  We introduce a natural variant of the parallel chip-firing game, called the
diffusion game. Chips are initially assigned to vertices of a graph. At every
step, all vertices simultaneously send one chip to each neighbour with fewer
chips. As the dynamics of the parallel chip-firing game occur on a finite set
the process is inherently periodic. However the diffusion game is not obviously
periodic: even if $2|E(G)|$ chips are assigned to vertices of graph G, there
may exist time steps where some vertices have a negative number of chips. We
investigate the process, prove periodicity for a number of graph classes, and
pose some questions for future research.
