  Although an intimate relation between entropy and diffusion has been
advocated for many years and even seems to have been verified in theory and
experiments, a quantitatively reliable study, and any derivation of an
algebraic relation between the two does not seem to exist. Here we explore the
nature of this entropy-diffusion relation in three deterministic systems where
an accurate estimate of both can be carried out. We study three deterministic
model systems, (a) the motion of a single point particle with constant energy
in a two-dimensional periodic potential energy landscape, (b) the same in
regular Lorentz gas where a point particle with constant energy moves between
collisions with hard disc scatterers and (c) motion of a point particle among
the boxes with small apertures. These models, introduced by Zwanzig, exhibit
diffusive motion in the limit where ergodicity is shown to exist. We then
explore the diffusion-entropy relation by an accurate calculation of both
diffusion and entropy for the aforementioned model systems. We estimate the
self-diffusion coefficient of the particle by employing computer simulations
and entropy by quadrature using Boltzmann's formula. We observe an interesting
crossover in the diffusion-entropy relation in some specific regions which is
attributed to the emergence of correlated returns. The crossover could herald a
breakdown of the Rosenfeld-like exponential scaling between the two, as
observed at low temperatures. Later, we modify the scaling relation to account
for the correlated motions and present a detailed analysis of the dynamical
entropy obtained via Lyapunov exponent which is rather an important quantity in
the study of deterministic systems.
