  We consider the quasi-deterministic behavior of systems with a large number,
$n$, of deterministically interacting constituents. This work extends the
results of a previous paper [J. Stat. Phys. 99:1225-1249 (2000)] to include
vector-valued observables on interacting systems. The approach used here,
however, differs markedly in that a level-1 large deviation principle (LDP) on
joint observables, rather than a level-2 LDP on empirical distributions, is
employed. As before, we seek a mapping $\psi_{t}$ on the set of (possibly
vector-valued) macrostates such that, when the macrostate is given to be $a_0$
at time zero, the macrostate at time $t$ is $\psi_{t}(a_0)$ with a probability
approaching one as $n$ tends to infinity. We show that such a map exists and
derives from a generalized dynamic free energy function, provided the latter is
everywhere well defined, finite, and differentiable. We discuss some general
properties of $\psi_{t}$ relevant to issues of irreversibility and end with an
example of a simple interacting lattice, for which an exact macroscopic
solution is obtained.
