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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  

Preprint Link:  scipost_202212_00053v2 (pdf) 
Code repository:  https://github.com/peterddrummond/xspde_matlab 
Date accepted:  20230522 
Date submitted:  20230310 02:08 
Submitted by:  Drummond, Peter 
Submitted to:  SciPost Physics Codebases 
Ontological classification  

Academic field:  Physics 
Specialties: 

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 timestep 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 forwardbackward equations. It can generate inputoutput 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.
Author comments upon resubmission
xSPDE3: extensible software for stochastic ordinary and partial differential equations
by Simon Kiesewetter , Ria R. Joseph, Peter D. Drummond
Dear Scipost,
Thanks for the response on this paper. The referee has requested minor changes. We appreciate these very useful suggestions.
In response to the requests, we have made a number of changes, both in the code and in the manual. In summary, our response is as follows:
(1) What is new in xSPDE3?
A summary of all the 12 major innovations is now included in the Introduction.
(2a) Equation 38, ..is presented as a "damped quantum harmonic oscillator". .. First, there is nothing "quantum" about this oscillator....It is a simple oscillator whose statistical properties are fully described by classical physics.
Quantum phasespace representations were pioneered by Nobel prize winners Schrodinger, Wigner and Glauber. In this representation the harmonic oscillator treatment given is fully quantum mechanical. There is a new section (2.8) to explain this background, with references.
(2b) .. there is no discussion about the damping constant ...it would be good if the authors could explain this in 1 or 2 sentences .
An explanation of the damping parameters, together with the original master equation, is now included. We thank the referee for this helpful suggestion.
(3) There is nothing quantum in this system. Moreover, similar inputoutput relations as in Equations 45 and 46 hold for purely classical systems.
This mapping uses quantum phasespace methods. The size of the vacuum fluctuations, and their dependence on the operators, comes from quantum theory. These issues are now explained and referenced. Such techniques are used to analyse quantum technology experiments.
(4a) 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 crossKerr nonlinearity. Can the authors confirm that? And what is the relevance of the particular type of multiplicative noise they assume?
A summary of the full derivation is now included. The nonlinearity is a parametric one, not a crossKerr nonlinearity. This type of nonlinearity is called a chi(2) effect. The noise terms are not assumed, but rather are derived rigorously from quantum mechanics.
(4b) ..the authors say that one can use this model to investigate purely quantum effects like entanglement, EPR paradoxes, and Bell violations...I was under the impression that the TWA cannot capture purely quantum effects like entanglement, Wigner negativities, etc. I'm sorry, but I do not understand how that is possible.
These can all be investigated using quantum phasespace stochastic equations. The relevant references are now included. Some quantum effects, like entanglement, can be treated using a truncated Wigner method, but the method used here is an exact one. These results are described in the referenced papers, and have been used in the quantum optics literature for many years. There are multiple textbooks and reviews available to describe how it is possible.
 Having used xSPDE myself and in my group, one problem we haven't solved is how to access the noise vector.
This is now available in XSPDE v3, using the p.auxfields defined parameter. When using errorchecking, the noise terms that are accessed are the coarse timestep noises, to maintain compatibility with xSPDE output file standards. Otherwise, all noise terms are accessible.
The details of how to do this are now explained in section (4.6).
Yours sincerely,
Simon Kiesewetter , Ria R. Joseph, Peter D. Drummond
List of changes
To summarise the changes,
(1) The Introduction is rewritten, listing the new features.
(2) A more complete set of references was added to reference earlier work
(3) A new section (2.8) is included to explain quantum phasespace methods
(4) There is a new section (4.6) to explain auxiliary fields and noise outputs
(5) The quantum harmonic oscillator and nonlinear examples are rewritten
(6) Changes and improvements are included in the xSIM and xGRAPH reference
Published as SciPost Phys. Codebases 17r3.44 (2023) , SciPost Phys. Codebases 17 (2023)
Reports on this Submission
Report #1 by Said Rodriguez (Referee 1) on 2023420 (Invited Report)
 Cite as: Said Rodriguez, Report on arXiv:scipost_202212_00053v2, delivered 20230420, doi: 10.21468/SciPost.Report.7082
Strengths
See my previous report for a list of strengths.
Weaknesses
No weaknesses.
Report
I read the authors' response and revised manuscript . The authors properly addressed all my questions and concerns. I appreciate the various improvements to the manuscript which the authors made. The improvements include several more detailed explanations, new references, a good overview of what is new in this version of xSPDE, and additional technical details. I am confident that this manuscript will become a valuable resource for many researchers simulating stochastic systems. I recommend publication of the current version of this manuscript in SciPost.