We consider random flights of point particles inside
n
n
-dimensional channels of the form
R
k
×
B
n
−
k
\mathbb {R}^{k}\times \mathbb {B}^{n-k}
, where
B
n
−
k
\mathbb {B}^{n-k}
is a ball of radius
r
r
in dimension
n
−
k
n-k
. The sequence of particle velocities taken immediately after each collision with the boundary of the channel comprise a Markov chain whose transition probabilities operator
P
P
is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the “microscopic” scale. Markov operators obtained in this way are natural, which means, in particular, that (1) the (at the surface) Maxwell-Boltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the so-called Knudsen cosine law, when this model is purely geometric, is the stationary distribution of
P
P
.
Our central concern is the relationship between the surface scattering properties encoded in
P
P
and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of
P
P
and compute, in the case of
2
2
-dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).