Theory of stochastic processes with multipole conservation

Dipole conservation imposes strict constraint on a single-particle hopping, allowing only a pairwise hopping of particles in the opposite directions. Zero-range stochastic processes can be generalized to incorporate such dipole conservation, or even a higher-order quadrupolar conservation. We write down models for totally asymmetric zero-range processes with both dipolar and quadrupolar conservations and identify certain exactly solvable stationary-state solutions, consisting of multi-dipolon and multi-quadrupolon excitations. Exact mapping of the multi-dipolon solution to the well-known Bethe ansatz solution of the totally asymmetric exclusion process is shown. The multi-quadrupolon solution is also mapped exactly to a constrained totally asymmetric exclusion process, which we solve by means of Bethe ansatz technique. In both cases the scaling of the energy gap is as L^-3/2 for the chain length L, suggesting a common Kardar-Parisi-Zhang dynamics taking place in both models.