  There is a relation between the irreversibility of thermodynamic processes as
expressed by the breaking of time-reversal symmetry, and the entropy production
in such processes. We explain on an elementary mathematical level the relations
between entropy production, phase-space contraction and time-reversal starting
from a deterministic dynamics. Both closed and open systems, in the transient
and in the steady regime, are considered. The main result identifies under
general conditions the statistical mechanical entropy production as the source
term of time-reversal breaking in the path space measure for the evolution of
reduced variables. This provides a general algorithm for computing the entropy
production and to understand in a unified way a number of useful
(in)equalities. We also discuss the Markov approximation. Important are a
number of old theoretical ideas for connecting the microscopic dynamics with
thermodynamic behavior.
