Asymmetric inclusion process.

We introduce and explore the asymmetric inclusion process (ASIP), an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process (ASEP). In both processes, random events cause particles to propagate unidirectionally along a one-dimensional lattice of n sites. In the ASEP, particles are subject to exclusion interactions, whereas in the ASIP, particles are subject to inclusion interactions that coalesce them into inseparable clusters. We study the dynamics of the ASIP, derive evolution equations for the mean and probability generating function (PGF) of the sites' occupancy vector, obtain explicit results for the above mean at steady state, and describe an iterative scheme for the computation of the PGF at steady state. We further obtain explicit results for the load distribution in steady state, with the load being the total number of particles present in all lattice sites. Finally, we address the problem of load optimization, and solve it under various criteria. The ASIP model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with (unlimited) batch service, and a tandem array of growth-collapse processes.