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Current fluctuations for the asymmetric simple exclusion process. (English) Zbl 0806.60099
Summary: We compute the diffusion coefficient of the current of particles through a fixed point in the one-dimensional nearest neighbor asymmetric simple exclusion process in equilibrium. We find $$D = | p - q | \rho (1 - \rho) | 1 - 2 \rho |$$, where $$p$$ is the rate at which the particles jump to the right, $$q$$ is the jump rate to the left and $$\rho$$ is the density of particles. Notice that $$D$$ vanishes if $$p = q$$ or $$\rho = 1/2$$. Laws of large numbers and central limit theorems are also proven.
Analogous results are obtained for the current of particles through a position travelling at a deterministic velocity $$r$$. As a corollary we get that the equilibrium density fluctuations at time $$t$$ are a translation of the fluctuations at time 0. We also show that the current fluctuations at time $$t$$ are given, in the scale $$t^{1/2}$$, by the initial density of particles in an interval of length $$| (p - q) (1 - 2 \rho) | t$$. The process is isomorphic to a growth interface process. Our result means that the equilibrium growth fluctuations depend on the general inclination of the surface. In particular, they vanish for interfaces roughly perpendicular to the observed growth direction.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics 82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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