Failure of logarithmic oscillators to serve as a thermostat for small atomic clusters

A logarithmic oscillator has the outstanding property that the expectation value of its kinetic energy is constant for all stationary states. Recently the ansatz that this property can be used to define a Hamiltonian thermostat has been put forward and a suggestion has been made that this logarithmic oscillator weakly coupled to a small system would serve as a thermostat as long as few degrees of freedom are involved as is the case in atomic clusters. We have applied these ideas to a cluster of four Lennard-Jones atoms and inspected two different models of coupling between the cluster and the logarithmic oscillator in three dimensions. In both cases we show that there is a clear generation of kinetic motion of the cluster center of mass, but that kinetic energy due to interatomic vibrations is not significantly affected by coupling to the logarithmic oscillator. This is a failure of the published ansatz, as the logarithmic oscillator is unable to modify the kinetic energy due to vibrations in small atomic clusters.