Abstract
We study a system of particles in the interval a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously new particles are injected at site 0 at rate (j > 0) and removed at same rate from the rightmost occupied site. The removal mechanism is, therefore, of topological rather than metric nature. The determination of the rightmost occupied site requires a knowledge of the entire configuration and prevents from using correlation functions techniques. We prove using stochastic inequalities that the system has a hydrodynamic limit, namely that under suitable assumptions on the initial configurations, the law of the density fields (φ a test function, the number of particles at site x at time t) concentrates in the limit on the deterministic value , interpreted as the limit density at time t. We characterize the limit as a weak solution in terms of barriers of a limit-free boundary problem.
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Acknowledgments
We thank Pablo Ferrari and John Ockendon for many useful comments and discussions. The research has been partially supported by PRIN 2009 (prot. 2009TA2595-002) and FIRB 2010 (Grant n. RBFR10N90W). A. De Masi and E. Presutti acknowledge kind hospitality at the Dipartimento di Matematica della Università di Modena. G. Carinci and C. Giardinà thank Università dell’Aquila for welcoming during their visit at Dipartimento di Matematica.
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Carinci, G., Masi, A.D., Giardinà, C. et al. Hydrodynamic limit in a particle system with topological interactions. Arab. J. Math. 3, 381–417 (2014). https://doi.org/10.1007/s40065-014-0095-4
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DOI: https://doi.org/10.1007/s40065-014-0095-4