  In this paper we consider a nonlinear Fokker-Planck equation with
asymptotically small parameters. It describes the diffusion of finite-size
particles in the presence of a fixed distribution of obstacles in the limit of
low-volume fraction. The equation does not have a gradient flow structure, but
can be interpreted as an asymptotic gradient flow, that is, as a gradient flow
up to a certain asymptotic order. We use this scalar equation as a simple
testbed model for more complicated systems of this kind. We discuss several
possible entropy-mobility pairs, illustrate their dynamics with numerical
simulations, present global in time existence results and study the long time
behavior of solutions.
