Current Fluctuations for the Boundary-Driven Zero-Range Process on Graphs: Microscopic Versus Macroscopic Approach and a Theory of Non-reversible Resistor-Like Networks

Abstract

We compute the joint large deviation rate functional in the limit of large time for the current flowing through the edges of a finite graph on which a boundary-driven system of stochastic particles evolves with zero-range dynamics. This generalizes one-dimensional results previously obtained with different approaches (Bertini et al. in J Stat Phys 123(2):237–276, 2006; Harris et al. in J Stat Mech Theory Exp 2005:P08003, 2005); our alternative techniques illuminate various connections and complementary perspectives. In particular, we here use a variational approach to derive the rate functional by contraction from a level 2.5 large deviation rate functional. We perform an exact minimization and finally obtain the rate functional as a variational problem involving a superposition of cost functions for each edge. The contributions from different edges are not independent since they are related by the values of a potential function on the nodes of the graph. The rate functional on the graph is a microscopic version of the continuous rate functional predicted by the macroscopic fluctuation theory (Bertini et al. in Rev Mod Phys 87(2):593–636, 2015), and we indeed show a convergence in the scaling limit. If we split the graph into two connected regions by a cutset and are interested just in the current flowing through the cutset, we find that the result is the same as that of an effective system composed of only one effective edge [as happens at macroscopic level and is expected also for other models (Akkermans et al. in Europhys Lett 103(2):20001, 2013)]. The characteristics of this effective edge are related to the “capacities” of the graph and can be obtained by a reduction using elementary transformations as in electrical networks; specifically, we treat components in parallel, in series, and in N-star configurations (reduced to effective complete N-graphs). Our reduction procedure is directly related to the reduction to the trace process (Landim in Probab Surv 16:143–227, 2019) and, since the dynamics is in general not reversible, it is also closely connected to the theory of non-reversible electrical networks in Balázs and Folly (Am Math Mon 123(7):657–682, 2016).