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. 
Julien Clinton Sprott, William Graham Hoover, and Carol Griswold   Hoover
cond-mat.stat-mech nlin.CD physics.class-ph physics.comp-ph