Langevin approach to the Porto system
<p>Plot of the dimensionless rotor-track potential energy <span class="html-italic">ψ</span> for <span class="html-italic">γ</span> = +1 as a function of <span class="html-italic">X</span> and Θ.</p> "> Figure 2
<p>Time dependence of angle Θ as a function of <span class="html-italic">T</span> for <span class="html-italic">γ</span> = +1 and the same values of all the input parameters as specified in the main text. In all three cases, the impact times as well as the random forces were identical. Only in time intervals between any two individual impacts, different integration time steps (1/32, 1/16, 1/10, 1/8 and 1/7 for curves a) to e), respectively) were used. The onset of chaos appears at about <span class="html-italic">T</span> = 1200. At <span class="html-italic">T</span> = 1000, the values of Θ obtained still coincided to 5 digits.</p> "> Figure 3
<p>The same for <span class="html-italic">X</span> as a function of <span class="html-italic">T</span> for the same values of all the input parameters. At <span class="html-italic">T</span> = 1000, the values of <span class="html-italic">X</span> obtained still coincided to 6 digits. Notice that all the curves increase with increasing time.</p> "> Figure 4
<p>Dependence of X on T for five different sequences of the random numbers involved for the ‘forward gear’ <span class="html-italic">γ</span> = +1.</p> "> Figure 5
<p>The same for the ‘reverse gear’ <span class="html-italic">γ</span> = −1. The same sequences of random numbers as in <a href="#entropy-06-00057-f004" class="html-fig">Figure 4</a> were used.</p> "> Figure 6
<p>Details of time-dependence of <span class="html-italic">X</span>(<span class="html-italic">T</span>) (<span class="html-italic">γ</span> = +1) for curve ‘c’ of <a href="#entropy-06-00057-f004" class="html-fig">Figure 4</a> before the first impact event in Θ occurs. Times <span class="html-italic">T</span> of the first six impacts in <span class="html-italic">X</span> were 4.4506, 5.0489, 5.9216, 9.7589, 9.9581, and 10.7401. The first impact in Θ appears at <span class="html-italic">T</span> = 216.8751. Worth noticing is how the impacts cause, because of the periodicity of <span class="html-italic">ψ</span>, sudden changes of <span class="html-italic">X</span> = <span class="html-italic">x</span>/<span class="html-italic">b</span> (on the vertical axis) by integers.</p> "> Figure 7
<p>Dependence of X on T for five different sequences of the random numbers involved for the ‘forward gear’ <span class="html-italic">γ</span> = +1. All the input parameters as well as the random forces were identical as in <a href="#entropy-06-00057-f004" class="html-fig">Figure 4</a>, but with the linear potential <span class="html-italic">c</span> · <span class="html-italic">X</span>, <span class="html-italic">c</span> = 0.01 added to <span class="html-italic">ψ</span>(<span class="html-italic">X</span>, Θ).</p> ">
Abstract
:1 Introduction
- Second, the present model and its present treatment, though they are perhaps more realistic than often in similar situations, are still just theoretical and their relation to Nature may be not as obvious as it might seem at the first sight.
2 Formulation of the problem
- the Θ(t) variable has its time-dependence determined also from a dynamic equation, and
- this approach respects existing connections between dissipation (friction) incorporated and by its effect decisive in the Porto model, and properties of the stochastic forces on the right hand side of the Langevin equations that were completely ignored in [6].
3 Numerical treatment
4 Numerical solution
5 Results
6 Conclusions
- Confirmation, at finite temperatures, that the rotor has a tendency to move along the track in prevailingly one direction only. This tendency was already found, in his simplified treatment corresponding inter alia to zero temperature, by Porto [6].
- Identification of another mechanism (in addition to that one found by Porto himself) also leading the unidirectional motion of the rotor along the train.
- Survival of the behavior even when the motion goes against a weak potential field.
7 Acknowledgement
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Bok, J.; Cápek, V. Langevin approach to the Porto system. Entropy 2004, 6, 57-67. https://doi.org/10.3390/e6010057
Bok J, Cápek V. Langevin approach to the Porto system. Entropy. 2004; 6(1):57-67. https://doi.org/10.3390/e6010057
Chicago/Turabian StyleBok, Jirí, and Vladislav Cápek. 2004. "Langevin approach to the Porto system" Entropy 6, no. 1: 57-67. https://doi.org/10.3390/e6010057