Logarithmic oscillator thermostat - Revision history

84.78.225.118: /* Practical applicability */

2020-12-22T11:54:28Z

Practical applicability

← Older revision		Revision as of 13:54, 22 December 2020
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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>.		<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>.

	==References==		==References==

Carl McBride: Added the authors to two references

2017-07-11T15:00:21Z

Added the authors to two references

← Older revision		Revision as of 17:00, 11 July 2017
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	:<math>H = \frac{P^2}{2M}+ T \ln \frac{\vert X \vert}{b}</math>		:<math>H = \frac{P^2}{2M}+ T \ln \frac{\vert X \vert}{b}</math>

	where <math>X</math> is the position of the logarithmic oscillator, <math>P</math> is its linear momentum, and <math>M</math> represents its mass. <math>T</math> is the desired [[temperature]] of the thermostat, and <math>b > 0</math> sets a length-scale.		where <math>X</math> is the position of the logarithmic oscillator, <math>P</math> is its linear momentum, and <math>M</math> represents its mass. <math>T</math> is the desired [[temperature]] of the [[Thermostats\|thermostat]], and <math>b > 0</math> sets a length-scale.
	==As a thermostat==		==As a thermostat==
	From the [[Virial theorem]]		From the [[Virial theorem]]
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	Having an infinite heat capacity is an ideal feature for a thermostat.		Having an infinite heat capacity is an ideal feature for a thermostat.
	==Practical applicability==		==Practical applicability==
	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 "Hamiltonian thermostats fail to promote heat flow" Communications in Nonlinear Science and Numerical Simulation, '''18'''~~, 12,~~ 3365-3372 ~~December~~ (2013)]</ref>		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://dx.doi.org/10.1038%2Fs41598-017-03694-w "Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat", Scientific Reports 7, Article number: 3460 ~~14 June~~ (2017)]</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>.

	==References==		==References==

150.244.119.82: /* Practical applicability */

2017-07-11T12:04:48Z

Practical applicability

← Older revision		Revision as of 14:04, 11 July 2017
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	Having an infinite heat capacity is an ideal feature for a thermostat.		Having an infinite heat capacity is an ideal feature for a thermostat.
	==Practical applicability==		==Practical applicability==
	<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>		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 "Hamiltonian thermostats fail to promote heat flow" Communications in Nonlinear Science and Numerical Simulation, '''18''', 12, 3365-3372 December (2013)]</ref>
	<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>		<ref>[https://dx.doi.org/10.1038%2Fs41598-017-03694-w "Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat", Scientific Reports 7, Article number: 3460 14 June (2017)]</ref>.
	<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>
	<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>
	<ref>[https://dx.doi.org/10.1038%2Fs41598-017-03694-w "Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat", Scientific Reports 7, Article number: 3460]</ref>

	==References==		==References==

150.244.119.82: /* Practical applicability */

2017-07-11T11:21:08Z

Practical applicability

← Older revision		Revision as of 13:21, 11 July 2017
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	<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>		<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>
	<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>		<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>
			<ref>[https://dx.doi.org/10.1038%2Fs41598-017-03694-w "Violation of the virial theorem and generalized equipartition theorem for logarithmic oscillators serving as a thermostat", Scientific Reports 7, Article number: 3460]</ref>

	==References==		==References==
	<references/>		<references/>

Carl McBride at 15:13, 16 April 2015

2015-04-16T15:13:19Z

@@ Line 1: / Line 1: @@
-{{Stub-general}}
+The '''Logarithmic oscillator''' <ref>[http://dx.doi.org/10.1103/PhysRevLett.108.250601 Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Logarithmic Oscillators: Ideal Hamiltonian Thermostats", Physical Review Letters '''108''' 250601 (2012)]</ref> in one dimension is given by (Eq. 2):
-The '''Logarithmic oscillator thermostat''' <ref>[http://dx.doi.org/10.1103/PhysRevLett.108.250601 Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Logarithmic Oscillators: Ideal Hamiltonian Thermostats", Physical Review Letters '''108''' 250601 (2012)]</ref>.
 ==Practical applicability==
 <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>
 <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>
 ==References==
 <references/>

Carl McBride: Created page with "{{Stub-general}} The '''Logarithmic oscillator thermostat''' [http://dx.doi.org/10.1103/PhysRevLett.108.250601 Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi..."

2015-04-16T13:55:48Z

Created page with "{{Stub-general}} The '''Logarithmic oscillator thermostat''' <ref>[http://dx.doi.org/10.1103/PhysRevLett.108.250601 Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi..."

New page

{{Stub-general}}
The '''Logarithmic oscillator thermostat''' <ref>[http://dx.doi.org/10.1103/PhysRevLett.108.250601 Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Logarithmic Oscillators: Ideal Hamiltonian Thermostats", Physical Review Letters '''108''' 250601 (2012)]</ref>.
==Practical applicability==
<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>
<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>
==References==
<references/>
;Related reading

[[category: statistical mechanics]]