Abstract
A heuristic law widely used in fluid dynamics for steady flows states that the amount of a fluid in a control volume is the product of the fluid influx and the mean time that the particles of the fluid spend in the volume, or mean residence time. We rigorously prove that if the mean residence time is introduced in terms of sample-path averages, then stochastic lattice-gas models with general injection, diffusion, and extraction dynamics verify this law. Only mild assumptions are needed in order to make the particles distinguishable so that their residence time can be unambiguously defined. We use our general result to obtain explicit expressions of the mean residence time for the Ising model on a ring with Glauber + Kawasaki dynamics and for the totally asymmetric simple exclusion process with open boundaries.
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Notes
As usual, we say that a property holds \(\mathbb {P}\)-almost surely (\(\mathbb {P}\)-a.s. for short) if it holds for all \(\omega \in \varOmega _o\in {\mathcal {F}}\) with \(\mathbb {P}[\varOmega _o]=1\).
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Zamparo, M., Dall’Asta, L. & Gamba, A. On the Mean Residence Time in Stochastic Lattice-Gas Models. J Stat Phys 174, 120–134 (2019). https://doi.org/10.1007/s10955-018-2175-x
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DOI: https://doi.org/10.1007/s10955-018-2175-x