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Hydrodynamics for asymmetric simple exclusion on a finite segment with Glauber-type source

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in Bardos et al (Commun Pa...

Scalar conservation law in a bounded domain with strong source at boundary

We consider a scalar conservation law with source in a bounded open interval $\Omega \subseteq \mathbb R$. The equation arises from the macroscopic evolution of an interacting particle system. The source term models an external effort driving the solution to a given function $\varrho$ with an intensity function $V: \Omega \rightarrow \mathbb R_+$ that grows to infinity at $\partial \Omega$ . We define the entropy solution $u \in L^\infty$ and prove the uniqueness. When V is integrable, u sati...

Equilibrium perturbations for stochastic interacting systems

We consider the equilibrium perturbations for two stochastic systems: the d-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order $N^{-α}$ to the equilibrium profile and speed up the process by $N^{1+κ}$ for parameters 0 < κ ≤ α. Under some additional constraints on κ and α, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in ...

Hydrodynamics for one-dimensional ASEP in contact with a class of reservoirs

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 $n^\theta$ for some $\theta<0$ 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 densit...

Quasi-static limit for the asymmetric simple exclusion

We study the one-dimensional asymmetric simple exclusion process on the lattice $\{1,…,N\}$ with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale $N^{-a}$ with $a>0$. We prove that at the time scale $N^{1+a}$ the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic b...

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