Introduction This contribution reports on recent progress [l]towards developing a quasi-classical simulation model for nuclear dynamical processes, such as can be generated in nuclear collisions at intermediate energies. The ultimate goal of such efforts is to develop a dynamical model for nuclear systems within the framework of classical mechanics. We here report a tentative first step consisting of simulating the non-interacting nucleon gas by means of a momentum-dependent repulsive two-body interaction potential to simulate the Pauli exclusion principle. Such an approach was first taken by Wilets et a1.. In that work, a repulsive momentumdependent Pauli potential was postulated and the parameters of an ordinary two-body potential were adjusted to fit certain gross nuclear properties. Although the model met with some success, it was never demonstrated that the phase-space distribution of the nucleons is actually well approximated. Since this property is expected to be important for the dynamical behavior of a colliding system, there is a need for scrutinizing the problem. Therefore, we have reconsidered the problem of determining an appropriate Pauli potential, and we have found that it is in fact possible to obtain a reasonably good reproduction of the important features of the Fermi gas, with a suitable choice of potential. Our Pauli potential is of Gaussian form and thus differs from that employed in ref. . We see our result as a possible first step in a program, of which the next step is the inclusion of a real two-body interaction for the purpose of describing both nuclear matter and finite nuclei. If this proves possible, the foundation is laid for an interesting dynamical model for the evolution of a highly excited nuclear system far from equilibrium, including its disassembly into multi-fragment final states.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987221
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General considerations We consider a system of N structureless identical classical particles in a Euclidian space of dimensionality D. The momentum and position of a given particle i, i = 1,..., N, are denoted by pi and qi, respectively. The particles are interacting via a momentum-dependent two-body potential, so the total energy of the system is represented by a Harniltonian of the form
Here Ti= pz/2m is the kinetic energy of particle i and the two-body interaction energy between particles i and Ji is Ej = =(pij, qij), where pij = [pi- pi[ and qij = Iqi - g,l. The prime on the summation in the interaction term is intended to remind of the convention that the sum over j should exclude the self-interaction term corresponding to j = i (although the inclusion of K; would be of no dynamical consequence). The dynamical evolution of the particles is governed by Hamilton's equations,
The momentum dependence of V produces a variable effective mass, so that the standard relationship between velocity and momentum, p, = m&, is replaced by the more general form p* = M . ql, where M(pl, ...,p ~q1,, ...,m) is the effective mass tensor. The ground state of the system has the minimum energy and it is therefore a solution to the 2 0 coupled equations p* = 0 and q, = 0. Such a state represents a frozen configuration in phase space, with the particles having finite momenta despite their vanishing velocities. Thus the associated effective mass tensor is divergent. This illustrates how the kinematics is strongly affected by the Pauli potential. In order to implement the qualitative concept of phase-space proximity, implied by the exclusion principle, it is necessary to define a metric in phase space. This can be accomplished in a convenient manner by introducing the dimensionless distance slz between the two phasespace points (PI, ql ) and (pz, 92),
where p12 and q1, are the distances between the two phase-space points when projections are made onto momentum and position space, respectively. The parameters p0 and qo are suitable characteristic scales for relative momentum and position and are further discussed below. In the present study, we present results obtained with Pauli potentials of Gaussian form,
Here V. determines the overall strength,
whereas p0 and qo determine the ranges in momentum and position space, respectively. This Gaussian form has the special advantage that V depends on p and q through the dimensionless separation s only, and yet V is separable with respect to momentum and position. In fact, V factorizes totally with respect to all 2 0 dimensions of phase space. The complete separability with respect to position makes the inclusion of neighboring cells in a periodic system especially economical - a distinct calculational advantage.
Roughly speaking, the parameters p0 and qo determine the size of an effectively excluded volume around each particle in phase space. Indeed, they can be thought of as the extensions in position and momentum of a spheroidal excluded volume around each particle. It should be noted that the range appearing in the two-body Pauli potential is twice the radius of the effective excluded volume around each particle, as is easily visualized by considering a system of hard spheres. In order that each particle block a phase-space volume of magnitude hD, we would therefore expect p. and qo to be related by poq0 E 2E, a sort of "uncertainty relation". [In fact, we should expect the product to be slightly smaller than that, with the exact number depending on the dimensionality D, because of the geometric fact that hard spheroids can not be packed to uniformity (whereas hard cubes can), but always leave a certain fraction of the space empty.] When considering an entire ensemble of similar systems, as we shall do, it is convenient to deal with the N-particle distribution function f (N)(pl,. . . , p ~q, l , ...,qN). Its form in thermal equilibrium is given by where H is given by (1) and T is the temperature. A statistical sample of representative Nparticle states can be generated by the Metropolis procedure . The reduced one-particle distribution function f(')(p,q) is of particular interest. If the ensemble is translationally invariant, then f ( l ) is independent of position and the spatial dependence of the reduced two-body distribution function f ( 2 ) is only via the difference A q = q - q'. The total energy of the system is then given by
where R = J d q denotes the total volume of the system. Since R + m, it is practical to introduce the energy per particle E = E I N , which is well-defined since N I R = p, the specified particle density. In practice, such a system can be approximated by a periodic system with a sufficiently large periodicity. Preparatory studies In order to gain some familiarity with the model, it is useful to consider the idealized situation where the two-body correlations can be neglected, f(') E f(')f('). This situation is expected to arise when the spatial range qo is Iarge in comparison with the typical spacing between neighboring particles (which is The ground state of the system is determined by the requirement that the energy be a minimum, for the specified density. It is especially simple to analyze the situation when the momentum range p0 is small compared with the characteristic momentum scale for f (p). By performing the variation p6.5 - Asp, it is elementary to show that the solution to this problem is given by a simple parabolic distribution. In order to approximate the momentum distribution for the corresponding Fermi gas, we demand that their second moments match, i.e. that they have the same mean kinetic energies. This requirement then determines the interaction strength V. in terms of the equivalent Fermi energy EF = P;/2m (where the *equivalent Fermi momentum PF is related to the density p through the relation pertaining to a standard cold Fermi gas):
This amounts to Vo/eF = 1.43, 1.13, 0.93 for D = 1, 2, 3, respectively. The Pauli potential prevents the particles from assembling at zero momentum, as they would in the absense of the momentum-dependent repulsion, and it actually produces a phase-space occupation that is fairly close to the correct quantum-mechanical one (which is unity up to the
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Fermi energy), see ref. 111. This result is encouraging, particularly since the present idealized treatment, in which the two-body correlations are neglected, will tend to overestimate the occupancy, so that there is reason to expect that a more refined treatment will produce even better results. It should be noted that there is little effect of increasing the momentum range p. from zero to a finite value, as long as it remains small in comparison with PF, which characterizes the momentum over which f ( p ) varies significantly. We therefore expect the analytical results obtained for p0 = 0 to be of relevance also for moderate finite values of po.
Figure 1. The figure shows the mean kinetic energy as a function of the temperature, for three different densities characterized by the Fermi kinetic energies TF = 18.5, 37, 74 MeV. The dashed curves are the exact Fermi-gas values. The results obtained from the Metropolis calculation with our standard Pauli potential (4) are shown as shaded bands whose widths correspond to the associated statistical error.
The nucleon gas After the above preparations, we now address the three-dimensional nucleon gas. As an approximation to the infinite system, we consider a sample of spatially periodic systems. Each elementary cell is of cubic shape and typically contains 64 particles. The periodicity condition is easily imposed by letting a leaving particle reenter the cell from the corresponding point on the opposite side with the same momentum. The interaction of the explicitel~considered particles with particles in neighboring cells is taken into account to sufficiently high order, typically the inclusion of two layers of neighboring cells suffices. [As already noted, this is relatively easy and economical to accomplish, due to the spatial factorization of the interaction.] A thermal sample of states is generated by employing the Metropolis procedure . The following set of parameters proves to yield quite satisfactory results and they have been adopted as a preliminary set of standard values:
7 4 0 MeV
a 12 Momentum
Figure 2. The figure displays the phase-space occupancy f ( l ) as a function of the magnitude of the for four different values of the temperature. The dashed curves momentum (in units of MeV.l~-~~s/fm), show the exact Fermi-gas values, while the histograms are calculated with our standard Pauli potential (4).
= 34.32 MeV , p. = 2.067 ~ e V . l 0 - ~ ~ s / ,f m qo = 6.00 fm .
= The strength V 0 has the value suggested by the above idealized treatment for D = 3, 0.9276 e ~ see , eq. (8). Furthermore, the product of the ranges is poqo = 1.886, which is quite consistent with our expectation of a value somewhat less than two times h. [In applications to nuclear systems, it must be recalled that each nucleon has a four-fold spin-isospin degeneracy. Since nucleons with different spin-isospin components do not interact via the Pauli potential ordinary nuclear matter would consist of four separate systems of the type discussed here.] The Pauli potential is only important when the relative momentum is comparable to, or smaller than, the range po. For the smallest relative momenta, the repulsion is so strong that the effective mass becomes temporarily negative. [As already pointed out in ref. , this
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feature makes the relationship between momentum and velocity multivalued and thus precludes a Lagrangian formulation.] In figure 1 we show the mean kinetic energy per particle as a function of temperature. Three different densities have been considered, corresponding to the standard Fermi energy and both twice and half that. [This corresponds to normal saturation density and densities 2.83 times higher and lower than normal, respectively.] This should cover the range of densities occuring in those nuclear collision processes towards which this work is oriented. It is seen that the mean kinetic energy tracks well the behavior of the Fermi gas as a function of both the temperature and the density. The more detailed behavior of the momentum distribution is illustrated in figure 2,which displays the phase-space occupancy f (p, q) as a function of the magnitude of the momentum. [In the present system f does not depend on the position, due to the translational invariance, and only depends on the magnitude of the momentum, due to the isotropy.] For the standard value of the Fermi momentum, a broad range of temperatures is considered, from close to zero to around the Fermi kinetic energy. The agreement with the results for the corresponding Fermi gases is remarkable. The most prominent deviations occur at the lower temperatures, where a peak in f appears at a small finite value of p. This is largely an artifact of the finite cell size employed: at low temperatures the particles prefer to organize themselves into fairly regular structures and this produces pronounced undulations in f . If an additional average were made over an interval in the periodicity of the order of the range go, these undulations would be reduced. With this in mind, we therefore consider the agreement between the model calculations and the Fermi gas results to be quite satisfactory.
Effective temperature For the present discussion it is important to recognize that the presence of a momentumdependent interaction blurs the concept of the kinetic energy. The quantity introduced above, Z'i = &p:, is based on the canonical momentum p;, as is appropriate in a Hamiltonian formulation. The corresponding kinetic energy per nucleon is given by
and is particularly useful for characterizing the momentum distribution of the nucleons in the system. An obvious alternative quantity is the "kinematic" kinetic energy K; = yq?,which is based on the velocity qi and appears in a Lagangian formulation. Its mean value is
and characterizes the velocity distribution. The analogous quantity ;pi . q; is of special significance in dynamical applications since it characterizes the momentum transport in the system. Indeed, its mean value is proportional to the trace of the stress tensor P = p < p q >, where p is the particle density. The standard pressure P is given by
where > indicates that an additional average over the D orthonormal directions A must be performed. [That n . P .?I is indeed the pressure in the direction ii is easily verified by the standard elementary consideration: the particle flux is given by and each particle deposits the momentum 2p, upon reflection, so the momentum deposit per unit area and time is p =ii.P.ii.]
In dynamical simulation studies of nuclear collisions it is of interest to determine the effective temperature in the system (which may depend on position if the system is not translational invariant). In the present context, this problem can be stated as follows: Imagine that we are presented with a configuration (or several configurations) sampled from a thermal ensemble with a definite but unknown temperature T. Our task is then to estimate the temperature characterizing the ensemble from which the given configuration is picked. More generally, one would like to calculate that temperature that would characterize the statistical equilibrium of the system at the given energy density, also when the actual system under scrutiny is far from equilibrium. In a standard classical gas with only momentum-independent forces that problem can be trivially solved by employing the (local) kinetic energy per particle, which is uniquely defined. Indeed, any of the above considered quantities, >, < mqi >>, or >, would yield the same effective temperature, since p = mq. However, the momentum-dependent interaction in our present system makes the situation more obscure, as explained earlier, and a more careful approach is called for. After having studied the situation for our present model system, we have found a method which may prove to be of general utility. Our prescription for extracting the effective temperature suggests itself when the standard thermodynamic relation P = p~ is invoked in conjunction with the above relation (12), namely
This expression provides a general method for extracting a useful effective temperature T,E for the system, applicable to any quasi-classical assembly of point particles. As an illustration, we show in figure 3 the effective temperature T,R calculated with this relation for samples of 150 states picked from thermal ensembles of various specified temperatures T. Not surprisingly, the correspondence is perfect.
Imposed temperature T (MeV) Figure 3. For a number of different temperatures T ,the effective temperature ree has been calculated by employing the proposed relation (13) to thermal samples of 150 configurations, each having 64 particles with the same spin and isospin.
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By restricting the average to particles located in a suitable region around a given position q of interest, the above prescription can readily be generalized to yield a local effective temper-
ature reR(q).The practical utility of this method remains to be determined from applications with dynamical simulation codes. Such efforts are presently underway. Summary In the present paper, we have discussed a system of classical particles interacting via a repulsive momentum-dependent two-body potential. We have illustrated some of the general features of such a system and, in particular, we have, for the first time, demonstrated that i t is possible to achieve a fairly good imitation of the nucleon gas, with a judicious choice of the parameters in a Gaussian potential. This result holds promise that it may be possible to develop an interesting model for intermediate-energy nuclear dynamics within the framework of classical equations of motion. Furthermore, we have proposed a simple prescription for extracting the local effective temperature that may prove useful in dynamical simulation studies.
This work was supported by the Director, Office of High Energy and Nuclear Physics of the Department of Energy under contract DE-AC03-76SF00098, the Consejo Nacional de Investigaciones Cientificas y Tecnicas, Argentina, and the Conselho Nacional Pesquisas e Tecnologia, Brazil. References 1. Part of the presented material has been published in C. Dorso, S.Duarte, and J. Randrup, Phys. Lett. B (1987) in press and more details may be found there.
2. L. Wilets, E.M. Henley, M. Kraft, and A.D. Mackellar, Nucl. Phys. A282 (1977) 341; L. Wilets, Y. Yariv, and R. Chestnut, Nucl. Phys. &OJ (1978) 359
3. N. Metropolis, A.W. Rosenblut, M.N. Rosenblut, A.H. Teller, and E. Teller, J. Chem. Phys. 2 (1953) 1087 4. D. Boal, work in progress, private communication (1987)