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Inverted pendulum driven by a horizontal random force: statistics of the never-falling trajectory and supersymmetry
by Nikolai A. Stepanov, Mikhail A. Skvortsov
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Mikhail Skvortsov |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2006.13819v2 (pdf) |
| Date accepted: | 2022-07-20 |
| Date submitted: | 2022-06-30 09:14 |
| Submitted by: | Skvortsov, Mikhail |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study stochastic dynamics of an inverted pendulum subject to a random force in the horizontal direction (Whitney's problem). Considered on the entire time axis, the problem admits a unique solution that always remains in the upper half plane. We formulate the problem of statistical description of this never-falling trajectory and solve it by a field-theoretical technique assuming a white-noise driving. In our approach based on the supersymmetric formalism of Parisi and Sourlas, statistic properties of the never-falling trajectory are expressed in terms of the zero mode of the corresponding transfer-matrix Hamiltonian. The emerging mathematical structure is similar to that of the Fokker-Planck equation, which however is written for the "square root" of the probability distribution function. Our results for the statistics of the non-falling trajectory are in perfect agreement with direct numerical simulations of the stochastic pendulum equation. In the limit of strong driving (no gravitation), we obtain an exact analytical solution for the instantaneous joint probability distribution function of the pendulum's angle and its velocity.
Author comments upon resubmission
We appreciate your efforts in searching Referees for our work at the edge of several different fields in mathematical/theoretical physics.
We thank the Referees for critically reading the manuscript and pointing out the points deserving better explanation.
In the revised manuscript, we respond to the main concerns of both Referees.
Among the main changes, we add a one-page proof of the uniqueness of a non-FT and extend discussion of how our approach is related to physics of inhomogeneous superconductors.
The derivation, results and figures remain intact.
We believe that provided changes improve the text, outlining several crucial points.
Some questions of both Referees are answered in the corresponding replies.
Sincerely,
N. Stepanov and M. Skvortsov
List of changes
1. A detailed discussion of the uniqueness of a non-FT in Whitney's problem is given in a new separate Appendix A (and a reference to this Appendix in Sec. 1, third paragraph from the end).
2. An explanation that the zero mode of the transfer-matrix Hamiltonian is intimately related to the existence of an non-FT is added (second paragraph of Sec. 2.3).
3. Discussion of the Lyapunov exponent, which is a measure of the divergence of trajectories, and its connection to higher modes of the transfer-matrix Hamiltonian is added (the second paragraph of Sec. 5).
4. We extended he last part of Conclusion by discussing the relationof the developed approach to inhomogeneous superconductors (the last paragraph of Sec. 5).
5. A number of new references is added with the new portions of the text: 34, 47-52.
Published as SciPost Phys. 13, 021 (2022)
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 2022-7-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2006.13819v2, delivered 2022-07-03, doi: 10.21468/SciPost.Report.5325
Report
I am fine with the response and I recommend the paper. As I previously mentioned I am not familiar with the mathematical formalism used to derive the results, but it is explained well in the Appendixes. Existence of such a bound trajectory for any drive does not look obvious to me.
Perhaps I would comment on the stability of this trajectory. I assume that longer and longer times require finer and finer tuning of the initial conditions. For the undriven problem, where such a non-failing trajectory is obvious, the time probably scales logarithmically with the error with the prefactor set by $1/\omega$. It looks like the situation remains the same in the linearized regime analyzed in Sec. 2.1. I think it might be interesting to comment how (at least qualitatively) this dependence changes in the nonlinear regime. I do not insist on this addition and leave this question up to the author's judgement.
Another perhaps related comment is that if we consider a real quantum problem by the uncertainty principle initial error is never zero and scales as roughly the Planck's constant. So I assume that quantum mechanics destroys the phenomenon and at best the system can be stabilized for a finite time. I understand that this might be a separate problem, but perhaps it is worth commenting about.
Author: Mikhail Skvortsov on 2022-07-21 [id 2674]
(in reply to Report 1 on 2022-07-03)
1) The question of the stability of the non-falling trajectory is in fact related to the Lyapunov exponent $\lambda$, which controls exponential divergence of trajectories. It is this quantity that comes as a prefactor in front of the logarithm of the time the Referee is interested in. In the case of weak drive, the Lyapunov exponent is indeed given by $\omega$, in accordance with the Referee. For arbitrary drive, $\lambda(\omega,\alpha)$ is calculated in the paper [N. A. Stepanov and M. A. Skvortsov, Lyapunov exponent for Whitney’s problem with random drive, JETP Lett. 112, 376 (2020), doi:10.1134/S0021364020190017; arXiv:2008.12013], which is cited as Ref. 47 in the last version of the manuscript. In the case of strong drive, $\alpha\gg\omega^3$, the Lyapunov exponent is $\lambda\approx 1.66\, \alpha^{1/3}$.
2) Quantum effects will certainly destroy classical description at large time scales, regardless of the presence of noise. For a physical pendulum such a question might be relevant. However, motivated by the physics of disordered superconductors where the pendulum equation arises with the coordinate playing the role of time (as discussed in the Conclusion part in the last version of the manuscript), we consider the pendulum equation of motion just as a mathematical equation and study its non-falling solutions.
Anonymous on 2022-07-03 [id 2627]
I am fine with the response and I recommend the paper. As I previously mentioned I am not familiar with the mathematical formalism used to derive the results, but it is explained well in the Appendixes. Existence of such a bound trajectory for any drive does not look obvious to me.
Perhaps I would comment on the stability of this trajectory. I assume that longer and longer times require finer and finer tuning of the initial conditions. For the undriven problem, where such a non-failing trajectory is obvious, the time probably scales logarithmically with the error with the prefactor set by $1/\omega$. It looks like the situation remains the same in the linearized regime analyzed in Sec. 2.1. I think it might be interesting to comment how (at least qualitatively) this dependence changes in the nonlinear regime. I do not insist on this addition and leave this question up to the author's judgement.
Another perhaps related comment is that if we consider a real quantum problem by the uncertainty principle initial error is never zero and scales as roughly the Planck's constant. So I assume that quantum mechanics destroys the phenomenon and at best the system can be stabilized for a finite time. I understand that this might be a separate problem, but perhaps it is worth commenting about.