Emergence of traveling density waves in cyclic multiparticle transport

Multiparticle flow through a cyclic array of K connected compartments with a preferential direction is found to be able to organize itself in traveling waves. This behavior is connected with the transition between uniform flow and cluster formation. When the bias in the system is large, the particles flow freely in the preferred direction, with all compartments being equally filled at all times. Conversely, when the bias is small the particles cluster together in one compartment. The transition between these two regimes is found to involve an intermediate state in which the flow exhibits a density peak traveling periodically around the system. We relate the emergence of this traveling wave to a Hopf bifurcation and analytically derive the critical value of the "symmetry parameter" at which this bifurcation occurs. This critical value proves to be independent of the number of compartments, but the width of the intermediate regime (and thus the chance of observing traveling wave solutions) decreases sharply with growing K. The reverse transition follows a different course and takes place at a significantly lower value of the symmetry parameter; it is an abrupt transition from a clustered state to a uniform flow without an intermediate regime of stable traveling waves.