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arXiv · 1410.3923

Transport and Equilibrium in Non-Conservative Systems

L. Chayes·H. K. Lei
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Abstract

We study, in finite volume, a grand canonical version of the McKean-Vlasov equation where the total particle content is allowed to vary. The dynamics is anticipated to minimize an appropriate grand canonical free energy; we make this notion precise by introducing a metric on a set of positive Borel measures without pre-prescribed mass and demonstrating that the dynamics is a gradient flow with respect to this metric. Moreover, we develop a JKO-scheme suitable for these problems. The latter ideas have general applicability to a class of second order non-conservative problems. For this particular system we prove, using the JKO-scheme, that (under certain assumptions) convergence to the uniform stationary state is exponential with a rate which is independent of the volume. By contrast, in related conservative systems, decay rates scale - at best - with the square of the characteristic length of the system. This suggests that a grand canonical approach may be useful for both theoretical and computational study of large scale systems.

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L. Chayes, H. K. Lei. 2014-10-15. Transport and Equilibrium in Non-Conservative Systems. https://arxiv.org/abs/1410.3923

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