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A dynamical system of the WR model in 1D

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A one-dimensional Widom—Rowlinson model with $q$ types of particles is considered from a dynamical point of view. Particles of a unit mass move along the line without noticing each other when they belong to the same type and experiencing elastic collisions when they belong to different types. A natural invariant (equilibrium) measure is where (i) the position distribution represents a shift-invariant point process formed by random (Poissonian/geometric) `series` of particles of a fixed type succeeded by series of other types separated from each other by hard-core exclusion intervals, and (ii) the velocities of different particles are IID . We show that an `equilibrium` dynamical system with such an invariant measure has extreme ergodic properties (generates a B-flow of an infinite entropy). Moreover, we check that a `non-equilibrium` flow, with an initial distribution of a general type, exhibits convergence to a limiting measure of the above form.

All concepts from the ergodic theory will be introduced on the spot, including a brief history of the Botzmann ergodicity conjecture (featured prominently in this year’s Abel prtize citation.)

This talk is part of the Probability series.

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