Abstract
We study the hydrodynamic behaviour of the asymmetric simple exclusion process on the lattice of size n. In the bulk, the exclusion dynamics performs rightward flux. At the boundaries, the dynamics is attached to reservoirs. We investigate two types of reservoirs: (1) the reservoirs that are weakened by \(n^\theta \) for some \(\theta <0\) and (2) the reservoirs that create particles only at the right boundary and annihilate particles only at the left boundary. We prove that the spatial density of particles, under the hyperbolic time scale, evolves with the entropy solution to a scalar conservation law on [0, 1] with boundary conditions. The boundary conditions are characterised by the boundary traces [3, 17, 20] at \(x=0\) and \(x=1\) which take values from \(\{0,1\}\).
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Notes
Here \(\psi \not \in {\mathcal {C}}_c({\mathbb {R}}^2)\), but it can be easily approximated by continuous functions.
Though \(\psi \) is compactly supported, the boundary terms cannot be omitted autonomously, since we need to take supreme over all \(\psi \) before send \(n\rightarrow \infty \).
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Communicated by Stefano Olla.
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This work has been funded by the ANR Grant MICMOV (ANR-19-CE40-0012) of the French National Research Agency (ANR).
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XU, L. Hydrodynamics for One-Dimensional ASEP in Contact with a Class of Reservoirs. J Stat Phys 189, 1 (2022). https://doi.org/10.1007/s10955-022-02963-x
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DOI: https://doi.org/10.1007/s10955-022-02963-x