Heat Conduction, and the Lack Thereof, in Time-Reversible Dynamical Systems: Generalized Nos\'e-Hoover Oscillators with a Temperature Gradient

We use nonequilibrium molecular dynamics to analyze and illustrate the qualitative differences between the one-thermostat and two-thermostat versions of equilibrium and nonequilibrium (heat-conducting) harmonic oscillators. Conservative nonconducting regions can coexist with dissipative heat conducting regions in phase space with exactly the same imposed temperature field.