### A deterministic displacement theorem for Poisson processes.

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In this work, we use the methods of nonequilibrium statistical mechanics in order to derive an equation which models some mechanisms of opinion formation. After proving the main mathematical properties of the model, we provide some numerical results.

We recover the Navier–Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a mesoscopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier–Stokes equation in a fixed time interval. The proof does not use nongradient methods or the multi-scale analysis due to the long range jumps.

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalized version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.

We study the upper tails for the energy of a randomly charged symmetric and transient random walk. We assume that only charges on the same site interact pairwise. We consider annealed estimates, that is when we average over both randomness, in dimension three or more. We obtain a large deviation principle, and an explicit rate function for a large class of charge distributions.

We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than $$\frac{7+\sqrt{41}}{2}$$ and obeys some natural regularity conditions.

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived markovian coupled process (ξt, ζt)t≥0 satisfies: (A) if ξ0≤ζ0 (coordinate-wise), then for all t≥0, ξt≤ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y,...

A kinetic collision operator of Landau type for Fermi-Dirac particles is considered. Equilibrium states are rigorously determined under minimal assumptions on the distribution function of the particles. The particular structure of the considered operator (strong non-linearity and degeneracy) requires a special investigation compared to the classical Boltzmann or Landau operator.

We answer some questions raised by Gantert, Löwe and Steif (Ann. Inst. Henri Poincaré Probab. Stat.41(2005) 767–780) concerning “signed” voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed)....