Fick Law and Sticky Brownian Motions
We consider an interacting particle system in the interval [1, N] with reservoirs at the boundaries. While the dynamics in the channel is the simple symmetric exclusion process, the reservoirs are also particle systems which interact with the given system by exchanging particles. In this paper we study the case where the size of each reservoir is the same as the size of the channel. We will prove that the hydrodynamic limit equation is the heat equation with boundary conditions which relate first and second spatial derivatives at the boundaries for which we will prove the existence and uniqueness of weak solutions.
KeywordsHydrodynamic limits Free boundary problem Sticky random walk
I greatly appreciate Prof. Errico Presutti for suggesting the problem and offering me a large number of useful ideas. I also would like to express my gratitude to Lorenzo Bertini, Paolo Butta, Anna De Masi, Pablo Ferrari and Frank Redig and Maria Eulalia Vares for their valuable comments and suggestions. In addition, I would like to thank the reviewers for their careful reading of my paper and for their insightful comments.
- 7.De Masi, A., Presutti, E., Tsagkarogiannis, D., Vares, M.E.: Exponential rate of convergence in current reservoirs. Bernoulli 21(3), 1844–1854 (2015). https://doi.org/10.3150/14-BEJ628
- 9.Knight, F.B.: On the random walk and Brownian motion. Trans. Am. Math. Sot. 103, 725–731 (1961)Google Scholar
- 13.Peskir, G.: A probabilistic solution to the Stroock-Williams equation. Ann. Probab. 42(5), 2197–2206 (2014). https://doi.org/10.1214/13-AOP865