If an additional time step is stored, either as the value of the state S at time t-1 as described by Margolus, or as a differential that permits the calculation of the prior state, then the automaton is reversible. If the state parameters are integers, the reversibility is exact. The acoustic wave model can be run backwards. The wave begins to regress as though a projector were running a film in reverse. If rising zero crossings are displayed, the roundoff error associated with the computation is shown as part of the display.
Lattice gases are specialized cellular automata that have been used to model dynamic systems.
They do this by describing the state space as a lattice of sites connected by a well- defined grid, usually rectangular, triangular or hexagonal, and allowing particles to move along the grid in specified directions. At any point in time a particle can move from its current resting place at an intersection of grid lines to a new lattice site according to specific rules that involve motion, collision and conservation of quantities pertinent to the system modeled. In many models, energy and momentum are among the conserved quantities. The dynamics result from the synchronous application of a local transition rule at each lattice site, the new values of a cell variable being a function of current values of the variables in cells belonging to a small neighborhood around the site. From the point of view of physics, it is straightforward to interpret the lattice as a discretized version of the physical space, the variables as occupation numbers of particles with a discrete repartition of internal states, and the evolution rules as propagation and collision rules for these particles.