We establish relationships between the generalized data processing theorems of
Zakai and Ziv (1973 and 1975) and the dynamical version of the second law of
thermodynamics, or, more precisely, the Boltzmann H-Theorem, 
which asserts that the Shannon entropy,
$H(X_t)$, pertaining to a finite-state Markov process $\{X_t\}$, is
monotonically non-decreasing as a function of time $t$, provided that the steady--state
distribution of this process is uniform across
the state space (which is the case when the process designates an isolated
system). It turns out
that both the generalized data processing theorems and the Boltzmann
H-Theorem can be viewed as special cases of a more general principle
concerning the monotonicity (in time) of a certain generalized information
measure applied to a Markov process. This gives rise to a new look at the
generalized data processing
theorem, which suggests to exploit certain degrees of freedom that
may lead to better bounds, for a given choice of
the convex function that defines the generalized mutual information.