Non-equilibrium and stationary fluctuations for the SSEP with slow boundary

Abstract

We derive the non-equilibrium fluctuations of one-dimensional symmetric simple exclusion processes in contact with slowed stochastic reservoirs which are regulated by a factor n −θ . Depending on the range of θ we obtain processes with various boundary conditions. Moreover, as a consequence of the previous result we deduce the non-equilibrium stationary fluctuations by using the matrix ansatz method which gives us information on the stationary measure for the model. The main ingredient to prove these results is the derivation of precise bounds on the two point space-time correlation function, which are a consequence of precise bounds on the transition probability of some underlying random walks.

Sharp Convergence to Equilibrium for the SSEP with Reservoirs

“…Our proof of the log-Sobolev inequatily is based on a simple, albeit powerful observation: one can combine the Glauber dynamics at the reservoirs with the exclusion dynamics at the interior of the interval in order to compare the exclusion process with reservoirs with a Glauber dynamics that acts on each site of the interval. This comparison principle holds true even if the reservoirs are slow [2,7], see Theorem 3.4. It is interesting to observe that our diffusive bounds on the log-Sobolev constant of the exclusion process in contact with reservoirs hold true exactly up to the same slow scale on which the reservoirs modify the hydrodynamic behavior of the system [6].…”

Hydrodynamical behavior for the generalized symmetric exclusion with open boundary

“…We observe that when α = 1 this model differs from the usual SSEP because it allows more than one particle per site and this difference, at the level of the microscopic dynamics, results in having a model that is not solvable by a matrix ansatz formulation unless α = 1 and, as a consequence, there is not much information about its non-equilibrium stationary measure. For α = 1, the matrix ansatz method developed by [5] allows getting information on its non-equilibrium stationary measure, which in turn enables one to obtain explicitly the stationary correlations of the system for any value of θ ∈ , see, for example, Section 2.2 of [7] and references therein.…”

“…The hydrostatics in the other cases is left open. The second problem is the analysis of the non-equilibrium fluctuations, which is well known for the case α = 1, see [7] and references therein. We also highlight that our replacement lemmas are the building blocks in order to analyse the equilibrium fluctuations and the large deviations principle for our model.…”

Long-time behavior of SSEP with slow boundary

“…Despite the simple structure of the model, it has attracted a lot of attention since then. The equilibrium/non-equilibrium fluctuations from the hydrodynamic limit are considered in [9,11]. The large deviation of the SSEP with slow boundary are investigated in [2,6,10].…”