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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 <ref>[http://arxiv.org/abs/1205.3478 Marc Meléndez Schofield "On the logarithmic oscillator as a thermostat", arXiv:1205.3478v1 (cond-mat.stat-mech) 15 May (2012)]</ref>. That is to say, if <math>H = E</math>, then the period of oscillation <math>t_{per}</math> increases with <math>E</math> according to <math>t_{per} \propto b e^{E/T}</math>. Furthermore, the maximum excursions of the oscillator also move outwards exponentially, <math>x_{max} \propto be^{E/T}</math>. This exponential scaling of time and length scales severely limits the practical applicability of the logarithmic thermostat <ref>[http://dx.doi.org/10.1103/PhysRevLett.110.028901 Marc Meléndez, Wm. G. Hoover, and Pep Español "Comment on “Logarithmic Oscillators: Ideal Hamiltonian Thermostats”", Physical Review Letters '''110''' 028901 (2013)]</ref>. Campisi ''et al.'' have defended that such a thermostat could work when interacting weakly with small atomic clusters <ref>[http://dx.doi.org/10.1103/PhysRevLett.110.028902 Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Campisi et al. Reply", Physical Review Letters '''110''' 028902 (2013)]</ref>, but further research has shown that the logarithmic oscillator does not generally behave as a thermostat even in that setting <ref>[http://dx.doi.org/10.1103/PhysRevE.89.021301 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)]</ref>. In addition, when two logarithmic oscillators with different values of <math>T</math> interact weakly with a system, they fail to promote heat flow
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 <ref>[http://arxiv.org/abs/1205.3478 Marc Meléndez Schofield "On the logarithmic oscillator as a thermostat", arXiv:1205.3478v1 (cond-mat.stat-mech) 15 May (2012)]</ref>. That is to say, if <math>H = E</math>, then the period of oscillation <math>t_{per}</math> increases with <math>E</math> according to <math>t_{per} \propto b e^{E/T}</math>. Furthermore, the maximum excursions of the oscillator also move outwards exponentially, <math>x_{max} \propto be^{E/T}</math>. This exponential scaling of time and length scales severely limits the practical applicability of the logarithmic thermostat <ref>[http://dx.doi.org/10.1103/PhysRevLett.110.028901 Marc Meléndez, Wm. G. Hoover, and Pep Español "Comment on “Logarithmic Oscillators: Ideal Hamiltonian Thermostats”", Physical Review Letters '''110''' 028901 (2013)]</ref>. Campisi ''et al.'' have defended that such a thermostat could work when interacting weakly with small atomic clusters <ref>[http://dx.doi.org/10.1103/PhysRevLett.110.028902 Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Campisi et al. Reply", Physical Review Letters '''110''' 028902 (2013)]</ref>, but further research has shown that the logarithmic oscillator does not generally behave as a thermostat even in that setting <ref>[http://dx.doi.org/10.1103/PhysRevE.89.021301 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)]</ref>. In addition, when two logarithmic oscillators with different values of <math>T</math> interact weakly with a system, they fail to promote heat flow
<ref>[https://doi.org/10.1016/j.cnsns.2013.05.010 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)]</ref>
<ref>[https://doi.org/10.1016/j.cnsns.2013.05.010 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)]</ref>
<ref>[https://dx.doi.org/10.1038%2Fs41598-017-03694-w 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)]</ref>. More recently, simulations have shown that logarithmic oscillators do not even respect the zeroth law of thermodynamics <ref>[https://www.nature.com/articles/s41598-018-30129-x Puneet Kumar Patra, and Baidurya Bhattacharya "Zeroth Law investigation on the logarithmic thermostat", Scientific Reports volume 8, Article number: 11670 (2018)]</ref>.
<ref>[https://dx.doi.org/10.1038%2Fs41598-017-03694-w 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)]</ref>.


==References==
==References==
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