Survival of interacting diffusing particles inside a domain with absorbing boundary

Suppose that a d-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density n0. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability P that no particles are absorbed during a long time T. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time T. As a result, P decays exponentially with T for a whole class of interacting diffusive gases in any dimension. For d=1 the stationary gas density profile and P can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that -lnP≃D0TLd-2s(n0), where D0 is the gas diffusivity, and L is the linear size of the system. We calculate the rescaled action s(n0) for d=1, for rectangular domains in d=2, and for spherical domains. Near close packing of the SSEP s(n0) can be found analytically for domains of any shape and in any dimension.