# Logarithmic oscillator thermostat

The **Logarithmic oscillator** ^{[1]} in one dimension is given by (Eq. 2):

where is the position of the logarithmic oscillator, is its linear momentum, and represents its mass. is the desired temperature of the thermostat, and sets a length-scale.

## As a thermostat[edit]

From the Virial theorem

one obtains

- .

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,

this implies that the heat capacity becomes

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

## Practical applicability[edit]

The averages considered above make sense only for times greater than the period of the logarithmic oscillator but, because of the logarithmic shape of the potential, the period increases proportionally to the exponential of the total energy ^{[2]}. That is to say, if , then the period of oscillation increases with according to . Furthermore, the maximum excursions of the oscillator also move outwards exponentially, . This exponential scaling of time and length scales severely limits the practical applicability of the logarithmic thermostat ^{[3]}. Campisi *et al.* have defended that such a thermostat could work when interacting weakly with small atomic clusters ^{[4]}, but further research has shown that the logarithmic oscillator does not generally behave as a thermostat even in that setting ^{[5]}. In addition, when two logarithmic oscillators with different values of interact weakly with a system, they fail to promote heat flow
^{[6]}
^{[7]}. More recently, simulations have shown that logarithmic oscillators do not even respect the zeroth law of thermodynamics ^{[8]}.

## References[edit]

- ↑ Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Logarithmic Oscillators: Ideal Hamiltonian Thermostats", Physical Review Letters
**108**250601 (2012) - ↑ Marc Meléndez Schofield "On the logarithmic oscillator as a thermostat", arXiv:1205.3478v1 (cond-mat.stat-mech) 15 May (2012)
- ↑ Marc Meléndez, Wm. G. Hoover, and Pep Español "Comment on “Logarithmic Oscillators: Ideal Hamiltonian Thermostats”", Physical Review Letters
**110**028901 (2013) - ↑ Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Campisi et al. Reply", Physical Review Letters
**110**028902 (2013) - ↑ Daniel Sponseller and Estela Blaisten-Barojas "Failure of logarithmic oscillators to serve as a thermostat for small atomic clusters", Physical Review E
**89**021301(R) (2014) - ↑ Wm.G.Hoover and Carol G.Hoover "Hamiltonian thermostats fail to promote heat flow", Communications in Nonlinear Science and Numerical Simulation,
**18**pp. 3365-3372 (2013) - ↑ Kai Chen, Dahai He, and Hong Zhao "Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat", Scientific Reports
**7**Article number: 3460 (2017) - ↑ Puneet Kumar Patra, and Baidurya Bhattacharya "Zeroth Law investigation on the logarithmic thermostat", Scientific Reports volume 8, Article number: 11670 (2018)

- Related reading