  This paper presents $N$-body and stochastic models that describe the motion
of tracer particles in a potential that contains a large population of extended
substructures. Fluctuations of the gravitational field induce a random walk of
orbital velocities that is fully specified by drift and diffusion coefficients.
In the impulse and local approximations the coefficients are computed
analytically from the number density, mass, size and relative velocity of
substructures without arbitrary cuts in forces or impact parameters. The
resulting Coulomb logarithm attains a well-defined geometrical meaning,
$\ln(\Lambda)=\ln (D/c)$, where $D/c$ is the ratio between the average
separation and the individual size of substructures. Direct-force and
Monte-Carlo $N$-body experiments show excellent agreement with the theory if
substructures are sufficiently extended ($c/D\gtrsim 10^{-3}$) and not
spatially overlapping ($c/D\lesssim 10^{-1}$). However, close encounters with
point-like objects ($c/D\ll 10^{-3}$) induce a heavy-tailed, non-Gaussian
distribution of high-energy impulses that cannot be described with Brownian
statistics. In the point-mass limit ($c/D\approx 0$) the median Coulomb
logarithm measured from $N$-body models deviates from the theoretical relation,
converging towards a maximum value $\langle \ln(\Lambda)\rangle \approx 8.2$
independently of the mass and relative velocity of nearby substructures.
