  Two aspects of conductive heat are focused here (i) the nature of conductive
heat, defined as that form of energy that is transferred as a result of a
temperature difference and (ii) the nature of the intermolecular potentials
that induces both thermal energy flow and the temperature profile at the steady
state for a 1-D lattice chain. It is found that the standard presuppositions of
people like Benofy and Quay (BQ) following Joseph Fourier do not obtain for at
least a certain specified regime of intermolecular potential parameters related
to harmonic (quadratic) potentials for nearest neighbor interactions. For these
harmonic potentials, it appears from the simulation results that steady state
solutions exist utilizing non-synthetic thermostats that couple not just the
two particles at the extreme ends of the lattice chain, but to a control volume
of $N$ particles located at either ends of the chain that does not accord with
the unique analytical solutions that obtains for single particle thermostatting
at the ends of the lattice with a different thermostatting algorithm that
utilizes coupling coefficients. If the method used here is considered a more
"realistic" or feasible model of the physical reality, then a re-evaluation of
some aspects of the standard theoretical methodology is warranted since the
standard model solution profile does not accord with the simulation temperature
profile determined here for this related model. We also note that the
sinusoidal temperature profile generated suggests that thermal integrated
circuits with several thermal P-N junctions may be constructed, opening a way
to create more complex thermal transistor circuits. A stationary principle is
proposed for regions that violate the Fourier principle $\mathbf{J_q.}\nabla T
\le 0 $, where $\mathbf{J_q}$ is the heat current vector and $T$ the
temperature.
