Pair-factorized steady states on arbitrary graphs

Stochastic mass transport models are usually described by specifying hopping rates of particles between sites of a given lattice, and the goal is to predict the existence and properties of the steady state. Here we ask the reverse question: given a stationary state that factorizes over links (pairs of sites) of an arbitrary connected graph, what are possible hopping rates that converge to this state? We define a class of hopping functions which lead to the same steady state and guarantee current conservation but may differ by the induced current strength. For the special case of anisotropic hopping in two dimensions we discuss some aspects of the phase structure. We also show how this case can be traced back to an effective zero-range process in one dimension which is solvable for a large class of hopping functions. PACS numbers: 89.75.Fb, 05.40.-a, 64.60.Ak ar X iv :0 90 4. 03 55 v1 [ co nd -m at .s ta tm ec h] 2 A pr 2 00 9 Pair-factorized steady states on arbitrary graphs 2 Stochastic transport of some conserved quantity, generically called “mass”, has recently attracted much attention due to a large variety of applications ranging from microscopic (intracellular) to macroscopic (highway) traffic [1]. Important examples of technological interest are granular flow [2] and granular clustering [3]. From a theoretical point of view, these systems are challenging, since they are in general out-of-equilibrium and allow for phase transitions even in one dimension [4]. An example is spontaneous symmetry breaking and the phenomenon of condensation which happens above some critical mass density and corresponds to jams in traffic or aggregation in granular media. Nevertheless, simple models such as the zero-range process (ZRP) [5], a dynamical version of the balls-in-boxes model [6], or asymmetric simple exclusion processes [7], can capture some important aspects of these problems while remaining analytically solvable. The non-equilibrium models are defined by specifying the dynamics rather than the probability of a microstate. Usually one proposes the transition rates between states, and the goal is to predict the existence and the properties of a stationary state. This is the state where macroscopic observables remain constant, although some currents may flow in the system. In this paper we study the reverse problem. Given a steady state which assumes a form factorized over pairs of sites that correspond to the links of an arbitrary graph, we search for a class of transition probabilities which lead to this state. This approach is motivated by the fact that knowledge of the stationary states of non-equilibrium models generically facilitates the discussion of the phase diagram, because a number of observables can be calculated analytically. For instance, the ZRP, defined in terms of particles hopping between sites of a lattice and interacting only if they are at the same node, has a steady state that factorizes over the sites of a lattice, or more generally, over nodes of an arbitrary graph. The factorization allows for a convenient mathematical treatment. A generalization of the ZRP that leads to pairfactorized steady states (PFSS) was proposed in [8] for a one-dimensional ring topology. It was shown that nearest-neighbor exponentially suppressed interactions plus some extra “pinning” (ZRP-like) potential result in a condensate that is spatially extended. In [9] the shape of the condensate was derived. It was also shown that the scaling of the extension of the condensate with the system size can be tuned via an appropriate competition between local and ultralocal hopping interactions. We shall show in this paper that the ZRP and the PFSS on a ring (where PFSS here should be understood as the corresponding processes leading to PFSS) are special cases of a more general setting. Beyond that, we shall consider non-local processes on an arbitrary graph, or processes with anisotropic hopping in two dimensions. In the latter case we shall show that it can be dimensionally reduced to a ZRP in one dimension with weights that contain the information on the pair-factorized stationary behavior in the second dimension. Therefore former results on PFSS on a one-dimensional ring topology [9] can be used to derive features of the condensation transition in the anisotropic case. The model. We consider a connected, undirected but otherwise arbitrary graph with N nodes (sites), and node degrees k1, ..., kN . We place M particles of unit mass Pair-factorized steady states on arbitrary graphs 3 on the nodes of this graph. “Particles” here stands for a generic mass that is involved in the transport process and has “bosonic” properties in the sense that mi ≥ 0 particles may be assigned to the same site i. The distribution of occupation numbers of nodes is denoted as ~ m = {m1, ...,mN}. The dynamics is defined as follows. We pick up a randomly chosen node i, and if it is not empty, a single particle departures with probability ui(mi| . . .), where the dots stand for occupation numbers of other sites (in general not necessarily nearest neighbors of node i). Next, the particle chooses a target site j with probability Wij ≡ W (i → j). The transition matrix W may be arbitrary, with the only assumption that all Wij ≥ 0 and ∑ jWij = 1 for any i. The hopping event is thus split into two steps: the departure from a site, determined by the function u, and the choice of destination site, determined by the rates Wij. We will next derive under which conditions on the hopping rate and the transition matrix the system reaches a steady state that assumes a pair-factorized form