (S. A.) thanks GANIL for the warm hospitality extended to him during his frequent visits. One of us (E. S.) thanks Tennessee Technological University for the warm hospitality extended to him during his visit . This work is supported in part by US-DOE grant DE-FG05-89ER40530 . REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11 . 12 . 13 . 14. 15 . 16 . 17. 18. 19.

G. F. Bertsch and S. Das Gupta, Phys. Rep. 160 (1988) 190. S. Ayik and C. Grégoire, Phys. Lett. B212 (1988) 269; and Nucl . Phys . A513 (1990) 187. J. Randrup and B. Rémaud, Nucl . Phys. A514 (1990) 339. P . G . Reinhard, E. Suraud and S. Ayik, Ann. Phys . 213 (1992) 204; and P. G. ;.ciz ~ard and E. Suraud, preprint GANIL-P9107 {1991), Ann. Phys. (N.Y.) (1992), in press . . Na anima, Prog . Theor . Phys. 20 (1958) 948 . R. W. Zwanzig, Quantum Statistical Mechanics, ed. P . H . E . Meijer {Gordon and Breach, New York, 1966) . H. Risken, The Fokker-Planck Equation (S ringer, Berlin, 1984j. M . Bixon and R. Zwanzig, Phys . Rev . 187 1969) 267. W. Bauer, G . F. Bertsch and S . Das Gupta, Phys. Rev. Lett . 58 (1987) 863. Ph. Chomaz, G. F. Burgio and J. Randrup, Phys. Lett. B254 (1991) 340; G. F. Burgio, Ph. Chomaz and J. Randrup, Nucl . Phys A529 (1991) 157; and F. Chapelle, et al., preprint LBL-30967 (1991) . E. Suraud, S. Ayik, J. Stryjewski and M. Belkacem, Nucl . Phys. A519 (1990) 171c ; E. Suraud, S. Ayik, M. Belkacem and J. Stryjewski, preprint GANIL-P9101., Nucl . Phys. A (1992), in press . C. Grégoire, B. Rémaud, F. Sebille, L. Vinet and Y. Raffray, Nucl. Phys. A465 (1987) 317. W. Cassing, V. Mettag, U. Mosel and K. Nüta, Phys. Rep. 188 (1990) 363 . P. Danielewicz, Ann. Phys. 197 {1990) 154 . M . Belkacemm, E. Suraud and S. Ayik, Preprint GANIL-92-07, submitted to Phys. RF;v. Létt . (1992) . J. Randrup and C . ~VI . Ko, Nucl . Phys . A343 {1980) 519; and 411 (1983) 537. J. Julien, et al ., Phys. Lett. B264 (1991) 269. S . Ayik, E. Suraud, J. Stryjewski and M . Belkacem, Z. Phys. A337 (1990) 413. W. Nörenberg, Phys . Lett. B104 (1981) 107.

,S. Ayik et al . I Tüe Boltzmann-Langevin model

C~~

üta, 2 . 3. 2 .

.

®renber and

.

ang, . Phys. A326 (1987) 69; and 328

(1987)

oilley, . Suraud, . Abe and S. Ayik, in pros. of XXX `Winter Meeting, or 'o, tal;~ (1992). . r é, . . Li and I~I. A. eidenYnuller, Phys. Rev. C27 (1983) 2063; and 3 (1986 209. . Abe, C . Grégoire and . Delagrange, J. Phys. (Paris) C4 (1986) 329. ush, G. . ertsch and . A . Brown, preprint LA-LTR91-=2748 and sub fitted to Phys . Rev. C (1991). . yi , Phys. Lett. 265 (1991) 47.