Several recent works have shown that the onedimensional fully asymmetric exclusion model, which describes a ystem of panicles hopping in a preferred direction with hard core interactions, can be solved exactb in the case of open boundaries. Here we present a new approach based on representing the weights of each mnfiguration in the steady state as a product of noncommuting matrices. Wth this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisQ ve'y simple algebraic NI= We obtain several explicit toms for these noncommuting matrices which are, in the general case. infinite-dimensional. Our approach allows exam expresions to be derived for the current and density profiles. Finally we discuss h'efly two possible generalizations of our results: the pmblem of panially asymmetric exclusion and the case of a mixture of two kinds of panicles.
These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow to calculate the fluctuations and large deviations of the density and of the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example non-Gaussian fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium. P non-equilibrium (C) = ? * Electronic address: bernard.derrida@lps.ens.fr
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A number of exact results have been obtained recently for the one-dimensional asymmetric simple exclusion process, a model of particles which hop to their right at random times, on a one-dimensional lattice, provided that the target site is empty. Using either a matrix form for the steady-state weights or the Bethe ansatz, several steady-state properties can be calculated exactly: the current, the density profile for open boundary conditions, the diffusion constant of a tagged particle. The matrix form of the steady state can be extended to calculate exactly the steady state of systems of two species of particles and shock profiles.
We consider the effect of a small cut-off ε on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off ε is to select a single velocity which converges when ε → 0 to the one predicted by the marginal stability argument. For small ε, the shift in velocity has the form K(log ε) −2 and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.
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