A Consistent Boltzmann Algorithm

. The direct ~imulation M?nte Carlo method for the Boltzmann equation is modified by an additional displacement m the advectiOn process and an enhanced collision rate in order to obtain the exact hard ~phere equationof state at all densities. This leads to consistent thermodynamic and transport properties m th_e ~ow density (Boltzmann) regime. At higher densities transport properties are comparable to the pred~c~wns of the _Enskog model. The algorithm is faster than molecular dynamics at low and moderate The direct simulation Monte Carlo (DSMC) method is a particle-based, numerical scheme for solving the nonlinear Boltzmann equation for hard spheres (HS) [1-3]. Rather than exactly calculating successive HS collisions, as in molecular dynamics (MD) [4], DSMC generates collisions stochastically with scattering rates and postcollision ve locity distributions determined from the kinetic theory of a dilute HS gas. DSMC encounters the usual inconsis tency of the Boltzmann equation; namely, it yields the transport properties for a dilute HS gas with diameter u, yet has an ideal gas equation of state (implying u = O) [5]. In this Letter a modification to DSMC is introduced which removes this inconsistency and recovers the exact HS equation of state at all densities with virtually no ad ditional computational cost. This consistent Boltzmann algorithm (CBA) has transport properties that are in simi lar (in some cases better) agreement with HS MD than Enskog theory [6]. In the standard DSMC method the positions and velocities {7\, vJ of the particles (mass m) are evolved in time by two steps: advection and collisions. During the advection step all particles are simultaneously propagated a distance vi8t, where the time step 8t is typically on the order of the mean collision time. The particles are sorted into (fixed) spatial cells of dimension 8x, which is typically on the order of .A, the mean free path. Within each cell pairs of particles are then randomly selected as possible collision partners with a HS collision probability that is dependent on their relative velocities. Once a pair is selected, the postcollision relative velocities are also stochastically determined, consistent with the conservation of momentum and energy. The collision is executed with the particles remaining in place. Since in the Boltzmann equation the advection process corresponds to that of point particles, the virial 0 = (Llvi rij) is zero, giving an ideal gas equation of state (Ll vi is the change in velocity of particle i, and rij is the line connecting the centers of the colliding particles). To obtain the correct HS virial, the CBA includes the extra displacement in the advection step that the particles would have experienced if they had collided as hard spheres [7],