Hamiltonian Thermostats Fail to Promote Heat Flow

Abstract

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 "phi-4" 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.

17 pages with four figures ; see also the many arXiv contributions mentioned in the references. Published in Communications in Nonlinear Science and Numerical Simulation 18, 3365-3372(2013); (received and accepted 10 May 2013, available online 17 May 2013)

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• Failure of Logarithmic Oscillators to Thermostat Small Atomic Clusters
• Heat pump without particle transport or external work on the medium achieved by differential thermostatting of the phase space
• Time-Irreversibility is Hidden Within Newtonian Mechanics
• Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat
• Singly-Thermostated Ergodicity in Gibbs' Canonical Ensemble and the 2016 Ian Snook Prize Award