  Recently, the author and Zia (2010) reported on exact results for a
far-from-equilibrium system in which two coupled semi-infinite Ising chains at
temperatures $T_h$ and $T_c$, with $T_h>T_c$, establish a flux of energy across
their junction. This paper provides a complete derivation of those results,
more explicit expressions for the energy flux, and a more detailed
characterization of the system at arbitrary $T_c$ and $T_h$. We consider the
two-point correlation functions and the energy flux $F(x)$ between each spin,
located at integer position $x$, and its associated heat bath. In the $T_h
\rightarrow \infty$ limit, the flux $F(x)$ decays exponentially into the cold
bath (spins with $x=1,2,...$) for all $T_c>0$ and transitions into a power law
decay as $T_c \rightarrow 0$. We find an asymptotic expansion for large $x$ in
terms of modified Bessel functions that captures both of these behaviors. We
perform Monte Carlo simulations that give excellent agreement with both the
exact and asymptotic results for $F(x)$. The simulations are also used to study
the system at arbitrary $T_h$ and $T_c$.
