  We consider and compare four Hamiltonian formulations of thermostated
mechanics, three of them kinetic, and the other one configurational. Though all
four approaches ``work'' at equilibrium, their application to many-body
nonequilibrium simulations can fail to provide a proper flow of heat. All the
Hamiltonian formulations considered here are applied to the same prototypical
two-temperature "phi-4" model of a heat-conducting chain. This model
incorporates nearest-neighbor Hooke's-Law interactions plus a quartic tethering
potential. Physically correct results, obtained with the isokinetic Gaussian
and Nose-Hoover thermostats, are compared with two other Hamiltonian results.
The latter results, based on constrained Hamiltonian thermostats, fail
correctly to model the flow of heat.

Chaotic Dynamics