  In all spatial dimensions $d$, we study the static and dynamical properties
of a generalized Smoluchowski equation which describes the evolution of a gas
obeying a logotropic equation of state, $p=A\ln\rho$. A logotrope can be viewed
as a limiting form of polytrope ($p=K\rho^{\gamma}$, $\gamma=1+1/n$), with
index $\gamma=0$ or $n=-1$. In the language of generalized thermodynamics, it
corresponds to a Tsallis distribution with index $q=0$. We solve the dynamical
logotropic Smoluchowski equation in the presence of a fixed external force
deriving from a quadratic potential, and for a gas of particles subjected to
their mutual gravitational force. In the latter case, the collapse dynamics is
studied for any negative index $n$, and the density scaling function is found
to decay as $r^{-\alpha}$, with $\alpha=\frac{2n}{n-1}$ for $n<-\frac{d}{2}$,
and $\alpha=\frac{2d}{d+2}$ for $-\frac{d}{2}\leq n<0$.
