  We study diffusion properties of an inertial Brownian motor moving on a
ratchet substrate, i.e. a periodic structure with broken reflection symmetry.
The motor is driven by an unbiased time-periodic symmetric force which takes
the system out of thermal equilibrium. For selected parameter sets, the system
is in a non-chaotic regime in which we can identify a non-monotonic dependence
of the diffusion coefficient on temperature: for low temperature, it initially
increases as temperature grows, passes through its local maximum, next starts
to diminish reaching its local minimum and finally it monotonically increases
in accordance with the Einstein linear relation. Particularly interesting is
the temperature interval in which diffusion is suppressed by thermal noise and
we explain this effect in terms of transition rates of a three-state stochastic
model.
