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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial T}{\partial U} = 0}

this implies that the heat capacity becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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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