Momentum-dependent potentials: Towards the molecular dynamics of fermionlike classical particles

We investigate classical Hamiltonian models for particles interacting with steep difFerential repulsive barriers both in coordinate and momentum space. The final aim is to de6ne a classical system of many particles behaving as fermions in many respects. In this paper we examine the appear- ance of the phase portrait of one- or two-particle systems to skim the essential features that would later be transcribed to the basic rules of a molecular dynamics algorithm. One of the remarkable properties of the phase portrait is the How from states that start far away with a wide range of momentum towards a narrow region in momentum — a virtual locking of momentum — in the vicinity of the steepest part of the barrier in momentum space. The central ideas are developed through two examples in one and two dimensions.


I. INTRODUCTION
The ancient distinction between theoretical and experimental physics traditionally furnished the unique scenario to locate the objects of research in this science. This frame is no longer unique in view of the increasing importance awarded to computational physics. In particular, experiments on molecular dynamics or Monte Carlo simulations of many-particle systems considerably enlarge the field of many-body physics; lying halfway between laboratory experiments and analytic theories, numerical experiments are capable of describing both qualitative and quantitatively the phenomenology of a variety of large physical objects, making room for comprehension of the detailed microscopic events responsible for observable efFects.
Since computational facilities may severely constrain the possibilities to invest efforts in numerical experiments for systems with many degrees of freedom, simplified models involving, for example, two-dimensional fluids together with a discrete dynamical map deserve special recognition. In particular, the dynamics of particles subject to piecewise constant potentialsbare hard disks in the simplest versioncan be dealt with numerically by means of a nondifferential, event driven algorithm that allows an excellent computational performance [1]. The phenomenology covered by N hard disk systems ranges &om hydrodynamic instabilities to nucleation phenomena, through transport processes and dynamical response [2]. In spite of the considerable power of this approach, systems of indistinguishable particles described by fully symmetrized or antisymmetrized state vectors are not amenable, in principle, to a molecular dynamics experiment [3]. Attempts to submit an N-fermion system to a dynam-ical simulation have been put forward by nuclear physicists aiming at descriptions of the evolution of excited and compressed nuclear matter. On the one hand, the fermionic molecular dynamics developed by Feldmeier [4] is a semiclassical map of the actual N-fermion motion on the manifold of Slater determinants of single-particle wave packets and includes the antisymmetrization effects both in the exchange matrix elements of the interaction and in the matrix of the symplectic metrics. It gives an interesting qualitative description of many phenomena occurring in nuclear reactions [5]; however, since the method relies on a mean Geld variational approach, it may be subject to the short-time regime as indicated by several authors (see, for example, [6] and references cited therein). On the other hand, the quantum molecular dynamics [3,8,9] together with the Vlasov-Uehling-Uhlenbeck or Boltzmann-Uehling-Uhlenbeck [10,11] approaches represent serious advances in this direction; the fermions are regarded as classical particles undergoing Hamiltonian motion in an efFective, i.e. , densitydependent, mean field and may participate in two-body collision events controlled by typical scattering cross sections, together with a criterion to rule out those processes forbidden by Pauli exclusion. Another useful viewpoint [12 -15] intends to mimic the antisymmetric character of the many-fermion wave function allowing the particles to repel each other according to a potential that depends on the relative momentum of the interacting pair. The momentum dependent potential may be parametrized so as to account for the equation of state of nuclear matter at zero temperature predicted by variational Monte Carlo methods [16]. It is worthwhile recalling that the Grst investigation of high energy nuclear dynamics, i.e. , above 300 -500 MeV per particle, was carried by Bodmer et al. [7]. In this work, a momentum dependent interac-1063-651X/95/51 (3) 1995 The American Physical Society tion was considered, however, as an artifact to reproduce the asymptotic two-body trajectory or deflection function since exclusion effects are not relevant at the given energies.
Although the momentum potential models have proven to work satisfactorily to describe many interesting phenomena in nuclea dynamics, such as fragmentation of hot compressed nuclear matter [17], several questions may be raised concerning the validation of this philosophy. To the best of our knowledge, the Hamiltonian dynamics induced by these potential models has not been investigated beyond the examples of phase space trajectories presented in Refs. [12,14]. It is consequently interesting to examine them, referring to simple dynamical systems in order to ascertain whether they account for exclusion effects. It seems appropriate then to examine the basic features of the dynamics of systems of particles that repel each other in momentum space according to a simple momentum-dependent potential. The most relevant aspects of such dynamics may then be translated into simple collision rules to be inserted into an event driven algorithm such as the one in Ref. [1]. A simple justification for the use of these potentials may rely on the fact that in a real quantum liquid, exchange correlations manifest themselves in the total energy by means of nonlocal matrix elements of the pair interaction; this nonlocality in turn gives rise to inertial effects, namely, to renormalization of the bare mass into an effective, momentum dependent, mass. These efFects can be taken care of by momentum-dependent interaction terms. This view could give support to the choice of momentum dependent potentials to face an approach to the dynamics of a quantum liquid with classical particles. The purpose of this work is to advance the understanding of the virtues and capabilities of momentum dependent potentials to provide some of the features of fermion repulsion. The final goal is the formulation of collision rules for molecular dynamics simulations of hard particle fluids, aiming at future descriptions of quantum liquids. This task will be the subject of a forthcoming paper.
The present article will be mostly concerned with momentum dependent fields that prevent particles from reaching a given set of momentum values; we then investigate a selection of continuous potential models such that in each of them, the hard disk repulsion is softened and represented by a barrier with smooth edges. For the sake of completeness, in Sec. II we discuss a one-dimensional model with quadratic coordinate and momentum potentials in a finite region of phase space, which can be tackled analytically. In Sec. III a one-dimensional model for soft disk repulsion together with a smooth momentum barrier or well is presented and discussed. Oneand two-particle systems in two dimensions, resembling closely the hard disk problem, are examined in Sec. IV and a full d.iscussion is the subject of Sec. V.

MODEL
In this section we present and discuss a simple onedimensional model for a particle moving in the presence of two short range potentials in both coordinate and momentum space. These potentials are of the form ( )~2 (1g') 0 otherwise for y = x or p. The two-dimensional phase space then splits into four difFerent regions, namely, (i) the four force-free domains~x~& 1 and~p~& 1; (ii) the vertical strips~x~( 1, and~p~& 1 where only the accelerating force is present; (iii) the horizontal strips~x~& 1, and~p~( 1 where the momentum field is nonvanishing; and (iv) the unit square where both potentials are active. The specific orbits in regions (i) -(iv) are described in what follows. Region (i) Th. e phase flow consists of horizontal straight lines corresponding to uniform motion with velocity x = p. Region (ii). The particle Hamiltonian is (2.2) Accordingly, the phase trajectories are either hyperbolas for the repulsive barrier or ellipses for the attractive well.
In the former situation, two separatrices show up at p = +v E, corresponding to an energy E = E Region (iii). With the orbits are straight lines. The only effect of the momentum dependent potential is to renormalize the particle mass according to the velocity equation x = p(1 -E"). (2.4) It is then clear that for a repulsive barrier, the effective mass changes sign if E")1, causing the velocity to lie antiparallel to the momentum. Region (iv). The Hamiltonian can be written as E. + E"(1 -E"), 2 2 (2.5) In this case the orbits present several possibilities according to the sign of E (E +E")/2 foreach energy E and to the signs of E,E",their shapes being either hyperbolas or ellipses.
In this model, the full features of the phase portrait for each choice of signs and magnitudes of the potential parameters can be explicitly worked out. In particular, fixed. points and separatrices can be located; it is easy to realize that the Hamiltonian flow is structurally unstable since the character of the (0, 0) fixed point changes according to the parameter set, &om a saddle to a stable elliptic point, the latter appearing when E (1 -E") is positive. In Fig. 1 we illustrate the phase diagrams for the four possible sign combinatioos and indicate the characteristic lines. The appearance of the orbits within the unit square depends also on the relative sizes of the strengths E and E". MOMENTUM-DEPENDENT POTENTIALS: TOWARDS THE. . .

257'
The most interesting outcome of this study is the existence of librations even in the case of purely repulsive barriers both in coordinate and in momentum space as shown in Fig. 1. It is worth noting that bound periodic motion takes place for positive E"larger than unity; since the effective particle mass is negative under such conditions, the velocity antiparallel to the momentum prevents the particle from escaping downhill the two-dimensional barrier on the energy surface. This eKect is intrinsic to 2 energy surfaces with other than purely kinetic, i.e. , p momentum dependence, and has been thoroughly illustrated for systems with spectrum generating algebras of the su (2)

III. SMOOTHED SQUARE POTENTIALS
In the following we discuss the phase portraits for a kind of particle Hamiltonians involving soft sphere repulsion and a momentum dependent field. These potentials are selected in such a way that their limiting form is a square well or barrier. The analytic form chosen for the current investigation is for y = x, p. Infinitely large values of either parameter 6 give rise to a perfect square barrier or well, whose y strength can also be adjusted controlling the corresponding intensity E". The particle Hamiltonian is then 3 2) everywhere. We shall consider only positive E and both signs for Ez.
Typical energy surfaces and contour plots are shown in Figs. [2][3][4] Fig. 2 (a) we display the energy surface corresponding to equal strengths E = Ez --3 (repulsive momentum barrier) and core widths b = bz --10. The nature of the fixed points can be better appreciated in Fig. 2(b), where the phase portrait is plotted; it can be noticed that the How here shown is equivalent to that in Fig. 1(b), while difFerent choices of the model parameters give rise to contour plots equivalent to those in Figs. 1(a), 1(c) or 1(d). In the present case, we realize that five fixed points appear along the p axis, namely, a saddle point at the origin of phase space at an energy equal to the sum  Fig. 6 we illustrate this situation for the choice E = E"=3, 6 = 6"=10 (a) in coordinate space and (b) in momentum space. The initial position and momentum have been fixed at xo = 2, p 0 = -2, kayo 0.
The outstanding feature in Fig. 6(a) is the significant deviation of the low impact parameter trajectories with respect to the reflection expected for a hard disk scattering center. Indeed, for y & 0.5 the particle acquires a negative y velocity. Even for an impact parameter equal to unity, the particle is deflected forward, revealing the existence of a nonvanishing repulsion within the finite surface width of the potential barrier. The trajectories are no longer deviated for impact parameters larger than 1.1. This analysis can be complementedregarding Fig.  6(b). In the following we show that for the lowest impact parameters, the two-dimensional momentum becomes frozen at the surface of the momentum dependent barrier. This behavior is consistent with the feature encountered in the one-dimensional model discussed in Sec.
III [cf. Figs. 2(b) and 5], namely, the fact that scattering trajectories impinging on the barrier lock the outcoming momentum at the edge of the momentum dependent field. A similar morphology is apparent for negative Ep's and will be discussed in the next subsection in the context of a localized momentum well. I + e~m [(+1 +2) +(w1 w2)ij Furthermore, we choose the Hamiltonian This Hamiltonian has been devised to mimic two particles immersed in the Fermi sphere characterizing a manyfermion system at low temperature; each particle momentum is thus allowed to vary within a narrow cell in momentum space. The well (4.3) simulates this situation for cells of average diameter equal to unity with a small border width bp. The magnitude of this width, rather than the size of the cell, takes into account the relative washing out of Pauli blocking due to an increase in temperature. For the sake of an illustration, we consider a geometrically symmetric configuration where the particles are initially located with opposite position and momentum vectors. The eight-dimensional system of Hamilton's equations of motion reduces then to the four-dimensional one for a single particle. The problem then resembles that depicted in Fig. 6 a particle hitting a repulsive barrier however, with the efFective mass characteristics of the example in Fig. 5. We show some spatial oneparticle trajectories in Fig. 7, for the parameter choice Ep: 3 p 0: 2 & and pyo = 0, for the particle under consideration, the remaining parameters remaining the same as in Fig. 6. We now see that the efFect of the repulsive barrier at the origin is preserved to the same extent; however, the bounce at the barrier surface does not correspond to a perfect reflection. One also verifies that after the particle has scattered its momentum saturates with modulus~p~= 1, consistent with the phase trajectories indicated in Fig. 5.
An interesting feature of the flow in these models is the fact that for some initial momentum values, incoming particles may bounce twice on the soft disk barrier. An example of this situation is shown in Fig. 8(a)

V. DISCUSSION
As stated in the Introduction, the current motivation to tackle a study of momentum dependent potentials is to establish the extent to which they are able to simulate the quantum behavior of indistinguishable particles. The majority of the potentials proposed here serve this purpose for fermionlike particles: while barriers inhibit a finite momentum domain, wells explicitly allow a finite momentum region excluding the remaining space. The surface width of the barriers or wells is a control parameter whose vanishing gives rise to a steep potential step.
One of the predominant characteristics of Hamiltonian flows of one-dimensional systems is the peculiar role played by the inflection point of the momentum field for the (x, y) [(p,p")] projection of a phase space orbit with initial momentum ( -2.7, 0) for a two-dimensional particle with the potential parameters of Figs. 6 and 7. This phenomenon has been observed with other versions of momentum dependent potentials, more specifically, for two different choices with the relative momentum as the variable [12,14]. Doubly bouncing orbits such as the one depicted in Fig. 8 are the two-dimensional versions of phase trajectories through two zero velocity lines, such as those appearing in Figs. 2(b) and 5 in the neighborhood of (+1, -1) and in the region of localized phase orbits with~z [ (1. (chosen as unity in the displayed examples). In fact, one can see that the density of horizontal flow lines is high in the neighbourhood of these points. As already mentioned in Sec. III, this means that phase trajectories starting in a Beld-free region but pointing towards the repulsive, soft disk barrier accumulate near the abovementioned inflection point after the bounce. The price to be paid on dynamical grounds is a lowering of the kinetic energy, which could be attributed to an increase in the effective mass that took place near the scattering center. A similar effect is encountered in the two-dimensional systems discussed in Sec. IV upon examination of the trajectories in either configuration or momentum space.
It is also worthwhile mentioning that the Gaussian momentum fields employed in nuclear dynamics calculations [14] possess this feature as well.
This picture suggests that, for example, a projectile hitting a Fermi sphere with whose particles it interacts, on the average, through a soft repulsion, becomes captured on the surface of the sphere. On the other hand, if a particle is blocked inside the Fermi sea within a narrow allowed momentum cell, whose edges furnish a nonvanishing contribution to the particle velocity via the derivative of a smoothed stepwise momentum field, soft disk scattering events tend to lock the particle momentum on the boundary of its cell.
In spite of its simplicity, we believe that the models presented in this article encourage further research along these lines, destined to carry, for example, classical molecular dynamical experiments in two dimensions and to compare the bulk properties and both the thermal and the dynamical responses of the many body system obtained by these means with the predictions of some quantum many body theorye.g. , a mean field theory where antisymmetrization effects have been fully taken care of. This work is presently in progress and the corresponding results will be presented elsewhere.