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Register a new userDerivation of viscous Burgers equations from weakly asymmetric exclusion processes
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We consider weakly asymmetric exclusion processes whose initial density profile is a small perturbation of a constant. We show that in the diffusive time-scale, in all dimensions, the density defect evolves as the solution of a viscous Burgers equation.
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The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Sepp{a}lainen and Balazs recently proved this conjecture for the nearest neighbor case. We extend their results to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.
We derive from a class of microscopic asymmetric interacting particle systems on ${mathbb Z}$, with long range jump rates of order $|cdot|^{-(1+alpha)}$ for $0<alpha<2$, different continuum fractional SPDEs. More specifically, we show the equilibrium fluctuations of the hydrodynamics mass density field of zero-range processes, depending on the stucture of the asymmetry, and whether the field is translated with process characteristics velocity, is governed in various senses by types of fractional stochastic heat or Burgers equations. The main result: Suppose the jump rate is such that its symmetrization is long range but its (weak) asymmetry is nearest-neighbor. Then, when $alpha<3/2$, the fluctuation field in space-time scale $1/alpha:1$, translated with process characteristic velocity, irrespective of the strength of the asymmetry, converges to a fractional stochastic heat equation, the limit also for the symmetric process. However, when $alphageq 3/2$ and the strength of the weak asymmetry is tuned in scale $1-3/2alpha$, the associated limit points satisfy a martingale formulation of a fractional stochastic Burgers equation.
We investigate the dynamics of a one-dimensional asymmetric exclusion process with Langmuir kinetics and a fluctuating wall. At the left boundary, particles are injected onto the lattice; from there, the particles hop to the right. Along the lattice, particles can adsorb or desorb, and the right boundary is defined by a wall particle. The confining wall particle has intrinsic forward and backward hopping, a net leftward drift, and cannot desorb. Performing Monte Carlo simulations and using a moving-frame finite segment approach coupled to mean field theory, we find the parameter regimes in which the wall acquires a steady state position. In other regimes, the wall will either drift to the left and fall off the lattice at the injection site, or drift indefinitely to the right. Our results are discussed in the context of non-equilibrium phases of the system, fluctuating boundary layers, and particle densities in the lab frame versus the frame of the fluctuating wall.
We study the one-dimensional asymmetric simple exclusion process on the lattice ${1,dots,N}$ 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 boundary rates. We consider two different types of boundary rates: the Liggett boundaries that correspond to the projection of the infinite dynamics, and the reversible boundaries, that correspond to the contact with particle reservoirs in equilibrium. The proof is based on the control of the Lax boundary entropy--entropy flux pairs and a coupling argument.
We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional lattice. The particles can be created/annihilated at the boundaries with time-dependent rate. These boundary dynamics are properly accelerated. We prove the hydrodynamic limit of the particle density profile, under the hyperbolic space-time rescaling, evolves with the entropy solution to Burgers equation with Dirichlet boundary conditions. The boundary conditions are characterised by boundary entropy flux pair.
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