Logarithmic oscillator thermostat

The Logarithmic oscillator in one dimension is given by (Eq. 2):

$H={\frac {P^{2}}{2M}}+T\ln {\frac {\vert X\vert }{b}}$

where $X$ is the position of the logarithmic oscillator, $P$ is its linear momentum, and $M$ represents its mass. $T$ is the desired temperature of the thermostat, and $b>0$ sets a length-scale.

As a thermostat

From the Virial theorem

$\left\langle X{\frac {\partial H}{\partial X}}\right\rangle =\left\langle P{\frac {\partial H}{\partial P}}\right\rangle$

one obtains

$T=\left\langle {\frac {P^{2}}{M}}\right\rangle$.

This implies that all expectation values of the trajectories correspond to the very same temperature of the thermostat, irrespective of the internal energy. In other words,

${\frac {\partial T}{\partial U}}=0$

this implies that the heat capacity becomes

$C_{V}:=\left.{\frac {\partial U}{\partial T}}\right\vert _{V}=\infty$

Having an infinite heat capacity is an ideal feature for a thermostat.

References

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