  Many modern techniques employed in physics, such a computation of path
integrals, rely on random walks on graphs that can be represented as Markov
chains. Traditionally, estimates of running times of such sampling algorithms
are computed using the number of steps in the chain needed to reach the
stationary distribution. This quantity is generally defined as mixing time and
is often difficult to compute. In this paper, we suggest an alternative
estimate based on the Kolmogorov-Sinai entropy, by establishing a link between
the maximization of KSE and the minimization of the mixing time. Since KSE are
easier to compute in general than mixing time, this link provides a new faster
method to approximate the minimum mixing time that could be interesting in
computer sciences and statistical physics. Beyond this, our finding will also
be of interest to the out-of-equilibrium community, by providing a new rational
to select stationary states in out-of-equilibrium physics: it seems reasonable
that in a physical system with two simultaneous equiprobable possible dynamics,
the final stationary state will be closer to the stationary state corresponding
to the fastest dynamics (smallest mixing time).Through the empirical link found
in this letter, this state will correspond to a state of maximal
Kolmogorov-Sinai entropy. If this is true, this would provide a more satisfying
rule for selecting stationary states in complex systems such as climate than
the maximization of the entropy production.
