Logarithmic oscillator thermostat: Difference between revisions

Revision as of 17:13, 16 April 2015

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

H = \frac{P^{2}}{2 M} + T \ln \frac{| X |}{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

⟨ X \frac{\partial H}{\partial X} ⟩ = ⟨ P \frac{\partial H}{\partial P} ⟩

one obtains

T = ⟨ \frac{P^{2}}{M} ⟩

.

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,

\frac{\partial T}{\partial U} = 0

this implies that the heat capacity becomes

C_{V} : = {\frac{\partial U}{\partial T} |}_{V} = \infty

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

Practical applicability

^[2] ^[3] ^[4] ^[5]

References

Related reading

[1] Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Logarithmic Oscillators: Ideal Hamiltonian Thermostats", Physical Review Letters 108 250601 (2012)

[2] Marc Meléndez Schofield "On the logarithmic oscillator as a thermostat", arXiv:1205.3478v1 (cond-mat.stat-mech) 15 May (2012)

[3] Marc Meléndez, Wm. G. Hoover, and Pep Español "Comment on “Logarithmic Oscillators: Ideal Hamiltonian Thermostats”", Physical Review Letters 110 028901 (2013)

[4] Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi "Campisi et al. Reply", Physical Review Letters 110 028902 (2013)

[5] 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)

[1]

[2]

[3]

[4]

[5]

@@ 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>.
+:<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==
 <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.110.028902 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/>

Logarithmic oscillator thermostat: Difference between revisions

Revision as of 17:13, 16 April 2015

As a thermostat

Practical applicability

References

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