Work in collaboration with B. Waclaw * and W. Janke* (* University of Leipzig), and J. Sopik (Jacobs University Bremen)

A variety of stochastic processes out-of-equilibrium may be summarized under the name of stochastic mass transport models, mass transport in the sense of microscopic dynamics that involves stochastic transport of some conserved quantity called ``mass'', from one point in space to another. Examples for such processes are traffic flow , in particular traffic in the cell , force propagation in granular media , aggregation and fragmentation of clusters, sandpile dynamics, protein transport in cells or molecular motors along microtubule filaments in biophysical systems. Usually the space continuum is replaced by a regular (low-dimensional) lattice, continuous time may also be replaced by discrete time steps, and the mass itself can take continuous or discrete values. Important special cases of mass transport processes are the zero-range process (ZRP), the asymmetric random average process, or the total asymmetric simple exclusion process ((T)ASEP).

Evans et al. [1] considered stochastic mass transport models of which the steady state takes the form of a pair-factorized state (PFSS) rather than a fully-factorized steady state, so that the probability of a configuration {m(i)} is a product over pairs (m(i), m(i+1)) of sites of a periodic chain. Here m(i) is a non-negative integer number of particles, each of unit mass at site i. In contrast to the zero-range process, the mass transport depends on the next neighbors as well as on the departure site. This leads to a change in the nature of the condensate which becomes spatially extended due to the short-range correlations. In this case the condensation transition can be shown to be related to the binding-unbinding transition of solid-on-solid interfaces. We shall consider variations of mass transport models with interactions leading to pair-factorizing steady states in higher dimensions.

We first present pair-factorizing measures on a d-dimensional hypercubic grid and on a complete graph which we have proven to be exact solutions of the stationary states of the system. This includes the special case of anisotropic systems in two dimensions with a fully factorizing measure in one dimension and a pair-factorizing measure in the second dimension. Such anisotropy may mimic systems which interact along chains and only very weakly between the chains as for multi-lane traffic with directed motion of particles. For this class we derive an effective one-dimensional description in terms of fully-factorizing steady states.

For a broad class of weights corresponding to a broad class of hopping rates in the one-dimensional case we analytically estimate the critical density of particles for which a condensate shows up for the first time. Since the anisotropic system in two dimensions can be mapped to an effective one-dimensional one, we can approximate the critical density also for this case. In addition, we determine the scaling of the extension of the condensate with the linear size. As it is analytically predicted and numerically confirmed, the condensate is still located at one site in one direction and extends almost over the whole string in the other direction. Therefore the formation of the condensate occurs as a combination of the spontaneous symmetry breaking (known for fully-factorizing measures) and interaction between neighbors (as known for pair-factorizing measures in one dimension).

For pair-factorizing steady states in two dimensions we study the phase diagram numerically by giving bounds on the critical density. The simulations show the formation of a condensate above a critical density, so that the condensate extends in each of the two directions with a size that scales with the square root of the linear system size. Below the critical density and for large system size we see a liquid phase characterized by a uniform distribution of particles For intermediate system size, the stationary state corresponds to a kind of interpolation between the liquid and the condensed phase, because we see the coexistence of multiple condensates each of which roughly carries the same number of particles. Here the condensate is interaction driven in both directions.

We use different algorithms for our simulations: one directly simulating the hopping rates, in terms of which the processes are originally defined, the other one simulating the weights in the partition function via a Metropolis algorithm. It can be shown that the stationary states of the system are the same as long as they are identified via the configurations of occupation numbers, but their dynamical features may be different as long as they are compatible with the occupation number distributions. Moreover, the simulations via the Metropolis algorithm considerably reduce the CPU-time in the condensed phase, since the simulation of the weights of the partition function amount to non-local (since random) hopping of the particles. Non-local hopping facilitates the merging of transient condensates of intermediate size and thus accelerates the formation of the final condensate that contains almost all particles of the system.

Finally, we will indicate extensions towards the effect of explicit symmetry breaking via the network topology, competing with the spontaneous one, as well as some dynamical properties concerning the formation and melting of condensates in such systems.

References:

[1] M. R. Evans, T. Hanney, and S. Majumdar, arXiv:cond-mat/0604664; Phys. Rev. Lett. 97, 010602 (2006).

Previous related work by some of us:

 [2] L. Bogacz, Z. Burda, W. Janke, and B. Waclaw, Chaos 17, 026112-1--6 (2007).

[3] B. Waclaw, L. Bogacz, Z. Burda, and W. Janke, Phys. Rev. E 76, 046114-1--9 (2007).