A stochastic particle system modeling the Carleman equation |
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Authors: | S. Caprino A. De Masi E. Presutti M. Pulvirenti |
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Affiliation: | (1) Dipartimento di Matematica Pura ed Applicata, Università dell'Aquila, 67100 L'Aquila, Italy;(2) Department of Mathematics, Boulder University, Boulder, Colorado;(3) Present address: Dipartimento di Matematica, Università di Roma La Sapienza, 00185 Rome, Italy |
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Abstract: | Two species of Brownian particles on the unit circle are considered; both have diffusion coefficient >0 but different velocities (drift), 1 for one species and –1 for the other. During the evolution the particles randomly change their velocity: if two particles have the same velocity and are at distance ( being a positive parameter), they both may simultaneously flip their velocity according to a Poisson process of a given intensity. The analogue of the Boltzmann-Grad limit is studied when goes to zero and the total number of particles increases like –1. In such a limit propagation of chaos and convergence to a limiting kinetic equation are proven globally in time, under suitable assumptions on the initial state. If, furthermore, depends on and suitably vanishes when goes to zero, then the limiting kinetic equation (for the density of the two species of particles) is the Carleman equation.Dedicated to the memory of Paola Calderoni. |
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Keywords: | Boltzmann-Grad limit Carleman equation stochastic interacting particle systems propagation of chaos |
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