  A logarithmic oscillator has been proposed recently to serve as a thermostat
recently since it has a peculiar property of infinite heat capacity according
to the virial theorem. In order to examine its feasibility by numerical
simulations, a modified logarithmic potential has been applied in previous
studies to eliminate the singularity at origin. The role played by the
modification has been elucidated in the present study. We argue that the virial
theorem is practically violated for the modified log-oscillator illustrated by
a linear dependence of kinetic temperature on energy. Furthermore, as far as a
thermalized log-oscillator is concerned, the generalized equipartition theorem
with respect to the position coordinate is broken if the temperature is higher
than a critical temperature. Finally, we show that log-oscillators fail to
serve as thermostats for its incapability of maintaining a nonequilibrium
steady state even though their energy is appropriately assigned.
