Non-equilibrium steady state of the symmetric exclusion process with reservoirs
Institute of Mathematics, Statistics and Computer Science, University of São Paulo（圣保罗大学数学、统计和计算机科学研究所）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；School of Mathematical and Statistical Sciences and the Center for Mechanisms of Evolution (Biodesign Institute), Arizona State University（亚利桑那州立大学数学与统计科学学院及进化机制中心（生物设计研究所））
Consider the open symmetric exclusion process on a connected graph with vertexes in $[N-1]:=\{1,\ldots, N-1\}$ where points $1$ and $N-1$ are connected, respectively, to a left reservoir and a right reservoir with densities $ρ_L,ρ_R\in(0,1)$. We prove that the non-equilibrium steady state of such system is $$μ_{\text{stat}} = \sum_{I\subset \mathcal P([N-1]) }F(I)\bigg(\otimes_{x\in I}\rm{Bernoulli}(ρ_R)\otimes_{y\in [N-1]\setminus I}\rm{Bernoulli}(ρ_L) \bigg).$$ In the formula above $ \mathcal P([N-1])$ denotes the power set of $[N-1]$ while the numbers $F(I)> 0$ are such that $\sum_{I\subset \mathcal P([N-1]) }F(I)=1$ and given in terms of absorption probabilities of the absorbing stochastic dual process. Via probabilistic arguments we compute explicitly the factors $F(I)$ when the graph is a homogeneous segment.
We provide a complete description of the equilibrium fluctuations for diffusive symmetric exclusion processes with long jumps in contact with infinitely extended reservoirs and prove that they behave as generalized Ornstein-Uhlenbeck processes with various boundary conditions, depending mainly on the strength of the reservoirs. On the way, we also give a general statement about uniqueness of the Ornstein-Uhlenbeck process originated by the microscopic dynamics of the underlying interacting particle systems and adapt it to our study.
Consider the symmetric exclusion process evolving on an interval and weakly interacting at the end-points with reservoirs. Denote by $I_{[0,T]} (\cdot)$ its dynamical large deviations functional and by $V(\cdot)$ the associated quasi-potential, defined as $V(γ) = \inf_{T>0} \inf_u I_{[0,T]} (u)$, where the infimum is carried over all trajectories $u$ such that $u(0) = \barρ$, $u(T) = γ$, and $\barρ$ is the stationary density profile. We derive the partial differential equation which describes the evolution of the optimal trajectory, and deduce from this result the formula obtained by Derrida, …
We study the hydrodynamic behaviour of the asymmetric simple exclusion process on the lattice of size $n$. In the bulk, the exclusion dynamics performs rightward flux. At the boundaries, the dynamics is attached to reservoirs. We investigate two types of reservoirs: (1) the reservoirs that are weakened by $n^θ$ for some $θ<0$ and (2) the reservoirs that create particles only at the right boundary and annihilate particles only at the left boundary. We prove that the spatial density of particles, under the hyperbolic time scale, evolves with the entropy solution to a scalar conservation law on $…
We consider the reversible exclusion process with reservoirs on arbitrary networks. We characterize the spectral gap, mixing time, and mixing window of the process, in terms of certain simple statistics of the underlying network. Among other consequences, we establish a non-conservative analogue of Aldous's spectral gap conjecture, and we show that cutoff occurs if and only if the product condition is satisfied. We illustrate this by providing explicit cutoffs on discrete lattices of arbitrary dimensions and boundary conditions, which substantially generalize recent one-dimensional results. We…
In this article, we consider a one-dimensional symmetric exclusion process in weak contact with reservoirs at the boundary. In the diffusive time-scaling the empirical measure evolves according to the heat equation with Robin boundary conditions. We prove the associated dynamical large deviations principle.
We consider the symmetric simple exclusion process evolving on the interval of length $n-1$ in contact with reservoirs of density $ρ\in (0,1)$ at the boundary. We use Yau's relative entropy method to show that if the initial measure is associated with a profile $u_0:[0,1] \to (0,1)$, then at explicit times $t^n(b)$ that depend on $u_0$, the distance to equilibrium, in total variation distance, converges, as $n \to \infty$, to a profile $\mathcal G(γe^{-b})$. The parameter $γ$ also depends on the initial profile $u_0$ and $\mathcal G(m)$ stands for the total variation distance $\|\mathcal N (m,…
We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in th…
In an exclusion process with avalanches, when a particle hops to a neighboring empty site which is adjacent to an island the particle on the other end of the island immediately hops and if it joins another island this triggers another hop. There are no restrictions on the length of islands and the duration of the avalanche. This process is well-defined in the low-density region, ρ<1/2. We describe the nature of steady states (on a ring) and determine all correlation functions. For the asymmetric version of the process, we compute the steady state current, and we describe shock and rarefaction …
We study the hydrodynamic limit for a model of symmetric exclusion processes with heavy-tailed long jumps and in contact with infinitely extended reservoirs. We show how the corresponding hydrodynamic equations are affected by the parameters defining the model. The hydrodynamic equations are characterized by a class of super-diffusive operators that are given by the regional fractional Laplacian with some additional reaction terms and various boundary conditions. This work solves some questions left open in \cite{BJGO2} about the same model.
We consider the Symmetric Simple Exclusion Process (SSEP) on the segment with two reservoirs of densities $p, q \in (0,1)$ at the two endpoints. We show that the system exhibits cutoff with a diffusive window, thus confirming a conjecture of Gantert, Nestoridi, and Schmid in \cite{Gantert2020}. In particular, our result covers the regime $p \neq q$, where the process is not reversible and there is no known explicit formula for the invariant measure. Our proof exploits the information percolation framework introduced by Lubetzky and Sly, the negative dependence of the system, and an anticoncent…
