  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.
