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

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