  We investigate a solvable model for energy conserving non-equilibrium steady
states. The time-reversal asymmetry of the dynamics leads to the violation of
detailed balance and to ergodicity breaking, as manifested by the presence of
dynamically inaccessible states. Two such systems in contact do not reach the
same effective temperature if standard definitions are used. However, we
identify the effective temperature that controls energy flow. Although this
operational temperature does reach a common value upon contact, the total
entropy of the joint system can decrease.
