Quasi-stationary description of fluctuations

Nuclear Physics A545 (1992) 71c-80c North-Holland, Amsterdam

NUCLEAR PH YSICS A

QUASI-STATIONARY DESCRIPTION OF FLUCTUATIONS A.BONASERAI), F.GULMINELLIz), P.SCHUCK3)

1 . Laboratorio Nazionale del Sud and INFN, sez. Catania, c.so Itdlia 5T, 951,29 Catania, ITALY ~. Dipartimento di Fisica and INFN, v.Celoria 16, ,~01~~ Milano, ITALY ~. Institute des Sciences Nucleaires, F-°3806 Grenoble-Cedex, FRANCE Hi~~T11,AC'T A general method to evaluate fluctuatio~is of the semiclassical o~ie body Wigner function, in the framework of Landa.u's theory of quasi-stationary fluctuations, is prese~ited. The model is valid far away from critical points, where the Boltzman~i Nordheim Vlasov equaüo~i represents a good approximation of the average Wig~ier functio~i. We discuss the different contributions of mean field fluctuations and fluctuations coming from the scattering processes, and apply ~.he model to the evaluation of the damping of collective modes. The relaxation of giant resonances bililt on excited states is also discussed. ~Ylt~~~llCt10Yl

iii recent years a great interest has been raised about the problem of fluctuations in nuclear physics. With the availability of HI experimental facilities at intermediate and relativistic energy, it is now possible to build up portions of nuclear matter far awây from equilibrium, at a density which is easily twice the saturation density and an excitation energy the,} can easil,-y reach th:~ bin~.ing energy per nucleon. In such a situation critical phenomena are expected, and it is probable that high order correlations and the dynamicsa propagation of fluctuations in the nuclear medium can lead to criti~~al behaviours, like nuclear multifragmentation and phase transitions. In principle the kinetic equations represent an ideal 'cool to analyze microscopically and self-consistently the dynamical evolution of a heavy ion collision . In particular, the time evolution of the one body semiclassical Wigner function is very well described by the Boltzmann Uehling Uhienbeck (BUU) or Boltzmann Nordheim Vlasov (BNV) equation 1 . However the BNV equation presents serious drawbacks for an application in the critical regime. First of all, the only correlations included among the particles are Pauli principle correlations and short range incoherent scatterings, while N-body i~~formations are lost . This prevents an application in the multifragment regirr~e . 0375-9474/92/$05 .

c~ i992 - Elsevier Science Publishers B.V . All rights reserved .

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~. ~n®aascra et ctl. I Qttasi-.stationary description ~fflactuations

Moreover , in order to derive the BNV equation from the BBGKY hierarchy for the density matrices, the dynamcal correlations caused by collisions are supposed to be very well localized both in space and in time . Then the two body distribution function is completely uncorrelated except at the very occurrence of a collision . Such assumption implies that the one body distribution function exhibits a weak dependence both on â (quasi homogeneous assumption) and on t (Markov process with ~ correlation function} and gives only the average behaviour at the one body level. As a consequence, the fluctuations induced by the scattering processes are averaged out before they can be propagated by the mean field, and the BNV equation cannot describe the approach to critical conditions . Even if in an actual HI collision no critical phenomenon is happening, it would be very useful to know the fluctuations of the exact microscopic distribution function, with respect to the Boltzmann approximation . As a matter of fact, in the computation of one body observables, the average distribution function f allows us to reproduce only mean values, while no informations on the variances of the observables is achieved . I~.ecently 2 we have introduced a method to calculate the fluctuations of the one body distribution function, based on the theory of quasi-stationary fluctuations . This has to be regarded as a minimal approach to the problem of fluctuations in nuclear physics, since only the fluctuations inherent in the average Boltzmann dynamics are explicitely evaluated, and at variance with other approaches s no attem?~t i5 loads to introduce the coupling to the correlated part of the two body" distribution function The model is briefly recalled in section 2., where some comparison with analytical results for a homogeneous gas are shown . Section 3 . addresses the problem of mean field fluctuations and shows some realistic applications to the damping of the Giant I~i~.~ole ILesonance at zero and finite temperature . Co~iclusions are finally presented in ce('.tion 4 . 2.

ese~iption ®f the M®del In the general case the cor cia,tto~i func;tio~~ of the fluctuations can be written as ~(ri~ ra, ki~ ~zv

t l , t2) _ (b f (ri ,

~ i~ ti)bf 1ra~ ~2~ t2))

wü.h bf(r,k,t) = f(~,k ;t) - f(~,k,t) . Here, f is the exact microscopic disLr-ib~ainn function which in principle should he obtained by solving the complete Liouville equation, and f is the solution of the BNV equation . The average () leas to be interpreted as a time average, and is equivalent to an ensamble average in the ergodic limit. In the nuclear context so far only ensamb!A ~~~~rag~~ have been employed s . Ii the fluctuation ~f is small it is possible to use flue tiirVâ- y of quasi-stationary fluctuations 4,s where the correlation function of fluctuations is the solution of a linearized Boltzmann equation ; therefore in this case it is, at least in principle, pos-

A. Bonasera et al. l Quasi-stationary description offluctuations

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sible to evaluate eq. 1 provided the initial condition, i.e . the equal time correlation function, is known . If we are dealing with an equilibrium gas the correlations between simultaneous fluctuations are propagated at distances of the order of the nucleon-nucleon interaction, which is regarded as zero in BNV theory. As a consequence, the equal time correlation function is a ~-function whose coefficient is the mean square fluctuation at the given phase space point . As it is well known s, in an ideal gas this is equal to the mean value of the distribution itself. The initial condition for a Maxwell gas reads Q12`0) - fo`1)~`rl - r2)~(~1 - ~2) where for simplicity we have written Q(ri, rz, kl , kz, 0) - Qiz(0) and fo(i) - fo(T~, p=, t) is the finite temperature equilibrium distribution function . In the Fermi Dirac case non zero range fluctuations appear as a consequence of antisymmetrization, that creates correlations among particles . However, the expression for the equal time correlation function at ri = iz is similar to eq. 2 Q(r, ~i~ ~z~ 0) v(0, ki )

= fo(r~ ~i)b(1~i - kz)b(r) ~+ Îo(r, ki)~(T, ki) _ _ fo (0~ ki )

(3)

Once the initial condition eqs. 2,3 is known, the finite time correlation function can be evaluated at any time by solving the linearized Boltzmann equation and this gives s,~,z Qrz(t) - (Qiz(0))e~P(-tIT) where T is the correlation time - of fluctuations and corresponds to the relaxation time of the perturbation 7 . Eqs . 2-4 show that the fluctuations of the distribution function are solely determined by the Boltzmann equation itself. This should not be surprising since ïn the equilibrium case we are dealing only with statistical fluctuations . In the non equilibrium situation the evaluation of the correlation function is more complicated . If the fluctuations are not too large, the correlation function again satisfies 4 a Boltzmann-like equation, but the initial condition will now be given by 4 (s) Qr2(0) - \ f 12/ - ~f 1f2~ + ~Î1) b ( r l - r2)~(~1 - ~2) which contains the average two body distribution function f iz . Thus, a solution of the second equation of the BBGKY hierarchy is required or some physically reasonable approximation for f iz has to be found . One very promising pos "i~;ility 3 is to make use of the fluctuation-dissipation theorem and to treat tlhe time evolution of the fluctuations as a Poisson ~~rocess, under the action of a random force as a proper generalization of the Boltzmann equation . Here we would like to examine, as a zero order approximation, the minimum fluctuations contained in the equation, without adding any extra physical ingredient

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A . ~onasera et al. I ~ltsasi-stationary description offluctuations

3r

5

1 1 .5 2 2.5 0 0.5 1 1 .5 2 2.5 k (fm- ~) k (fm-1 )

Fib . 1 : Equal time correlation function veasus momentum for an homogeneous gas in equilibrium at a t.emperat.ure T = ß0 MeV and a density p = .15 fm -3 . Full line: numerical correlation function . Left side: Maxwell Boltzana~api . Right side: Fert~ii Dirac . Crosses : Eq. 2,3 respectively.

to the simple Eoltzmann equation for the one body distribution function . This means we put (6~ {f~~~ _ ~f~f2> neglecting all the simultaneous correlations between different phase space points . With this approximation the correlation function of the fluctuations can be still be calculated with no other information but the Boltzmann average distribution function . The numerical procedure is straightforward : the distribution function is initialized with the test particle method 1 and the BNV equation is solved in the standard way i . The procedure is repeated twice with different initial condition for f. Using these two runs we can compute the fluctuation ô fat each time step, and this in turn gives the correlation function z via Eq. 1 . Equation 6 corresponds to assume small deviations of the real distribution function with respect to the Boltzmann solution . T}~is assumption is certainly not at all valid near to a critical point, when long range correlations play an important role and even the Boltzmann average distribution function f is incorrect . We thus expect this method to give the exact solution in equilibrium, and to provide a good approximation only far away from critical points, iii particular ibis should be a reasonable approximation in the stuây of small oscillations around equilibrium. An example of a comparison between the analytical result Eq. Z, 3 and the equilibrium equal time correlation function, numerically obtained with our model as a particular case from Eq. 1, is given in Fig. 1. The method is seen to reproduce very accurately the analytical expression, both for the classical and the Fermi Dirac distribution .

A. Bonasera et al. l Quasi-stationary description of fluctuations

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r

102

r

0 .5

1

Es (MeV/u)

5

10

Fig . 2 : Relaxation time extracted from the correlation function at different excitation energies in a~i homogeneous Ferrai gas . Square : quadrupole deformation, with an amplitude determined by the GQR fenomenology for a C'a nucleus . CI'OSS

Diamond : equilibrium gas at a. temperature T=8 .37 MeV .

Sylllb0l : equilibrium gas at T=20 MeV . The ~iur~~erical error bars are within the dimensio~is

of the symbols . Full line : eq . 7 from ref . 7 .

With the approximation given in eq. 6, ey. 4 is expected to be satisfied also in a non equilibrium dynamics ~. This is confirmed by our calculations 2 , and it turns out that the relaxation time ~ is solely determined by the excitation energy of the system â::d not by the form this energy takes, in agreement with Fermi ïiyuid theory ~ . As a matter of fact, an excited Fermi gas has a relaxation time given by ~ 1 T

r..

E D -}- ET

Equation 7 means that a given T can be obtained by giving a quadrupole deformation ED or a thermal excitation ET or a combination of the two such that the total excitation energy E = ED + ET is a constant . The same result is obtained in our model, as it is shown by Fig. 2. Here we have plotted the slope of the correlation fun~;ti~n calculated by the numerical solution of eq. 4 for different equilibrium and non equilibrium ~q_uadrupole deformation) initial conditions, in comparison with eq. 7. The pa,r~,rnetPr~ of t.hP calcul~,tior. ~Fermi Lr~rgy, Yauli blacking . . .) are chosen such to compare to the analytical calculation of ref. 7. 3 . Damping of Collective 1®~Iotion

All the model applications that we have s}sown in the previous section referred I,o an homogeneous gas, i .e. the effect of the mean field was neglected . In this

~, ~mtcasc~ra ct ctl. I Qaaasi-stationary dcscrption offluctuations

itec

3vv 250!-

0

1

2 E'(MeV/u)

3

4

Fig . 3: Full 1î11e: slope of the correlation function versus excitation energy for a compressed Ca alucleus evolved by V11sov dynamics. Symbols : time at which the first emitted test particle escapes frolal t.lle 1lclcleus, 1gä111 1s 1 function of compression energy.

simplified situation the main source of fluctuations is given by the collision integral, as it has already been discussed by other authors s . However in our approach mean field fluctuations can be treated unambigously on the same ground, by simply substituting the Boitzmann equation with the Vlasov one. In the following we will study the case of small oscillations of a Cd nucleus and follow its time evolution for a time T ~ 50U fm/c . Particular ~.ttention must be given to the numerical procedure so to hw~.-e smooth solutions witl-i very good energy conservation, and a true non oscillating ground state with a Q(t~ = 0 correlation fiinction . VVe use 20000 test particles per run, and energy is conserved out of 0.2% in 500 ftn~c. A Skyrme-like mean field of compressibility 200 MeV is used . Giving an arbitray small compression plus deformation to the system, a non zero mass number (i.e. integrated over r and p space) correlation function results . lts behaviour is near t~o an exponential decay in agreement with eq. 4, and the anechanism responsible for the decay 2 is the escape of particles from the excited s,ucleus, as it. can be seen from an inspection of Fig. 3 . Acre, the relaxation time extracted from the correlation function, in a Vlasov calculation, is plotted as a function of the excitation energy deriving from different initial compressions. The linear decrease is due to an almost linear increase of the escape rate. As we have shown in ref. 2 this can he interpreted as the Landau Damping for finste nuclei . The nucleus is coherentl~r oscillating with a certain frequency w~, and the const.i t.uents nucleons are oscillating as well . If a particle is in phase with the coherent field, the average `vork of the field on the nucleon will be non zero. Such

A . Bonasera et al. l Quasi-stationary description of fluctuations

77c

a work can be either positive or negative, which is the main difference with reâl damping due to viscous forces . Since there are more particles with a frequency w < w~ than particles with w > w~, a net transfer of energy from the colitctive to the single particle degrees of freedor~i results, and this leads to the damping of the collective motion . What happens in practice z is that particles in phase with the coherent field are in resonance with it, therefore the amplitude of their motion increases with time until the mean field is uncapable of keeping them in the nucleus, and the collective energy transferred to the particle is definitely lost . By extracting the initial deformation energy of Mue system from giant resonances systematics, it is possible to connect these Landau
PF(P(r))~ The relaxation times extracted from the rate of, particle escape and those extracted from the damping of the quadrupole momentum in p space are in very good agreement for all the temperatures studied . This means that even with the inclusion of the collision integral, the main source ~f t;"~e damping is still given by the escape iî~~ partiLr leS frc~rr the ~~iï. t~:(1 nllif:l~ll~ . From fig . 4 we can also see that the relaxation time is a decreasing function of the excitation energy, that is no saturation of the width is observed up to 3 MeV of initial temperature. Moreover, even at the higher temperature the extracted width t = ~./r -"- 2 MeV is much lower than the measured width$. This implies that the opening of phase space at high temperatures is not the dominant effect in the increasing of the widths, also if the neglected temperature effect on the collision integral could at least partly explain this discrepancy. 4. Conclusions

In this paper we have presented a new method ~ to e=.~aluate fluctuations in the

7$c

A . ~onasera et al. l Quasi-,stationary description offluctuations

200 300 t (fm/c)

400

500

Fig . 4: Giant Dipole Resouauce of a Sn nucleus . Upper part: quadrupole momentum as a function of t,iteee (full line) and fit with a damped harmonic oscillator with decay time T=258 fm/c (dashed line) . Lower part: member of particles escaped from flee nucleus versus time and corresponding slope in Ease/c for different. initial temperatures of the GDR . Full line: T=0 MeV . Dashed line: T=1 .5 MeV . Dashed dot,t.ed line: T= 3 MeV .

Mine evolution of the average one body distribus_.ion function, both in the classical and in the quantum case . The good agreement with the analytical results from fluctuation theory ~~' ensures that the model is physically well founded and that almost no numerical noise is added to the real physical fluctuations . Since time averages are used instead of ensamble averages, only two full ensamble BlETV runs are needed to compute the correlation function of fluctuations . This allows complete six dimensional calculations and realistic applications t,o nu clear dynamics . In the limit of spatial homogeneity the main source of fluctuations is given by the collision integral, but mean field fluctuations can be treated una bigously- in the same way. In this collisionless situation we have shown that a non zero correlation function can be obtained, and we have discussed the role of Landau Damping in the relaxation of collective modes . The inclusion of two body collisions in a self consatent calculation for collective modes is seen to give some contribution to the damping, especially for giant resona,nces built on excited states . In this case we obtain widths which exhibit a monotonically increasing behaviour with the excitation energy, but that are sistematically smaller than the ones shown by experimental data . The model breaks down if the fluctuations get too large, i.e. approaching a critical region . As a consequence, a comparison with the experimental widths of the distributions of one body= ~~}~servables in HI collisions will be very interesting, since it could give informations about the approach to critical phenomena.

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References 1 . A.Bonasera, F.Gulminelli, Phys .Lett . 275B(1992)24, P.Schuck et al. Progr . Part. Nucl . Phys. 22(1988)181 . 2. A.Bonasera, F .Gulminelli, P.Schuck, Phys.Rev .C, in press 3. S .Ayik, C .Gregoire, Nucd.Phys.513A,187 (1990) . E.Suraud, S .Ayik, J .Stryjewski, M.Belkacem, Nucl .Phys .519A,171c (1990) . P.Chomaz, G .F.Burgio, J. Randrup, Nucl.Phys.514A,339 (1990), and 529A,157 (1991) . 4 . E.M. Lifshits and L .P. Pitaevskü, Physical Kinetics (Pergamon press, New York, 1981) . 5. L.D.Landau and E .M.Lifshits, Statistical Physics, (Pergamon Press, New York, 1981) . 6. S.Ayik, E.Suraud, J .Stryjewski, M .Belkacem, Zeit.Phys.337A,413 (1990) . 7. G .F .Bertsch, Zeit.Phys.289A103, (1978) . 8. A .Van Der Woude "Giant Resonances", ed . J . Speth (World Scientific, Singapore)1990 . 9 . S .Ayik et al . MSU reprint 1991, in press . 10 . A .Smerzi, A .Bonasera, M .Di Toro, Phys. Rev . 44C(1991)1713 . 11 . A .Bonasera, F.Gulminelli, P.Schuck, Proc . 8th Winter Workshop aii Nuclear Dynamics, Jackson Hole, Wyoming (1992), W .Bauer et al . eds .