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## 4.5.3 Comparison to Lattice Gas Models

LGrM closely resembles LGM [Frisch:86a], [Margolis:86a] in some respects. First, for two-dimensional applications, the region of space in which the particles are to move is discretized into a triangular lattice-work, upon each node of which can reside a particle. The particles are capable of moving to neighboring cells at each tick of the clock, subject to certain simple rules. Finally, two particles arriving at the same cell (LGM) or adjacent cells (LGrM) at the same time, may undergo a ``collision'' in which their outgoing velocities are determined according to specified rules chosen to conserve momentum.

Each of the particles in LGM has the same magnitude of velocity and is allowed to move in one of six directions along the lattice, so that each particle travels exactly one lattice spacing in each time step. The single-velocity magnitude means that all collisions between particles are perfectly elastic and that energy conservation is maintained simply through particle number conservation. It also means that the temperature of the gas is uniform throughout time and space, thus limiting the application of LGM to problems of low Mach number. An exclusion principle is maintained in which no two particles of the same velocity may occupy one lattice point. Thus, each lattice point may have no more than six particles, and the state of a lattice point can be recorded using only six bits.

LGrM differs from LGM in that it has many possible velocity states, not just six. In particular, in LGrM not only the direction but the magnitude of the velocity can change in each collision. This is a necessary condition because the collision of two macroscopic particles is always inelastic, so that mechanical energy is not conserved. The LGrM particles satisfy a somewhat different exclusion principle: No more than one particle at a time may occupy a single site. This exclusion principle allows LGrM to capture some of the volume-filling properties of granular material-in particular, to be able to approximate the behavior of static granular masses.

The determination of the time step is more critical in LGrM than in LGM. If the time step is long enough that some particles travel several lattice spacings in one clock tick, the problem of finding the intersection of particle trajectories arises. This involves much computation and defeats the purpose of an automata approach. A very short time step would imply that most particles would not move even a single lattice spacing. Here we choose a time step such that the fastest particle will move exactly one lattice spacing. A ``position offset'' is stored for each of the slower particles, which are moved accordingly when the offset exceeds one-half lattice spacing. These extra requirements for LGrM automata imply a slower computation speed than expected in LGM simulations; but, as a dividend, we can compute inelastic grain flows of potential engineering and geophysical interest.

Next: 4.5.4 The Rules for Up: 4.5 An Automata Model Previous: 4.5.2 Comparison to Particle

Guy Robinson
Wed Mar 1 10:19:35 EST 1995