Shock fluctuations in the asymmetric simple exclusion process
We consider the one-dimensional nearest neighbors asymmetric simple exclusion process with rates \(q\) and \(p\) for left and right jumps respectively; \(q<p\). \textit{P. A. Ferrari}, \textit{E. Presutti}, \textit{E. Scacciatelli} and \textit{M. E. Vares} [Stochastic Processes Appl. 39, No. 1, 89-105 (1991; Zbl 0749.60094) and ibid., No. 1, 107-115 (1991; Zbl 0749.60095)] have shown that if the initial measure is \(\nu_{\rho, \lambda}\), a product measure with densities \(\rho\) and \(\lambda\) to the left and right of the origin respectively, \(\rho < \lambda\), then there exists a (microscopic) shock for the system. A shock is a random position \(X_ t\) such that the system as seen from this position at time \(t\) has asymptotic product distributions with densities \(\rho\) and \(\lambda\) to the left and right of the origin respectively, uniformly in \(t\). We compute the diffusion coefficient of the shock \(D = \lim_{t \to \infty}t^{-1} (E(X_ t)^ 2 - (EX_ t)^ 2)\) and find \(D = (p-q) (\lambda - \rho)^{-1} (\rho (1-\rho) + \lambda (1 - \lambda))\) as conjectured by \textit{H. Spohn} [Large scale dynamics of interacting particles (1991; Zbl 0742.76002)]. We show that in the scale \(\sqrt t\) the position of \(X_ t\) is determined by the initial distribution of particles in a region of length proportional to \(t\). We prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities \(\rho\) and \(\lambda\). This is the so-called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.
nearest neighbors asymmetric simple exclusion process
scientific article; zbMATH DE number 639361