Logarithmic oscillator thermostat: Difference between revisions
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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 | |||
:<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. | |||
==As a thermostat== | |||
From the [[Virial theorem]] | |||
:<math>\left\langle X\frac{\partial H}{\partial X} \right\rangle = \left\langle P\frac{\partial H}{\partial P} \right\rangle </math> | |||
one obtains | |||
:<math>T = \left\langle \frac{P^2}{M} \right\rangle </math>. | |||
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, | |||
:<math>\frac{\partial T}{\partial U} = 0</math> | |||
this implies that the [[heat capacity]] becomes | |||
:<math>C_V := \left. \frac{\partial U}{\partial T} \right\vert_V = \infty </math> | |||
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> | <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> | <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>[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> | |||
==References== | ==References== | ||
<references/> | <references/> | ||
Revision as of 16:13, 16 April 2015
The Logarithmic oscillator [1] in one dimension is given by (Eq. 2):
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle H={\frac {P^{2}}{2M}}+T\ln {\frac {\vert X\vert }{b}}}
where is the position of the logarithmic oscillator, is its linear momentum, and represents its mass. 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 T} is the desired temperature of the thermostat, and 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 b > 0} sets a length-scale.
As a thermostat
From the Virial theorem
- 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 \left\langle X\frac{\partial H}{\partial X} \right\rangle = \left\langle P\frac{\partial H}{\partial P} \right\rangle }
one obtains
- 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 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.
Practical applicability
References
- ↑ 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)
- Related reading