Hamiltonian thermostats fail to promote heat flow

•Computational thermostats stabilize either kinetic or configurational temperature.•Computational thermostats based on Hamiltonian mechanics fail away from equilibrium.•Thermostats based on the Nosé–Hoover approach are best at and away from equilibrium.

Hamiltonian mechanics can be used to constrain temperature simultaneously with energy. We illustrate the interesting situations that develop when two different   temperatures are imposed within a composite Hamiltonian system. The model systems we treat are ϕ4ϕ4 chains, with quartic tethers and quadratic nearest-neighbor Hooke’s-law interactions. This model is known to satisfy Fourier’s law. Our prototypical problem sandwiches a Newtonian subsystem between hot and cold Hamiltonian reservoir regions. We have characterized four different Hamiltonian reservoir types. There is no tendency for any of these two-temperature Hamiltonian simulations to transfer heat from the hot to the cold degrees of freedom. Evidently steady heat flow simulations require energy sources and sinks, and are therefore incompatible with Hamiltonian mechanics.