Logarithmic oscillator thermostat

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

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

where 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 X} is the position of the logarithmic oscillator, 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 P} is its linear momentum, 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 M} 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

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 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 H = E} , then the period of oscillation 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_{per}} increases with 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 E} according to 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_{per} \propto b e^{E/T}} . Furthermore, the maximum excursions of the oscillator also move outwards exponentially, 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 x_{max} \propto be^{E/T}} . 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 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} interact weakly with a system, they fail to promote heat flow [6] [7].

References

  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)
  6. ↑ 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)
  7. ↑ 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)
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