Classical kinetic theory simulations using smoothed particle hydrodynamics James C. Simpson* and Matt A. Wood Department of Physics and Space Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, Florida 32901 Received 26 December 1995 The technique of smoothed particle hydrodynamics SPH is used to simulate a variety of three-dimensional systems comprised of elastic spheres contained in a box with perfectly reflecting walls. Particle interactions are determined solely by the conservative SPH body forces, from which the potential energy function is derived. This function is followed to monitor the conservation of the total energy as various initial nonequilibrium velocity distributions are quickly randomized by particle collisions. The resulting equilibrium speed and velocity distributions are found to agree with those predicted by kinetic theory. The algorithm conserves the total energy to within 0.02%. The pressure exerted on the box walls and the mean free path between collisions are comparable with those expected for a system of rigid particles. The problem of two isolated systems that are allowed to mix after an impenetrable partition is removed is also simulated with acceptable results. Finally, the equilibrium spatial distribution of the particles is considered and a semiemperical relationship derived for this multiply anticorrelated distribution. S1063-651X9601608-X. PACS numbers: 02.70.-c, 02.70.Ns, 05.20.Dd I. INTRODUCTION Advances in computational science have made computer simulations an integral part of investigations of classical sta- tistical mechanics see 1 for a general review. The starting point for such simulations is usually the interaction potential. A number of different choices have been used, beginning with the pioneering work of Alder and Wainwright 2,3 us- ing infinitely hard spheres surrounded by square well poten- tials, and followed by the introduction of the continuous Lennard-Jones potential 4,5. In this paper we do not specify the potential a priori, but rather start with the funda- mental fact that the momentum distributions in classical sta- tistical mechanics do not depend on the exact nature of the interaction between the particles within a system, and so can be expressed in a form applicable to all bodies 6. There- fore, any convenient type of interactions may be chosen for simulations of these distributions provided that no quantum effects are to be considered. A useful computational choice turns out to be the tech- nique of smoothed particle hydrodynamics SPH in which the particles may be thought of as macroscopic soft spheres see 7 for a general review. The SPH particle interactions are repulsive and purely hydrodynamic, depending on the local pressure, density, temperature, and interparticle dis- tance. Even though SPH has usually been used to model astrophysical situations with complicated geometries, such as mass-transferring binaries 8,9, stellar collisions 10, and the origin of the earth-moon system 11, it has many quali- ties that also make it well suited for basic molecular dynam- ics simulations. First, the conservative and continuous SPH interparticle forces are usually truncated at a definite separation distance, avoiding the need to compute long-range forces as well as the difficulties that can be associated with discrete particle interactions such as hard spheres or square wells 12. Sec- ond, the fundamental requirements of any SPH code to effi- ciently locate nearest neighbors and advance their phase- space coordinates means that it is straightforward to modify existing SPH code for simple molecular dynamics simula- tions 13. Third, all the phase-space information about every particle in the system is always available for statistical analy- sis, so systems can be studied in considerable detail. These are more than mere coincidences, but the simple explanation is that SPH already is a molecular dynamics technique that has been adapted to the needs of modeling the average properties of a fluid by using macroparticles and a smoothing procedure with a statistically significant number of neighbors usually 30 neighbors for each particle for three-dimensional systems. Hoover et al. 14 explore this application of SPH for molecular fluids. But here we use SPH to simulate rarefied classical sys- tems, so the need for a large number of neighbors is auto- matically eliminated, even though it is possible for any num- ber of particles to collide simultaneously. When this does occasionally happen, the forces are just the sums due to the individual binary collisions because SPH interactions are handled pairwise. Now, if these systems are restricted to some numerical volume where wall collisions are included by reflecting the particles’ normal velocities at specified boundaries, we then have an ideal method for performing simulations of classical systems governed by Maxwell- Boltzmann statistics. In the simulations presented below, we use 5000 marble-sized particles of arbitrary mass in a 1 m 3 box with perfectly reflecting walls. The particles are soft but elastic, and capable of storing elastic potential energy during collisions. We note that SPH particles do not have any spin angular momentum, so they should be regarded as smooth spheres, with only the momenta parallel to the lines joining particle centers altered during collisions. The rest of this paper is organized as follows. In Sec. II we give a brief introduction to the theory of SPH. The details of the numerical method are given in Sec. III. The interaction potential is derived from the momentum equation in Sec. IV. *Electronic address: simpson@hubble.pss.fit.edu, wood@kepler.pss.fit.edu PHYSICAL REVIEW E AUGUST 1996 VOLUME 54, NUMBER 2 54 1063-651X/96/542/20777/$10.00 2077 © 1996 The American Physical Society