Universal current fluctuations in the symmetric exclusion process and other diffusive systems

Abstract

-We show, using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs are the same on an arbitrary large finite domain in dimension d as in the one dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward.Introduction. -When a system is connected for a long time to two heat baths at unequal temperatures or to two reservoirs of particles at different densities, it reaches a nonequilibrium steady state, with a non vanishing current of heat or of particles. This current fluctuates with time and the study of its fluctuations and of its large deviations has become a central aspect in the theory of non equilibrium systems [4,5,12,16,19,20,23,24,26,27,29,34,35]. A quantity which characterizes these fluctuations is the probability P (Q t ) of observing an energy or a number of particles Q t flowing through the system during a time window t. A notorious property of these fluctuations is known as the fluctuation theorem [15,17,18,28] which establishes a general relation between P (Q t ) and P (−Q t ), starting from some time reversal symmetry of the microscopic dynamics. Apart from simple models, however, it is usually difficult to predict the whole shape of the distribution P (Q t ).

Current fluctuations in boundary driven diffusive systems in different dimensions: a numerical study

“…We proceed with the study of the diffusion coefficient and the current fluctuations in a GEP where maximally two (interacting) particles can occupy each lattice site. The first three cumulants of the current distribution are calculated by combining the AP with the results from [24]. In one and two dimensions the first three cumulants obtained from kMC are in agreement with the predicted values.…”

“…In our simulations of the SSEP we consider contacts with the reservoirs that are macroscopic in size. Complete convergence of the Fano factor to the analytical prediction of [24] is found in two dimensions. For three dimensions the data indicate convergence for large system sizes.…”

“…This is especially important because many experimental systems are higherdimensional. The results from [24] indicate that it is, indeed, possible to do this. So far, only a few studies have addressed this question.…”

“…If the diffusion is isotropic, which is the case considered here, one can [22]. Akkermans et al studied current fluctuations in higher-dimensional diffusive systems [24]. The shape of the system and the contacts with the reservoirs are taken arbitrary, but macroscopic in size.…”

“…Using the MFT, Akkermans and co-workers studied current fluctuations in diffusive systems connected to two reservoirs [24]. They showed analytically that for a system of arbitrary (but fixed) dimension, the ratio of the cumulants of the current distribution is independent of the shape of the system and the shape of the contacts with the reservoirs.…”