  Entropy production in stochastic mechanical systems is examined here with
strict bounds on its rate. Stochastic mechanical systems include pure
diffusions in Euclidean space or on Lie groups, as well as systems evolving on
phase space for which the fluctuation-dissipation theorem applies, i.e.,
return-to-equilibrium processes. Two separate ways for ensembles of such
mechanical systems forced by noise to reach equilibrium are examined here.
First, a restorative potential and damping can be applied, leading to a
classical return-to-equilibrium process wherein energy taken out by damping can
balance the energy going in from the noise. Second, the process evolves on a
compact configuration space (such as random walks on spheres, torsion angles in
chain molecules, and rotational Brownian motion) lead to long-time solutions
that are constant over the configuration space, regardless of whether or not
damping and random forcing balance. This is a kind of potential-free
equilibrium distribution resulting from topological constraints. Inertial and
noninertial (kinematic) systems are considered. These systems can consist of
unconstrained particles or more complex systems with constraints, such as
rigid-bodies or linkages. These more complicated systems evolve on Lie groups
and model phenomena such as rotational Brownian motion and nonholonomic robotic
systems. In all cases, it is shown that the rate of entropy production is
closely related to the appropriate concept of Fisher information matrix of the
probability density defined by the Fokker-Planck equation. Classical results
from information theory are then repurposed to provide computable bounds on the
rate of entropy production in stochastic mechanical systems.
