  A logarithmic oscillator has the outstanding property that the expectation
value of its kinetic energy is constant for all stationary states [1]. Recently
the ansatz that this property can be used to define a Hamiltonian thermostat
has been put forward [2]. The latter publication suggests that the logarithmic
oscillator weakly coupled to a small system would serve as a thermostat as long
as few degrees of freedom are involved such as 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 3D. 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 are not significantly affected by coupling to the logarithmic
oscillator. This is a failure of the published ansatz [2], as the logarithmic
oscillator is unable to modify the kinetic energy due to vibrations in small
atomic clusters.
