  We explore the consequences of a deterministic microscopic
thermostat-reservoir contact mechanism. With different temperature reservoirs
at each end of a two-dimensional system, a heat current is produced and the
system has an anomalous thermal conductivity. The microscopic form for the
local heat flux vector is derived and both the kinetic and potential
contributions are calculated. The total heat flux vector is shown to satisfy
the continuity equation. The properties of this nonequilibrium steady state are
studied as a function of system size and temperature gradient identifying key
scaling relations for the local fluid properties and separating bulk and
boundary effects. The local entropy density calculated from the local
equilibrium distribution is shown to be a very good approximation to the
entropy density calculated directly from velocity distribution even for systems
that are far from equilibrium. The dissipation and kinetic entropy production
and flux are compared quantitatively and the differing mechanisms discussed
within the BGY approximation. For equal temperature reservoirs the entropy
production near the reservoir walls is shown to be proportional to the local
phase space contraction calculated from the tangent space dynamics. However,
for unequal temperatures, the connection between local entropy production and
local phase space contraction is more complicated.
