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xSPDE3: extensible software for stochastic ordinary and partial differential equations
by Simon Kiesewetter , Ria R. Joseph, Peter D. Drummond
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Submission summary
| Authors (as registered SciPost users): | Peter Drummond · Simon Kiesewetter |
| Submission information | |
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| Preprint Link: | scipost_202212_00053v1 (pdf) |
| Code repository: | https://github.com/peterddrummond/xspde_matlab |
| Date submitted: | 2022-12-19 00:07 |
| Submitted by: | Kiesewetter, Simon |
| Submitted to: | SciPost Physics Codebases |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
The xSPDE toolbox treats stochastic partial and ordinary differential equations, with applications in biology, chemistry, engineering, medicine, physics and quantum technologies. It computes statistical averages, including time-step and/or sampling error estimation. xSPDE can provide higher order convergence, Fourier spectra and probability densities. The toolbox has graphical output and χ 2 statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. All equations may have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector field component, and at either end of any interval.
Current status:
Reports on this Submission
Report 1 by Said Rodriguez on 2023-1-31 (Invited Report)
- Cite as: Said Rodriguez, Report on arXiv:scipost_202212_00053v1, delivered 2023-01-31, doi: 10.21468/SciPost.Report.6645
Strengths
- Excellent presentation of an excellent computational toolbox for solving stochastic ordinary and partial differential equations
- Very well written and organized manuscript
- Fair amount of theoretical background
- Plenty of examples illustrating the capabilities of the toolbox
- Sufficient technical details about algorithms, analysis tools, and other options of the toolbox
- The xSPDE software itself is powerful, user friendly, and easy to build upon
Weaknesses
- No significant weaknesses
- There are some minor points that could be improved. I discuss these in the list of requested changes.
Report
The manuscript presents a detailed overview of a computational toolbox, called xSPDE3, for solving stochastic ordinary and partial differential equations. The software is powerful, versatile, and especially user friendly. I strongly believe that xSPDE3 will become an invaluable resource for many researchers and students who wish to understand the effects of noise on dynamical systems. I strongly support the publication of this manuscript. It is an excellent guide for the toolbox itself, which is readily available for downloading.
While the authors have my unconditional support for the publication of this work in SciPost Physics, I would still like to make a few suggestions. I provide these in the list of "requested changes".
Requested changes
1. For those of us familiar with xSPDE, which was much more briefly presented in a previous publication in SoftwareX, the main question is: What is new in xSPDE3? What are the main improvements and additions? It would be good to have their essence mentioned in the abstract and introduction, and then described in a bit more detail wherever relevant in the manuscript.
2. Equation 38, one of the simplest models used to introduce the capabilities of the toolbox, is presented as a "damped quantum harmonic oscillator". There are a couple of minor semantics issues here. First, there is nothing "quantum" about this oscillator. It is a simple oscillator whose statistical properties are fully described by classical physics. Second, there is no discussion about the damping constant in this equation. Of course, I understand that a is complex and its real part describes the damping and the imaginary part describes the oscillation frequency. But it would be good if the authors could explain this in 1 or 2 sentences . The extra clarity could be especially useful for students, since this is one of the first problems they will encounter in the manuscript.
3. Related to the previous point: Right below Equation 46 , the authors argue that the system (the same one of a simple classical oscillator) is essentially quantum mechanical. I disagree. There is nothing quantum in this system. Moreover, similar input-output relations as in Equations 45 and 46 hold for purely classical systems.
4. The example in section 4.7.5. seems to be a new one. It is great that the authors included it! However, there are too few details for the vast majority of users to understand what is going on here. The authors give a few generic references to books where similar equations are used. But it would be very valuable if the authors could also explain, in this manuscript, where these equations come from. To begin, what does each term describe? what is a1, a2, c, and lambda? The equations seem to describe two damped oscillators which are coupled both linearly and nonlinearly. In the parlance of optics, the nonlinear coupling would be called a cross-Kerr nonlinearity. Can the authors confirm that? And what is the relevance of the particular type of multiplicative noise they assume?
Then, the authors say that one can use this model to investigate purely quantum effects like entanglement, EPR paradoxes, and Bell violations. I'm sorry, but I do not understand how that is possible. There are only 2 equations for mean fields plus fluctuations. I guess these equations can be derived from a full quantum model by applying the truncated Wigner approximation (TWA). But then, if that is the case, how do quantum effects come about? I was under the impression that the TWA cannot capture purely quantum effects like entanglement, Wigner negativities, etc. But maybe my assumption is wrong. If so, it would be valuable for the authors to explain this, since I think my assumption is shared by many in the community.
5. Having used xSPDE myself and in my group, one problem we haven't solved is how to access the noise vector. Sometimes, one would like to know the exact value of the noise field at each time step. Is there a way to retrieve that a posteriori?