Logarithmic oscillator thermostat

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

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

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

$⟨X\frac{\partial H}{\partial X}⟩=⟨P\frac{\partial H}{\partial P}⟩$

one obtains

$T=⟨\frac{P^2}{M}⟩$.

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

$C_V:={\frac{\partial U}{\partial T}|}_V=\mathrm{\infty }$

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

References

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