Talk:Supersymmetric theory of stochastic dynamics

Supersymmetry, Chaos, and Wikipedia

This page is about a theory that establishes a close relation between the two most fundamental physical concepts, supersymmetry and chaos. The story of this relation has two major parts. The first is the well celebrated Parisi-Sourlas stochastic quantization of Langevin SDEs. The second is the more recent generalization of this procedure to SDEs of arbitrary form. At the first sight, it may look like it is too early for the second part to be on a wikipage. On the other hand, without this part there is no supersymmetry-chaos relation because Langevin SDE are never chaotic. Their evolution operators have real and non-negative spectra. As a result, partition functions of Langevin SDEs never exhibit exponential growth in time that would signify the key feature of chaotic behavior - the exponential growth of the number of closed trajectories.

Needless to say that notability for a general audience is one of the wikipedia requirements for a theory to have its own wikipage and that it is the connection of supersymmetry to the ubiquitous chaotic behavior in Nature that makes STS notable for a reader that has no background in mathematical/theoretical physics.

To assure that the supersymmetry-chaos relation is suitable for wikipedia, creation of this page had to wait until the material had been published a sufficient number of times in Physical Review, Annalen der Physik and a few other scientific peer-reviewed journals. By wikipedia regulations, this material is no longer an “original research” because it is now an opinion of not only a handful of authors recently working on this subject but also (at least partly) of the reviewers and editors of the above journals. This is why the tagging of this page for deletion (see the top of this talk page) was ruled in favor of keeping this page.

By now, I have been almost the sole editor of this page and the presentation is most likely biased. Please help by editing the page or discussing possible ways to improve it on the talk page.

Vasilii Tiorkin (talk) 15:07, 12 December 2021 (UTC)

Links from other Wikipages

I took the liberty of linking many other Wikipedia pages to this one, and there are two main reasons why I believe this is the right thing to do.

The first reason is that these links can be made. As a theory of SDEs, STS is applicable across all branches of modern science. Furthermore, as a theory explaining chaos and 1/f noise, it holds relevance in virtually every scientific field. This is an inherently multidisciplinary subject, even from a purely mathematical perspective. If done properly, this page could become one of the most referenced scientific pages on Wikipedia, which would only benefit from a more structured representation of scientific knowledge.

The second reason is that these links must be made. The theoretical physics community is currently divided into two major groups: the quantum and the classical. This division is stark -- typical theorists in dynamical systems are often unable to access papers on topics like string theory because the mathematical tools used in modern quantum field theories are conceptually far more complex than those used in dynamical systems. STS, in essence, could serve as a practical bridge across this gap. By cross-fertilizing ideas and enabling theorists from different disciplines to speak the same mathematical language, this theory has the potential to enhance the efficiency and productivity of scientific knowledge. Imagine a neuroscientist using knotted Wilson loops on anti-de-Sitter space to explain a qualitative aspect of the collective dynamics of neuronal electrochemical potentials. It may sound far-fetched now, but who knows? Something like this might turn out to be true. This is why I believe STS is worth promoting, and linking it to relevant Wikipedia pages helps further this objective.Vasilii Tiorkin (talk) 16:50, 29 January 2025 (UTC)

Cleanup

I'm less than entirely happy with the current state of this article. It's unreadable to those with an ordinary education in mathematics. I'm thinking that it would be a wise idea to split this article into two, or maybe three. with the first preliminary case dealing with just the Parisi-Sourlas N=2 supersymmetry.

There is at least one reason pointing on that this may not be a good idea. Namely, if we do that and create a separate page for Langevin SDEs, then we should do the same with E.Gozzi et.al work on the extension of the Parisi-Sourlas method to classical dynamics. And there are other classes of SDEs that this method has been extended to, before it was realized that the topological supersymmetry exists in all SDEs.

Perhaps, we can instead use the already existing page on Stochastic_quantization and move some discussion there. This is the Parisi-Wu proposition to use Parsi-Sourlas approach to Langevin SDEs and provide quantum field theories with a-la holographic description. We can move the pathintegral gauge-fixing picture of the Parisi-Sourlas approach there. However, the discussion of the Parisi-Sourlas method as a prototypical cohomological topological field theory, relevant to STS, may not fit there, well, for "a la" political reason.

The Parisi-Sourlas proposition is basically this. To address an SDE, say, this one,

${\dot {x}}(t)=-\nabla U(x(t))+\eta (t)x(t)+\xi (t)$

where U is Langevin potential, the last term being the noise, and $\eta (t)$ being the probing field that can be used later to probe the system, we can construct the following "partition function",

$Z_{PS}(\eta )=\iint \delta ({\dot {x}}+\nabla U-\eta x-\xi )JP(\xi )DxD\xi$

where the pathintegral is taken over noise configurations and closed trajectories in the phase space (periodic boundary conditions for x (x(0)=x(T))), $P(\xi )$ is a functional representing the probability of a noise configuration, which is typically assumed Gaussian white and normalized $\iint P(\xi )D\xi =1$, and $J=Det{\frac {\delta ({\dot {x}}(t)+\nabla U(x(t))-\eta (t)x(t)-\xi (t))}{\delta x(t')}}$ is the Jacobian. Because of the periodic boundary conditions, the (infinite) number of the noise variables equals the number of the system variables so that the Jacobian is nonzero.

Thinking of $Z_{PS}$ as of a generating functional is a conceptual mistake. The point is that this object is not the partition function of the system. It is the Witten index representing the partition function of the noise. It is independent of $\eta$. As a result, the response correlators (with the probing field introduced at the level of the SDE) all vanish:

$\left.{\frac {\delta ^{n}Z_{PS}(\eta )}{\delta \eta (t_{1})...\delta \eta (t_{n})}}\right|_{\eta =0}=0$

To see this, we proceed down the standard path and introduce the Faddeev-Popov ghosts to represent the Jacobian and arrive at

$Z_{PS}(0)=\iint _{P.B.C}e^{-\{Q,\Psi \}}=Tre^{-T{\hat {H}}}(-1)^{\hat {F}}=\sum _{i}\iint \langle i|e^{-\{Q,\Psi \}}(-1)^{F_{i}}|i\rangle$

Here $Q$ is the topological or BRST symmetry and $\Psi$ is a gauge fermion, the sign of the functional integration denotes functional integration over all the necessary fields including the F-P ghosts with P.B.C. as explicitly signified by the subscript. The sign of functional integration without this subscript in the right expression stands for pathintegration over open paths connecting the arguments of $\langle i|and|i\rangle$.

Now, some of eigenstates are supersymemtric singlet, $\langle \theta |,and|\theta \rangle$, such that $\iint \langle \theta |\{Q,X\}|\theta \rangle =0$ for any functional X, and pairs of nonsupersymmetric doublets $\langle \zeta |Q$, and $|\zeta \rangle$ and $\langle \zeta |$, and $Q|\zeta \rangle$. For any pair of nonsupersymmetric doublets $\langle \zeta |\{Q,X\}Q|\zeta \rangle =\langle \zeta |Q\{Q,X\}|\zeta \rangle$. Using this, and recalling that $\{Q,X_{1}\}\{Q,X_{2}\}=\{Q,X_{1}\{Q,X_{2}\}\}$, one can easily see that $\iint _{P.B.C}\{Q,X\}e^{-\{Q,\Psi \}}=0$ and consequently, (${\bar {\chi }}$ is one of FP ghosts)

$Z_{PS}(\eta )=\iint _{P.B.C}e^{-\{Q,\Psi \}-\int _{O}^{T}dt\eta (t)\{Q,i{\bar {\chi }}(t)x(t)\}}=Z_{PS}(0)$

In fact, in the Literature on stochastic quantization, it is more typical to see correlators like this

$\iint _{P.B.C}x(t_{1})...x(t_{n})e^{-\{Q,\Psi \}}\neq 0$

which camouflages the subtle but fundamental mistake of using Witten index as the partition function. In the above correlation function, some closed trajectories will contribute negatively and this makes no sense from the point of view of stochastic dynamics.

In order to fix this mistake, one should switch to antiperiodic conditions for ghosts,

Let me add an example here. First, we can rewrite the above PS functional as

$Z_{PS}(\eta )=\iint \delta ({\dot {x}}+\nabla U-\eta x-\xi )JP(\xi )DxD\xi =\iint Ind(\xi ,\eta )P(\xi )D\xi$,

where $Ind(\xi ,\eta )=\sum _{\alpha \in solutions}signJ(x_{\alpha }(\xi ,\eta ))$ is the sum over all the solutions of SDE with this particular $\xi ,\eta$. This is called the index of the map. It is a topological constant independent of $\xi ,\eta$, $Ind(\xi ,\eta )=Ind$ (for closed phase spaces it equals Euler characteristic), which shows once again that $Z_{PS}(\eta )=Z_{PS}(0)=Ind\iint P(\xi )D\xi$ is the partition function of the noise up to a topological factor.

Consider the simplest example with a 1D phase space, R, and in the deterministic limit of weak noise. Then, the solutions to SDE are constant values at the critical points of the Langevin function, which we assume Morse-type, i.e., with isolated critical points (min's and max's). One now has

$\iint _{P.B.C}x(0)e^{-\{Q,\Psi \}}=\sum _{\alpha \in Mins}x_{\alpha }-\sum _{\alpha \in Maxs}x_{\alpha }$.

This value does not make much sense from the physical point of view. If we turn to anti-peridiotic boundary conditions, however,

$\iint _{A-P.B.C}x(0)e^{-\{Q,\Psi \}}/\iint _{A-P.B.C}e^{-\{Q,\Psi \}}=\sum _{\alpha \in Mins\cup Maxs}x_{\alpha }/\#(Mins\cup Maxs)$,

i.e., the mean x averaged over all the closed solutions of SDE, which makes sense.

This discussion suggests that the traditional stochastic quantization with p.b.c. for ghosts is only valid when the Langevin function (or action in higher dimensional models) has only one minimum/vacuum. Such is the simplest situation where perturbative corrections tell all the story. In this case, however, there is no need for fermions -- there are no fermions in the only minimum/vacuum and all fermionic loops vanish identically. Vasilii Tiorkin (talk) 17:51, 3 June 2024 (UTC)

and it is very hard to find a paper on stochastic quantization that says it explicitly. In other words, almost all papers on this subject makes this mistake. Therefore, we do not want to speak of the Witten index on that page on stochastic quantization pointing this out. We should keep the Witten-index interpretation of the Parisi-Sourlas pathintegral on this page. Perhaps, in a shortened form. Well, we can decide later how to proceed. Vasilii Tiorkin (talk) 20:09, 2 June 2024 (UTC)

Thanks for the detailed reply. I will be preoccupied until July, and after that will have to ruminate, so maybe August. The article stochastic quantization should probably not be extended to include supersymmetric results. I note that topological supersymmetry is currently a red link. Again, the overall point here is that wikipedia articles should review general topics for the "general audience". That is, articles should be written so that they can be understood by the kinds of people who are interested in reading about such things. Current wikipedia articles on quantization topics are slim. There's also geometric quantization; it is also almost a stub. So topological field theory is a reasonable start of an article; but cohomological field theory is a red link. No rush, it might take years or decades to add details. So it goes. 67.198.37.16 (talk) 22:07, 2 June 2024 (UTC)

Just saying "oh la de dah its just BRST quantization" is useless, given the rather poor condition of the current version of the BRST page.

STS is a member of cohomological field theories, the class of models featured by topological supersymmetry. In general, the topological supersymmetry can not be recognized as a BRST symmetry. In some cases, however, it can. Stochastic dynamics is one of such cases, with the caveat that a nontrivial reinterpretation of the very meaning of the gauge must be invoked (discussed below) Vasilii Tiorkin (talk) 20:09, 2 June 2024 (UTC)

I spent all day yesterday, whacking on it, to get at least the informal description mostly coherent. The formal mathematics description is a train wreck. This article then goes on to invoke (-1)F which is currently a freakin stub that got nominated for AfD, and Witten index which is also a stub. Both of those articles need to be fixed first.

I currently have this super hand-wavey sketch. It need to be fleshed out. Start with Stochastic differential equation#Use in physics which currently states

Therefore, the following is the most general class of SDEs:

${\dot {x}}(t)=F(x(t))+\sum _{\alpha =1}^{n}g_{\alpha }(x(t))\xi ^{\alpha }(t),\,$

where $x\in X$ is the position in the system in its phase (or state) space, $X$, assumed to be a differentiable manifold, the $F\in TX$ is a flow vector field representing deterministic law of evolution, and $g_{\alpha }\in TX$ is a set of vector fields that define the coupling of the system to Gaussian white noise, $\xi ^{\alpha }$.

I guess that phase space X can be replaced by a symplectic manifold or a Poisson manifold. The section Stochastic differential equation#SDEs on manifolds is underwhelming as currently written. This needs to be fixed/expanded and the various deficiencies corrected. ...

No quite. The phase space is not required to have a symplectic form, which would make it even dimensional. The phase space here can be any dimension -- recall the famous Lorenz weather model, which is 3-dimensional. Or Langevin SDEs, whose phase space dimension can be anything. The same is with the Poisson structure. Its existence is not explicitly required at any step.

Anyway $F\in TX$ so the Lie derivative ${\mathcal {L}}_{F}$ makes sense. If we freeze time. I don't entirely understand what happens when the gaussian noise is added. Also, since X is supposed to be symplectic,

It is not supposed to be symplectic (see above)

it seems like there should be relationships to either a Poisson bracket or maybe a Schouten–Nijenhuis bracket or something, who knows, a detailed reference is needed. Then, write

$M=\exp -{\mathcal {T}}\int _{t_{0}}^{t}{\mathcal {L}}_{F}$

where ${\mathcal {T}}$ is the time-ordering operator, and the $\exp -{\mathcal {T}}\int {\text{stuff}}$ is explained very poorly in product integral and in State-transition matrix, both of which are in woeful shape, and a slightly better in Magnus expansion and perhaps best explained in ordered exponential and a worthy special case in Dyson series.

I attempted to make some minimalist repairs to these five articles in the last 48 hours, but each one requires many days of work to whip into shape. Magnus expansion defines

$\exp -{\mathcal {T}}\int _{t_{0}}^{t}A$

but wants A to be an NxN time-dependent matrix. That's OK, as long as we are careful to then say $A=\left.{\mathcal {L}}_{F}\right|_{p}$ for a point $p\in TX$ (I think this is correct, I'd like to have an actual reference for this.) Thus,

$\left.M\right|_{p}=\exp -{\mathcal {T}}\int _{t_{0}}^{t}\left.{\mathcal {L}}_{F}\right|_{p}$

is well-defined, assuming that ordered exponential is spruced up, and maybe some extensions of Magnus expansion to a suitable manifold setting. All is cool so far. The operator M is being called the "Stochastic evolution operator", it seems.

A few more details are needed... there needs to be some kind of averaging over the gaussian noise. I do not currently understand how to do this correctly and formally.

The way to do it in the pathintegral and operator representations are respectively Chapter 4 and 3 is Ref. 9

I never read a formal, mathematical treatment of the Langevian eqn, so I have lots of little questions that math people who enjoy rigor would ask. Next, we have to argue that

$M=\exp -(t-t_{0})H$

for some Hamiltonian-like H which is the "Stratonovich interpretation of SDEs" .. something something Stratanovich integral which I don't currently understand.

Stratonovich and Ito interpretations differ in the exact form of the Fokker-Planck equation, which comes from the ambiguity of ordering of operators, similar to the same problem in quantum theory where it is resolved by Wyels symmetrization which assures that the resulting Hamiltonian is Hermitian. Not sure what link(s) can be useful here.

Equivalently, this is a "(bi-graded) Weyl symmetrization" ... I assume that the bi-grading refers to the Gerstenhaber algebra and I assume that Gerstenhaber appears there because everything is being done on a Poisson manifold, but this is unclear.

Poisson manifold is not relevant here.

More generally, the set of mathematical concepts relevant to STS is that of the cohomological or Witten-type topological field theories. The Parisi-Sourlas construction for Langevin SDEs came before the formulation of TFT, an it was the predecessor for the model in Ref.26, which, in turn, was the beginning of TFTs. All TFT look like they are gauge-fixing of an empty theory.

The Poisson superalgebra has the other kind of grading. I bitched about that on Talk:Graded ring a few days ago, too. For Weyl, perhaps we need to point at Moyal product, but maybe instead the deformation quantization is the other article. After getting all this untangled, we can finally write

$W={\text{Tr}}(-1)^{F}M$

which is the Witten index after handwaving that $M=e^{-(t-t_{0})H}=e^{-\beta H}$ but its unclear how to do the handwaving. I'm also not sure how (-1)F got in there, except maybe it has something to do with ... beats me.

The term quantization here is a slang. It is not a quantum system.

Meanwhile, at the bottom of Langevin equation#Path integral we've got the nascent sketch of a path integral formulation. Apparently, the Parisi-Sourlas supersymmetrizes that up to N=2. I guess ??? It goes something like this: treat the Langevin equation as if it were a constraint, create a Lagrange multiplier for

$\delta ({\dot {x}}(t)-F(x(t))-\sum _{\alpha =1}^{n}g_{\alpha }(x(t))\xi ^{\alpha }(t))$

This suffers from all the conventional issues during quantization, because it looks like a gauge fixing term, which is why BRST is invoked.

yes. The gauge symmetry here is the fact that the noise partition function is independent of the variables of the system, so we gauge-fix it using the SDE as a constraint. In result, we get a pathintegral representation of the object in terms of the variables of the system and which is a representative (up to a topological factor) of the partition function of the noise -- the Witten index.

I understand BRST, but I don't understand this particular leap. There is finally one more (one last?) leap; take something that resembles the BRST charge

$Q_{BRST}=c^{i}\left(L_{i}-{\frac {1}{2}}{{f_{i}}^{j}}_{k}b_{j}c^{k}\right)$

and something something something $H=[Q,{\overline {Q}}]$ and claim this is exactly the same H needed to construct the Witten index.

Clearly I'm totally lost by here. And if I understand correctly, this is "merely" for the Parisi-Sourlas results. There's a whole lot of connect-the-dots here that remain unconnected, for me. Assuming that I'm mathematically average, then other wikipedia readers will be just as lost. This is the basic problem, here. 67.198.37.16 (talk) 05:42, 30 May 2024 (UTC)

Folks I would like to drop in with an external unbiased perspective, of someone with a lengthy background in mathematical physics. I think the view here is a bit lost on just how overly technical and non-wiki this page is; I just read the archived discussion for deletion, and to be honest this page is hanging by a thread. Don't get me wrong: I love highly advanced wiki pages to exist for a wider audience. I myself contributed to many Langlands Program-related pages, which are far too formal. However the tone and style of this page does not belong on wikipedia... moreover the talk page itself looks like an argument between two doctoral candidates. Instead, of the practical editorial-discussion it ought to be. If this page is to be accepted on the wiki – it has to be *massively* simplified. And believe me; if you can simplify Artin Reciprocity, you can simplify this. The fact it is written by practically a single -albeit dedicated and knowledgable author- is a major red flag. Furthermore, that there are barely a handful of journals which published on this topic, is even worse. This is a fascinating topic; and it merits to be known by a general audience. However the fact this page is linked in almost every chaos or stoachastic-related page... means it needs to be a whole lot less 'original research'-y. My suggestion is: focus less on the rigour. That belongs in journals. Instead simplify it to the critically best readable form. Otherwise; a deletion or merger discussion, would need to be reopened. 185.146.221.25 (talk) 09:29, 23 June 2024 (UTC)

Totally agreed. Even as an expert in an adjacent area, I find parts of this article to be essentially incomprehensible. The problem seems to be that it consists of an interesting backbone consisting of a combination of topics that definitely deserves a (or possibly several) Wikipedia article(s) (stochastic analysis on manifolds, Parisi-Sourlas interpretation, Witten Laplacian, etc), but that this is suffused with wildly speculative interpretations due to the author of the article (I. Ovchinnikov, who goes by 'Vasilii Tiorkin' on this site). Disentangling the two would be a lot of work, which I unfortunately do not have the time or inclination for. This is why I originally nominated the article for deletion. Hairer (talk) 14:05, 26 June 2024 (UTC)

(OP) Quite frankly I dove quite deep to understand the mathematics of this article, which are right up my ally. But the truth is there is only a handful of solid equations; and a serious lack of action principles, or exact geometric constructions. Simplest put: I am struggling to even grasp what the basic 'definition' section of this theory would be. It seems quite unclear to the author either... whom is unfortunately more concerned with touting the possible implications for such a theory. Without properly explaining what it is. As an original research paper; this would have a great deal of trouble being accepted (which is why I gather, the only few citations which directly support this theory, are by a single author...). As a wikipedia page? This is rather unacceptable. Again, don't get me wrong, I think this topic is fascinating, foundational, and fundamentally important. However the author is wholly within original research territory here - and a deeper second review. I very much think is necessitated. Just think of the amount of people who pass by this article without *any* prior knowledge in the field; how incomprehensible is it to them...?

I normally don't criticize so harshly: but the problem is the author took the liberty of linking this page, to every single possible related topic. And every time I come across it I am somewhat irritated anew. Especially since it seems to get only worse over the years (and I have known this page for I think about five years now...).

The simple fact remains - this theory is not widely accepted, nor even discussed, by the mathematical physics community at the moment. As such, being its own page; and such a verbose one, is perhaps not the right way to go. So long as it is the product of a single author (granted with many decades of prior related supporting literature): it is very much original research. Furthermore; the page itself, is too poorly formatted and styled to remain as such. 2001:861:44C2:AAA0:CCEB:5D4E:686B:5566 (talk) 18:52, 29 June 2024 (UTC)

I apologize for not addressing this discussion sooner -- somehow, I missed the notification about it. Let me respond to the points raised now; better late than never.

Regarding the alleged issue of "original research": Wikipedia's definition of "original research" -- and I’ve taken the time to familiarize myself with it -- differs significantly from that used in Academia. Apparently, this is the origin of confusion here. On Wikipedia, something qualifies as "original research" if it has not been published in peer-reviewed scientific journals. A published material, on the other hand, represents not only the opinions of the authors but also those of the editors and reviewers, and by Wikipedia's standards, it is no longer considered "original research". This undoubtedly applies to the content of this page, as the material (here I refer to the theory of general form SDEs) has been published many times in many journals (PRD,PRE,Annals of Physics, Annalen der Physik...). One example is the latest paper in the world’s leading scientific journal on chaos:Ubiquitous order known as chaos. The following analogy may be helpful in this context: an actor typically earns a personal Wikipedia page after appearing in just one Hollywood movie. By comparison, the general SDE theory discussed on this page has already "appeared" in over a dozen "movies".

Regarding the fact that the theory is not widely accepted: this is simply not true. The (Parisi-Sourlas) part of the theory related to Langevin SDEs is half a century old and is a matter of many textbooks. As to the overreaching theory of the general form SDEs, it is certainly younger. However, as discussed earlier, it has matured well beyond the "original research" category as defined by Wikipedia's regulations. Consequently, it is fully appropriate for inclusion on a Wikipedia page.

As to the fact that there are not that many authors who work on this theory at the moment, Wikipedia has no regulations on the number of authors who are currently actively working on the subject.

The idea of creating a separate page dedicated to the (Parisi-Sourlas) supersymmetric approach to Langevin SDEs has two significant drawbacks. First, this class of models has no connection to chaos, making it less appealing as material for a Wikipage, as a broader audience is unlikely to find supersymmetry interesting without real-world applications. Second, to maintain consistency, separate pages would also need to be created for the supersymmetric theories of classical mechanics (developed by Gozzi in the mid-1990s) and several other special classes of SDEs. However, over time, these separate pages would inevitably need to be merged, as they represent different realization of the same theory. The most natural solution is to present the supersymmetric theory of Langevin SDEs and more general classes of SDEs on a single page -- a suggestion originally made by a moderator during a discussion initiated seven years ago by M. Hairer.

I totally agree with the main point of this discussion: the page is overly technical. I do not find it satisfactory either. Let me try and simplify the content in the next few months to make it more accessible to a broader audience.Vasilii Tiorkin (talk) 17:06, 26 January 2025 (UTC)

Apologies for the double response. Let me address another raised concern about the "speculative interpretations by the author": with all due respect, this is a misunderstanding. To clarify:

-- The interpretation of wavefunctions as differential forms, and the broader connection between supersymmetry and algebraic topology, was established by E. Witten in his seminal Morse Theory and Supersymmetry.

-- The interpretation of stochastic quantization by Parisi-Sourlas as a topological field theory was established by L. Baulieu and B. Grossman in Physics Letters B (1988) A topological interpretation of stochastic quantization.

-- In the context of classical mechanics, a similar interpretation is due to E. Gozzi and M. Reuter in Physics Letters B (1990) Classical mechanics as a topological field theory.

-- The interpretation of Faddeev-Popov ghosts as differentials used in the butterfly effect was introduced by R. Graham in EPL (1988) Lyapunov Exponents and Supersymmetry of Stochastic Dynamical Systems.

-- Even the identification of chaos as a topological supersymmetry breaking can be partly credited to D.Reulle (see page 893 of Dynamical Zeta Functions and Transfer Operators). Even though he was not aware of the supersymmetric structure underlying his construction, he introduced the concept of "pressure" for the purpose of chaos quantification. In application to SDEs, this pressure is the (exponent of minus of the real part of the) ground state eigenvalue of the stochastic evolution operator and nontrivial pressure associated with chaos in random systems (>1) is equivalent to the spontaneous breakdown of the topological supersymmetry in STS.

These works constitute the majority of the interpretations within STS. The contribution from the "author" is limited to putting the pieces together and recognizing that this supersymmetry exists not only in specific cases like Langevin SDEs, Kramers equation, or classical mechanics, but in all SDEs -- the step forward that became possible in big part due to the work by A. Mostafazadeh on pseudo-Hermitian evolution operators with complex spectra (see, e.g., Pseudo-supersymmetric quantum mechanics and isospectral pseudo-Hermitian Hamiltonians.

If anyone finds these interpretations speculative, they are encouraged to publish a paper with alternative interpretations in a peer-reviewed scientific journal. This is how Wikipedia works: if such interpretations appear, we will reflect them on this wikipage. In this context, it is worth noting that the first paper on the STS of the general-form SDEs appeared in 2011, providing ample time -- 14 years -- for alternative interpretations to appear in the literature. So far, none have emerged. Vasilii Tiorkin (talk) 00:26, 27 January 2025 (UTC)

Hello there, I put the recent tags on the article. First, allow me to congratulate you on the monumental task of creating this page; as a fellow researcher in mathematical physics, I recognise the depth of your work on this topic. With this being said – I think you are missing the point. This article is utterly incomprehensible to a lay reader... and impenetrable to most specialists. The topic is bleeding-edge level research: and has yet to gain foothold as a widely-practiced mainstream field. This is similar again to the Langlands analogy: however if you take a look at that page (which is criminally underdeveloped for a similar reason, and I personally worked on), you will find an equally if not more technical topic; reduced to an arguably semi-understandable level.

The biggest problem is that you practically wrote this entire page. And it just so happens most of the references, are your own publications. That's a big no-no. We need many more publications from other authors and other opinions. Equally as less reliance on yours for the core equations. Otherwise; this is strongly OR. The fixes here are thus – massive simplification to encyclopedic format. And, other publications from many other authors working *specifically* on STS. Not adjacent stochastic-dynamics fields, like BRST or parisi-sourlas. Wikiledia is a tertiary source: not a primary one. It is not your job to show your research is 'well supported'. That's for the journals. Instead you must show, from secondary sources (is there a single news publication about this?), that it exists whatsoever...!

Ninety percent of this article is your own construction. You have to show there are other people verifying this and contributing, for it to be fully accepted. I personally love the elegance of this idea mathematically-speaking. But you must first define your constitutive equations properly; for us to know what you're talking about! Without relying on derivations of derivations from other works by your own papers (at least on the wiki). This article is written like a dissertation, and you defend it like a doctorate. This is not the purpose of wikipedia; to serve as a personal publication ground for your own work, even if it's well-supported. It needs to become an encyclopedia page. Which, I might add, will garner much more popular recognition as apreciation for it...! My tip, start with a 'definition' section, like every other mathematics article on the wiki.

Keep up the good works. And, for what its worth; I understand your struggle to bring a hyper-complex topic to a wider audience. I've followed the evolution of this page for years, and hope it continues to improve so simplify. Cheers!

P.S

The fact that chiefly you edit this page even under a pseudonym, and that this is a theory which is largely published in your own papers; goes beyond original research... into the territory of 'neutral point of view' issues. 2001:861:44C2:F4B0:3945:C7F0:A21E:AEFD (talk) 15:58, 28 January 2025 (UTC)

Great, thanks! Yes, I plan on working on the page in the nearest future. I will take your suggestions into account and will certainly make the page shorter -- something I planned on doing for a long time.

But lets resolve the issue with the tags first. As we already agreed, a claim that has been published in peer-reviewed journals is no longer "original research" from the point of view of Wikipedia. The "original research" tag itself clearly states it: " ...improve it by verifying the claims made and adding inline citations... ". In other words, the issue of "original research" is resolved by adding citations, which in case of the scientific subject means nothing else but publications in peer-reviewed journals, no matter who published it or when. Therefore, tagging this page as "original research" is clearly a result of misunderstanding. Please remove this tag.

Another issue that you raised is that the page may have issues with the "neutral point of view" concept. This is not the case either. Look, in order for a point of view to be "not neutral", there must first exist (at least) two mutually exclusive opinions on the same subject. The subject of this wikipage is that there is a close relation between supersymmetry and chaos. There is not a single paper in the scientific literature that reports an alternative opinion that, say, such relation does not exist. In other words, at this very moment, the opinion of any editor of this wikipage cannot in principle be "not neutral" -- there is simply no alternative that can be overlooked. But even if one day such an alternative opinion appears, we will certainly reflect it on this wikipage so that it does not have an issue with the "neutral point of view" rule. Thus, even if there is a Wikipedia tag for "neutral point of view", it would again be a result of a misunderstanding if someone tries to put it on this wikipage. Please do not do it.

Your claim that "It needs to become an encyclopedia page" is not entirely accurate. The key point is this: While Wikipedia is primarily an encyclopedia in the traditional sense, advancements in information technology have expanded its functionality beyond that of conventional encyclopedias. For example, certain events receive dedicated Wikipedia pages within a day of occurring, whereas in the 19th century, it could take decades for a subject to be included in an encyclopedia. In this context, Wikipedia’s guidelines explicitly state: Wikipedia can report your work after it is published and becomes part of accepted knowledge; however, citations of reliable sources are needed to demonstrate that material is verifiable, and not merely the editor's opinion. Here, the term "accepted" does not mean universally accepted by everyone in the dynamical systems community or that of the traditional approach to SDEs. Rather, it refers to acceptance by at least some part of the scientific community, such as editors and reviewers of peer-reviewed scientific journals, as discussed earlier in relation to the misunderstanding of Wikipedia’s definition of "original research".

You, Dr. 2001:861:44C2:F4B0:3945:C7F0:A21E:AEFD, and Dr.Hairer,

Sorry, Dr. Hairer, for misspelling your name in the original commit. Please blame it on the exhaustion of running around screaming about chaos -- it’s tiring and makes me feel like an idiot. The more I do it, the more I suspect I am one....Vasilii Tiorkin (talk) 15:10, 31 January 2025 (UTC)

pointed out a few times that I am not using my real name as a Wikpipedia's editor. Ok, lets take a look at the editor names in the latest edits of, say, Stochastic_differential_equation: Tensorproduct, Edoarad, "Me, Myself, and I are Here", 5.103.110.218, DamianoBrigo2, Madamadore00, 78.135.226.220 ... When I was picking the name for my Wikipedia account I was under impression that most Wikipedia editors use pseudonyms. I am simply doing what most editors do, and I am by no means hiding from anyone. I am Igor Ovchinnikov, the guy who believes that chaos is the topological supersymmetry breaking, who thinks that this is very important, and who speaks about it everywhere, including Wikipedia.

You also mentioned that this page contains many self-citations. I cite my own work only because I have no other option -- I am not aware of any alternative references on the relationship between chaos and supersymmetry. If you know of any, I would sincerely appreciate you pointing them out. You have my word that as soon as other researchers begin publishing on this topic -- and I have no doubt they will, with this Wikipedia page helping to facilitate that -- I will gladly replace my self-citations with newer papers by other authors. My intention is not self-promotion (I have already left academia, so academic recognition is not that important to me) but rather the advancement of a theory that will benefit everyone -- and one that I cannot stop thinking about :)

Moreover, I am very much aware of the fact that the way I see this topic may be biased, which is basically true universally for everybody. I state that clearly in the beginning of this talk page: ... presentation is most likely biased. ... I also add, however, Please help by editing the page or discussing possible ways to improve it on the talk page... because I want to find those who also find this topic very interesting just as I do. This is partly why I decided to create this wikipage in the first place. Hopefully, as time goes on and more people join the discussion it will help develop a collective, balanced picture of this most important truly multidisciplinary topic.

Let me also add that, in my view, STS also carries an inherently political dimension. The acceptance of this theory would inevitably lead to a redistribution of influence among scientific subcommunities, and it is only natural to expect that some may harbor personal biases against it. At the same time, it is also natural for the scientific community to resist new ideas -- this skepticism is a healthy aspect of the immune system needed to filter quacks out. From this perspective, criticism is invaluable. Unfortunately, this immune system does not always function as it should. For example, it remains largely silent for the last few decades on the unscientific concept of self-organized criticality. Lets do our best and make sure that it did not develop an autoimmune response to STS.Vasilii Tiorkin (talk) 16:09, 29 January 2025 (UTC)

Hello again, and thanks for you taking the time. Firstly, in the interest of partial transparency; I am that same "Dr." 2001:861:44C2:AAA0:CCEB:5D4E:686B:5566, also I might add there is no need to 'dox' yourself on my account. The fact that you write under a pseudonym (and that I remain anonymous entirely) is perfectly fine and even encouraged. The only problem is, that if you are the lead researcher on this topic, and you cite yourself... that inevitably veers into original research and neutrality problems. Imagine if, for example; Robert Langlands created the 'Langlands Program' page. Would be rather strange no, haha.

I agree, it is funny. But even funnier would be the idea of Robert Langlands submitting a paper titled "The Langlands Program" to a peer-reviewed journal. Right? By the same logic, me publishing in peer-reviewed journals should also be funny, and this cannot be right. To resolve this apparent inconsistency, we must look at this wikipage from a more accurate angle -- this is a wikipage on the Parisi-Sourlas approach to SDEs. I did contribute to the extension of this theory on general form SDEs. But the theory is not mine, it is theirs.Vasilii Tiorkin (talk) 21:25, 30 January 2025 (UTC)

Your self-awareness and passion for this project are obvious: and are signs of an intelligent mind in my book. At the same time I cannot really help but feel that to my perspective -and most other readers- this page is highly anomalous on wikipedia. Because, you are the main contributor, because you cite yourself a bit more than is reasonable (though I understand yours is almost the only work extant in this field, which is kind of the problem). Original research can also apply to things well-published in journals; which are entirely primary sources by a single author, and have no secondary sources to verify them.

And this is where the problem is. If I wrote a book about some-such mechanics, and published a hundred peer-reviewed papers about it myself; it would be well-cited, but still original research. I want to see other sources talking about this theory. And, I want to see other editors; as is obligatory in the mathematical sciences equally as wikipedia, support your claims. Because the burden of proof cannot lie with the average reader. Most of all I want this page to continue existing on the wiki: and be a simplified and good resource for a wide audience. Right now; even myself having an extensive research history in stochastic analysis, symmetry-groups, and topological gauage-theories... I am having difficulty understanding what the 'action principle' would be here.

The OR tag is a warning for people to know that they are reading an article which is not considered fully known or accepted by the wider scientific community (i.e is not mainstream). You could say the same of string theory, but the difference is there are hundereds of researchers working on that. And as mentioned before; since you integrated this page into many adjacent-topic articles, this is a necessary tag for people who stumble here from chaos theory or the butterfly effect. The problem of original research is that you, and almost only you, have written this entire article. And almost only you have publications in this field... if you do not see the immediate conflict of interest here; I don't know what else to say. If you still think my tag is unfair, you can remove it yourself, I won't get in your way. Or apply for third-party conflict resolution. But insofar as I see: it needs to remain. Which is why I also edited it after this discussion. Because very few readers would be aware of all this behind-the-scenes stuff, thus would take it at face value. When I said it needs to be an "encyclopedia article"; I meant, a wiki page. A simple, understandable, widely readable summary of a topic. Not a doctoral dissertation for a personal theory. Elegant as it may be. Even if it is already published in peer-review.

I'm sorry to give you a hard time, and I don't want to be 'that guy', who just roadblocks stuff because it is different. I have struggled myself with such interactions many times. I just see fundamental problems that have bothered me for years with this article (that I haven't seen anywhere else), and decided to take action. My extent of editing wikipedia is mostly relegated to occasional fixes and tags whilst reading to improve the wiki. Sometimes I take on larger projects, such as the Langlands Program page, or Automorphic Forms. But here I see a situation which needs deeper resolution; and it seems no one else has the time or the understanding required to improve it. I really want to see this page get to a point where it is a well-established topic, and that many people flock to it for its depth and novelty. Which is why I didn't try to reopen the deletion discussion. Right now; it's not there yet. And the fundamental issues I present, to my perspective, are well-founded. I will not do any further edits, as I do not wish to waste my or anyone's time. You have my full opinion; and once again I commend your efforts for this work. All I can say now is, do what you think is best. 2001:861:44C2:F4B0:6943:85A2:A355:572A (talk) 10:37, 30 January 2025 (UTC)

I am thankful for the discussion. Let me just say a few more words to dot the i's.

Look, we all know that a job of a scholar is to seek out new knowledge and share it with those who might find it useful. In the 16th century, publishing books was the only option for the said sharing. Today, we have a wide range of options -- peer-reviewed journals, YouTube, web pages, Scholarpedia, and yes, Wikipedia. So I viewed Wikipedia as a practical tool for sharing valuable information, not as a "hall of fame." I’m just a guy who accidentally stumbled upon something shiny and is now running around trying to show it to everyone -- only to find, to his frustration, that no one seems to care much. :)

But I think I finally managed to understand your point. The issue is not in the "original research", so that the tag is still not very fair. The issue is in the absence of secondary sources. I was actually under impression that primary sources is enough because the wikipage itself is a secondary source. Hmmm, this theory was never too popular: a dozen of authors in 80's, a few more authors in 90'ies, and in 00-x it just died out. It may take a little longer before a secondary source shows up, but I am absolutely sure some day it will.

Ok, does this sound reasonable? Let's give this page a little more time -- perhaps a year or two. If no secondary sources appear by then, I will personally delete this page and, honestly, with a sense of relief too. In the meantime, I can try and rewrite it as we discussed. Vasilii Tiorkin (talk) 21:25, 30 January 2025 (UTC)

I think again the point is to be repeated, that wikipedia is in fact a tertiary source. We gather here primary sources and other people's opinions about those in secondary sources; then agglomerate them without additional interpretation in this, somewhat mechanical aggregation tool. At least that's what it's supposed to be when it works correctly according to the rules. To this end it's apparent to me the tag is still very warranted. Chiefly because lay readers need to know from the outset, this is mostly the work of a single author! Believe me; there are much much worse articles than this. So again I don't want to barrel down too harshly on yours: because I think we need as many advanced scientific articles here as possible. For what it's worth, I personally think your discovery is fascinating, from a mathematical perspective. I've even worked on some parallel research in stochastic systems which implies similar structures. I do not think deletion is warranted; unless you want to, of course. I can sense the stress you have from attempting to expound so defend your work, and can thoroughly empathise. The problem here is just a technical one. We need secondary sources. We need simpler direct derivations. We need other points of view or contributors; and somewhat humbler claims. Most of all we need this page to be fully comprehensible! If you manage to achieve this simplicity: I think you will see much more widespread adoption. We mathematicians often forget that the very tools we use to understand the cosmos can equally be a screen of obfuscation. Even to our own intellect. Try to look at this page from outside and say: 'if I were an external reader, what is the simplest most elegant form I would want it to be'. That's my criterion for my own work at least.

Thank your for your time as work, good luck, and I hope you continue to improve. 2A01:CB1E:69:53E4:0:31:7CA0:A801 (talk) 12:22, 31 January 2025 (UTC)

Lorenz attractor

Is it possible to use the Lorenz attractor as a concrete example? Without any noise, each tangent space to the phase space splits into an unstable manifold and a stable manifold and a center manifold. Although one can always integrate along a particular flow, there are going to be saddle points and bifurcations; I don't know how to describe them or think about them correctly. That's without noise. With noise .. ?

Strange attractors can be best understood as unstable branched manifolds from the Topological Theory of Chaos https://onlinelibrary.wiley.com/doi/pdf/10.1002/9783527639403.fmatter . they are not topological manifolds and in STS they are represented by a non-supersymmetric ground states.

Unrelated to the above, I also don't understand what part of all this looks "gauge invariant" and thus why the Langevin eqn looks like a "gauge fixing term". 67.198.37.16 (talk) 02:51, 31 May 2024 (UTC)

Thank you for comments ! Detailed discussion of the relation among the Parisi-Sourlas perpresentation of Langevin SDEs, cohomological TFTs, and the relevant mathematical concepts can be found in the first part of Ref.22. Vasilii Tiorkin (talk) 03:31, 2 June 2024 (UTC)

OK.The lede to this article makes grand claims about solving chaos in some general form, and so if it is solved in some general way, there should be an example or two showing how it actually works in some prototypical case. 67.198.37.16 (talk) 22:13, 2 June 2024 (UTC)

One example is the astrophysical kinematic dynamo limit of the Dynamo_theory. This is the effect of the exponential growth of the magnetic field at early stages of the formation of galaxies. People knew that this phenomenon must be related to the chaoticity of the underlying flow of the ionized interstellar matter. But no such connection could be rigorously established because, in the general case, magnetodynamics includes diffusion and this corresponds to a stochastic version of the underlying flow and no definition of chaos for stochastic dynamics existed previously (in fact, even no definition of chaotic deterministic flows existed previously). STS provided such a definition of chaos for SDEs and this enabled to establish that the above conjecture is correct and the kinematic dynamo can be viewed as a result of the chaoticity of the corresponding SDE describing the flow of the matter (Ref.34, a numerical investigation can be found here https://iopscience.iop.org/article/10.1088/2399-6528/aac94a).

Another example is the explanation of Self-organized_criticality - a wide spread belief among numerical experimenters that some stochastic dynamical systems on the border of deterministic chaos have a mysterious tendency to fine-tune themselves into a phase transition into chaos. This point of view contradicts many well-estabilshed things in physics and math including the critical phenomena theory and the very scientific method itself. STS explained that on the border of determinstic chaos there exists a phase where stochastic dynamics has those peculiar properties (instanton-dominated dynamics) previously viewed through the prism of a "mysterious tendency".

But even more generally, as I mentioned before, no definition of chaos existed before even for deterministic dynamics, let alone stochastic dynamics, while all natural dynamical systems are always stochastic. It does now within STS Vasilii Tiorkin (talk) 01:11, 3 June 2024 (UTC)

Sorry, but grandiose sentences like "no definition of chaos existed before even for deterministic dynamics [...] It does now within STS" are precisely what creates the "immune reaction" you mention in your other comment. Your claim is simply nonsense. There are various quite sensible definitions of "chaos" that go back to the sixties and seventies (positive Lyapunov exponents, mixing, etc). Just because you have a definition you're fond of doesn't mean that people who came up with these notions were idiots. (This is in the same vein as your previous nonsensical claim of having 'solved' the 'Itô-Stratonovich dilemma'.)

Anyway, this is all just fluff. If there's a general link to 'chaos', then you should take the simplest possible example of chaotic system (cat map or whatever) for which you can formulate some simple mathematical theorem illustrating the connection to STS and explain that, not just make grand claims. Hairer (talk) 17:44, 29 January 2025 (UTC)

Hello there, just dropping by from the other parallel discussion haha. I have to say I second what Hairer is saying; though perhaps not as harshly. It is these kinds of "ultimate statements" that give an instant knee-jerk reaction, and make us question everything you write. 'Extraordinary claims require extraordinary evidence' after all... even if it is a pop-science platitude. And from my mathematical point of view; I simply do not see the supporting derivations and proofs to the enormous claims made! Forgive me, but your elaborate defenses and saying things like 'self-organised criticality is against the scientific method itself (paraphrasing)' make you sound rather unprofessional in this matter. Even if there is a modicum of truth to it. Going back to my prior analogies, string theory and the langlands program are bleeding-edge research topics, which are practiced by hundreds the world over. That is why they are recognised as such - even if they are as-yet thoroughly unproven! You're going to need to lower the lofty claims; and give us a solid basic and straight to the point mathematical construction... before we can say that this is valid for the general public. After all: very few people have the necessary background to be able to see this critically, and most will not understand a thing whatsoever. Which after all I gather is what you want; for people to grasp the theory! 2001:861:44C2:F4B0:6943:85A2:A355:572A (talk) 10:58, 30 January 2025 (UTC)

You guys are right -- I should shift down a gear and exercise restraint. In my defense, I think I slipped into this defensive mode and have been struggling to snap back. This is why I sometimes mistake criticism of the page for attacks on the theory itself. Moreover, I am not a mathematician, and the mathematical community follows stricter standards of communication. So, forgive my occasional lack of formality.

As to quite sensible definitions of "chaos" (topological mixing, tranistivity and other trajectory-based things), these are not definition per se. These are properties of chaos. Not to mention that these are not generalizable to stochastic dynamics and all natural dynamical systems are stochastic. We even wrote a paper on this issue some time ago. So, I am still convinced that no definition of chaos existed previously and I believe this is actually the reason why mathematicians in the dynamical systems theory often avoid using this term, so "chaos" is mostly used in physics community. Anyways, let me come up with an improved and more balanced version of the page in two weeks so we could have a more concrete subject for discussion.Vasilii Tiorkin (talk) 20:21, 31 January 2025 (UTC)

Glad you take feedback well, even if it may be harsh sometimes. Insofar as the definition for chaos is concerned; it is almost always deterministic as opposed to stochastic, because of strange attractors. What you're talking about is 'stochastic chaos' which is inherently non-deterministic because of the 'forgetfulness' of stochastic processes (although those can have limit-cycles as well). This is the more difficult and important version to define. To my knowledge there is indeed no rigorous analytical-functional definition of this type of chaos as yet (though many parametric descriptions have been made like sensitivity etc). Generally speaking if you wish to make these kinds of claims, which is not a crime, make sure they are swiftly followed by a clear and direct exemplar equation which supports them in published research. Correct me if I'm wrong; but the term STS does not seem to appear anywhere but in your papers right? If you could find another external source which uses this exact theory that would help much. Good hunting! 2A01:CB1E:69:53E4:0:31:7CA0:A801 (talk) 09:19, 1 February 2025 (UTC)

I’ve grown a thick hide and am proud of it -- but don’t test it too much :)

A few more words on chaos, the previous definition of deterministic (continuous-time dynamical) chaos I like is called "non-integralibity of flow in the sense of dynamical systems theory" (and there are other senses of course). It is a formal way of saying -- just as you pointed out -- that the so-called global unstable manifolds are not topological manifolds but what is informally called "strange attractors".

As far as the very concept of definition is concerned, however, a definition must not only capture the lowest level essence of phenomenon, but also lead to the explanation of its key qualitative properties. In case of chaos, this key property is 1/f noise -- the universal experimental signature of chaos. No previous definitions of chaos explained it, while top.susy.breaking seemingly does: Goldstone theorem>gapless excitation>effective field theory as CFT>operator product expansion>power-law correlators.

Richard Feynman, who said about hydrodynamical chaos (and I am paraphrasing) "turbulence is the most important unsolved problem of classical physics", apparently was most certainly not satisfied with previous definitions of chaos. I am absolutely sure he would be satisfied with top.susy.breaking.

I forgot to point out that this great scientist was with us until 1988 and was well aware of all the previous definitions of chaos from the 60-ties mentioned by Hairer.Vasilii Tiorkin (talk) 15:01, 1 February 2025 (UTC)

Actually, I look at the Ito-Stratonovich dilemma mentioned by Hairer from the same practical angle, but lets talk about it in a few weeks.

As to the term STS, I may be the only one using it. But as I said, to the best on my knowledge, since Tailleur, Tanase-Nicola, Kurchan (2005)[1] on Kramers equation, no one is using supersymetric approach to dynamics. There is no one out there to call it any name, not only STS. But it has got to have some name and STS is short and accurate.

I believe, however, the question of the exact name is not that important at the moment. Recall that even some well-estabiished theories have more than one name sometimes, e.g., gauge-gravity duality and AdS/CFT correspondence.Vasilii Tiorkin (talk) 14:09, 1 February 2025 (UTC)

Sorry, there is yet another issue I forgot to address. It is the fact that tradtionally chaos is discussed mostly in the context of deterministic dynamics. Well, deterministic dynmaics is just a limiting case of stochastic dynamics. And many things in math and phsyics are defined as a certain limit. Fractals, for example, or traditional view on SDEs. In case of deterministic chaos one can do this: add a little noise, caclulcate what is interesting, and send the noise intensity to zero at the end. Therefore, if we are in possession of a practial definiton of chaos for stochastic models, we automatically have it for deterministic models. there are, of course, formal complications in the strict deterministic limit: the spectrum of the stochastic evolution operator may not be bounded from below or may not be descrete. But this is no concern for physicists and engineers like myself.Vasilii Tiorkin (talk) 16:22, 1 February 2025 (UTC)

The point about the name of the theory was not for terminology's sake. So, by your self-admission, you are the only person working on this specific theory? If you want to understand the entire point of this discussion; that is it... There needs to be at the very least one other source which verifies this thing is 'a thing'. Not similar, adjacent, or research this is based on. That specifically, "Supersymmetric theory of stochastic dynamics", is a verified existing object by itself. That is the problem of 'original research': if you, and only you work on it, so no one else is (peer review does not fulfill this criterion)... that's where it's at.

Well, it has got a few dozen citations, but, yeah, just as we spoke previously, lets wait a little longer an remove the page if other source do not appear, say, within a year or two.Vasilii Tiorkin (talk) 17:41, 1 February 2025 (UTC)

As said, I think removal is a bit excessive. There's some good work here. Just try to find some other direct sources for STS and simplify it as discussed; I think it should be good then :) 2A01:CB1E:69:53E4:0:31:7CA0:A801 (talk) 20:05, 1 February 2025 (UTC)

Insofar as the definition for non-deterministic stochastic dynamical systems which exhibit chaos. I agree, there is no fundamentally analytical defintion, that I'm aware of. Yet insofar as I see it you have not exactly rigorously supplied it either.

It is rigorously established that Top.Susy.Breaking is equivalent to the view on chaos in random systems proposed by D.Reulle -- spectral radius of generalized transfer operator is larger than unity. But, you probably mean I did not rigorously supplied it on this page.Vasilii Tiorkin (talk) 17:41, 1 February 2025 (UTC)

Anyway, we can go on forever. I'm happy if I was able to help this article improve; and I hope you're joking about "don't test it too much"...

Of course, there is a smiley there trying to deliver the message of this good line from an old movie which translates roughly as "smile more often -- the worst things in this world are done with serious faces..." Sorry if it did not sound right, though.Vasilii Tiorkin (talk) 17:41, 1 February 2025 (UTC)

I think I've done all i can in this situation, and I'll let you continue the work. Have a good one. 2001:861:44C2:F4B0:ED79:1B62:A032:1EB5 (talk) 16:40, 1 February 2025 (UTC)

New version

I've just committed a new version. It's significantly different and notably shorter. Also, the story is more structured and consistent: it begins with the definition and gradually unfolds into the breaking of the topological supersymmetry and 1/f noise at the very end. Please let me know if you think I am moving into a right direction.Vasilii Tiorkin (talk) 21:47, 4 February 2025 (UTC)

Be careful not to imply more generality than necessary / reasonable. For example, the first paragraph in the mathematical part starts with an SDE on a topological manifold with smooth vector fields. Since a topological manifold can admit several inequivalent smooth structures, this is already a bit of a can of worms: I would simply start with a smooth manifold so it's clear what we're talking about.

Further down, the sentence "GTO is unique and up to intrinsic ambiguities associated with pathintegral representation of evolution operators known in the theory of SDEs as Ito-Stratonovich dilemma" has no mathematical meaning / content. You should probably just say from the start that the SDE should be interpreted in Stratonovich form (which is of course the only interpretation that has intrinsic meaning on manifolds). The last two paragraphs are also mathematically meaningless. You should be much more careful at distinguishing between expressions that have a proper mathematical definition and those that are just suggestive, but without any well-defined meaning, like the path integral manipulations. Hairer (talk) 08:24, 5 February 2025 (UTC)

Yes, this is a good point. Thank you. Changed to smooth manifold.

As to interpretaitons/ambiguities, it is my fault that I did not make it sufficiently clear that there is no ambiguity in the mathemtical perspective section of the page. The construction is unique and there is no ambiguity that can be removed by specifying an interpretation. I stessed this point in the new version. I even tried to avoid using the term SDE. Please take a look.

Ambiguitites only appear in the pathingetral representation of evolution operator. The pathintegrals are by definition a continuous time limit of a descrtete-time evolution picture, which is the direct analogue or, rather, equivalent to the traditional understanding of SDEs as a continuous-time limit of step like evolution. Which interpretation of stochastic dynmaics one choses to use must be specified only in pathintegrals or in the traditional picture of SDEs, but not in the construction in the first part of the page. I believe this may be an important piece of understaning. Please give it another thought.Vasilii Tiorkin (talk) 13:58, 5 February 2025 (UTC)

Hello again. First, congratulations on the speedy work! I would say this is definitely in the right direction; keep giving direct derivations with proper explanations of each object, that is the way on the wiki. I still think a lot more simplification is needed; but it's on the right track. The next thing would be inline citations for each formulation, and a bit of lowering of generalised claims without direct non-tangential cited support (like the last section). I would suggest two things: first a brief introduction section before the history, explaining in the simplest possible linked terms, the empirical-mathematical basis of the theory for a general audience. This would serve as the necessary background for professional readers to have proper formal context too. A single paragraph should do. Secondly, and quite importantly, the lede paragraphs must be thoroughly cited with at least a couple of secondary sources. Keep up the good work – this reads much closer to a wiki page.

P.S

The tag removal is a bit early; and the article very much still qualifies for a "too-technical" template for a general audience, which I won't add. But please be aware this is still impenetrable to 99.9% of wiki readers; which is a paramount consideration, this is an encyclopedia after all... 2A01:CB09:D03E:21CB:0:40:7657:F501 (talk) 12:39, 8 February 2025 (UTC)

Thank you for the suggestions. Will work on it next.Vasilii Tiorkin (talk) 13:00, 8 February 2025 (UTC)

I put some effort into making the page sound neutral and encyclopedic. I think we can remove the tone tag now. If you do not agree, please put it back or, better yet, please edit the page to improve the way it sounds.Vasilii Tiorkin (talk) 15:35, 7 February 2025 (UTC)

Manifold?

The article currently states:

A continuous-time non-autonomous dynamical system can be defined as,

$${\dot {x}}(t)=F(x(t))+(2\Theta )^{1/2}G_{a}(x(t))\xi ^{a}(t)\equiv {\mathcal {F}}(\xi (t)),$$

where ${\textstyle x\in X}$ is a point in the phase space which can be assumed to be a closed smooth manifold

There are several "minor" problems here. First, the use of upper/lower summation indexes implies the existence of a metric. So perhaps it should say Riemannian manifold instead of "smooth manifold"? Or perhaps there is no metric, and the upper-lower notation is being used to denote the contraction of a vector and a one-form? So I guess $G_{a}$ will be a vector, and $\xi ^{a}$ will be a one-form? Or perhaps the other way around? Maybe even perhaps this will end up being a multiple of the tautological one-form?

This page has a lot of problems (some of which you point out below) and is indeed basically unreadable. I don't think this particular point is a problem though: here the $G_a$ are just a collection of vector fields and the $\xi^a$ are a collection of (presumably i.i.d.) white noises. No Riemannian structure required. Hairer (talk) 16:27, 8 May 2025 (UTC)

Exactly. There is no need for metric. The metric may be important, however, at the analysis of the spectra of the evoluiton operator. Namely, the leading term in the diffusion part of the evoluiton operator is $G_{a}^{i}G_{a}^{j}\partial _{ji}^{2}$ so that $G_{a}^{i}G_{a}^{j}$ (summation over a) can be looked upon as a metric on $X$. If we would like to make sure that the spectrum of the evolution operator is limited from below, or rather, the real parts of the eigenvalues are limited from below, then this metric must have certain properties. For example, if this metric is positive-definite everywhere on $X$, the evoluiton operator is elliptic and this means the spectrum must be bounded from below (at least for compact $X$).

This question is related to the choice of the phase space too, e.g., for non-compact phase spaces one may need to include into the Hilbert space only integrable modes/eigenstate as is done sometimes in field theories. I guess there is a lot to think about for a mathematician. In a meantime, the simplet way to proceed is to focus on models that have bounded spectrum, which effectively removes the necessity to address the metric-spectrum relation. This is exactly what is done at the moment -- the part speaking about the spectra says that we focus only on "physical models" with bounded spectra.Vasilii Tiorkin (talk) 13:57, 9 May 2025 (UTC)

I'm also confused by ${\textstyle x\in X}$ is a point in the phase space, Now, by convention, here in Wikipedia, phase space is taken to be flat; whereas if some wikipedia article wants to talk about mechanics in a more general setting, a wikilink is made to a symplectic manifold. At any rate, a "point in phase space" has both a position and a momentum, and so am I supposed to understand that $x=(q,p)$? But then there is all this talk of tangent vector fields, so maybe $x$ is supposed to be a point in the base space? That is, $x=q$ is just the coordinate, not (coordinate,momentum)? Do recall that all tangent bundles are always symplectic manifolds. So are your tangents spaces tangent to the symplectic manifold? Or just to the base space?

Dynamical system theory is a generalization of classical mechanics/ Hamilton dynamics. The flow vector field is not required to be a symplectomorphism and the equations of motion are not required to follow from a least action principle. The flow vector field is arbitrary and the phase space does not have to be even-dimensional. There are no momenta, as, e.g, in Lorenz model which is 3d.

A classical mechanical model is a special case of this picture when the flow vector field is given by a gradient of a Hamilton funciton and a symplectic form splitting dynamical variables into the coordinate-momentum pairs. In this situation, $X$ becomes the phase space from the point of view of classical mechnics. Thus, adopting this term for $X$ does the matching with the terminology accepted in classical mechanics.

The term base space may not be the best choice because this term is used in nonlinear sigma models to indicate what in our case is "time" (in string theory it is worldsheet). Moreover, from the nonlinear sigma model point of view, $X$ is the "target space".

Next, and perhaps this is quibbling: do you really mean "closed", or do you mean compact? Lots of interesting spaces have cusps. Is the goal here to avoid spaces with cusps?

This is an interesting question and I believe is it linked to the metric-spectrum relation above. It is something to be explored. For now, we avoid this issue by focusing on physical models with bounded spectra.

Finally, the definition you give looks more or less exactly like the Ito process, generalized to a general Riemannian manifold setting. I assume this is intentional. But then this should be stated up front, not leaving the reader to guess. The bits and pieces have a name. $F(x(t))$ is called the drift velocity, and $(2\Theta )^{1/2}G_{a}(x(t))$ is half the square root of the diffusion tensor. Then I guess its normally appropriate to say something along the lines of "${\dot {x}}(t)$ is a Feller process" or something like that. Assuming that's the intent here.

Exactly, this is a generalization of Itô_calculus#Itô calculus for physicists on general $X$ and general noises. And $(2\Theta )^{1/2}G_{a}(x(t))$ is indeed half the square root of the diffusion. This is why $G_{a}^{i}G_{a}^{j}$ can be viewed as a metric on $X$ defining the diffusion that comes from noise. Because of this, I initially believed that the corresponding diffusion operator must be the Hodge-Laplacian or Bertlami Laplacian that are often used in various generalizations of stochastic dynamics to Riemmanian spaces. But it turned out to be wrong. The diffusion part of the evolution operator is given by yet another Lapacian, $\sim {\hat {L}}_{G_{a}}{\hat {L}}_{G_{a}}$, which in the general case is not Hodge-Laplacian but it is, quite fortunately, also $d$-exact, just like the Hodge-Laplacian.Vasilii Tiorkin (talk) 13:57, 9 May 2025 (UTC)

Rather than answering me directly, here, I'm asking that the article be fixed up to resolve these questions. I think I can guess my way through, to the right answers; I've got a working imagination. But one aspect of Wikipedia is to be precise enough that readers should not be left guessing. 67.198.37.16 (talk) 06:53, 8 May 2025 (UTC)

Thanks for pointing this out. Let me do some more thinking how to reflect this on the page.Vasilii Tiorkin (talk) 13:57, 9 May 2025 (UTC)

I am struggling to figure out how to move this conversation forward. I posed what had hoped would be some fairly simple questions, and I posed them suggestively, so as to leave a lot of wiggle-room. Let me try asking again, in plainer terms. There are two questions.

First, we seem to have a dot product, but I cannot tell what space(s) that dot product is acting on. Harier suggests its "just a product" of a vector and some white noise, but I don't know how to take a product of a vector and white noise. The whole point of abstract Wiener space and Fokker–Planck equation and Cameron–Martin space is to explain "what that means".

If I understand it correctly, you are talking about this term $G_{a}\xi ^{a}=G_{1}\xi ^{1}+G_{2}\xi ^{2}+...$, where $G$'s are different vector fields on $X$ while $\xi$'s are scalars representing noise. This expression is well defined because vector fields over $X$ are linear objects and one can construct any linear combinations from them, which the above expression actually is.

By "what this means" you probably refer to Ito-Stratonovich dilemma, i.e., the well-known ambiguity in the evolution operator when the noise is white. This discussion is too long for the talk page. However, I am actually finalizing a paper on it. Let me later provide a ref to it once it is published.

In a meanwhile, the assumption of white noise is not a requirement for STS. Noise can be colored and non-Gaussian and the model will still have topological supersymmetry. So this issue is not of primary importance.

Second: you say "phase space" but when I look at the context, I think you are talking about Configuration space (physics) and not phase space. This seems to be confirmed when you say "there is no momenta". Now, by definition, half of phase space is momenta, so this is not reassuring. Clarifying this should not require appeals to Laplace-Beltrami or Hodge stars or duals, and certainly not to sigma-models of any shape or form. It was meant to be a simple question.

In case of classical mechanics, the equations of motions in the configuration space are second order in time derivative. For example, for a 1d harmonic oscillator ${\ddot {x}}=-\omega ^{2}x$. To bring the equation of motion to the canonical form, that is, first order in time-derivative, momentum variables are introduced for each dynamical variable. In the above example, ${\dot {p}}=-\omega ^{2}x,\;{\dot {x}}=p$, which can be given as ${\dot {\tilde {x}}}^{i}=\Omega ^{ij}\partial _{j}H({\tilde {x}})$, where $H=p^{2}/2+\omega ^{2}x^{2}/2$, $\Omega$ is the symplectic form, and ${\tilde {x}}$ are dynamic variables from the phase space that include both the variables from the configuration space and their momenta. In this way, and in classical mechanics only, the dimensionality of the phase space is always even and equals two dimentionalities of the configuration space.

Dynamical systems theory is a generalization of classical mechanics. Its equations of motion are first order in time derivative from the definition. They are already in canonical form, that is, there is only the phase space but no configuration space from the classical mechanics point of view. The phase space is not necessarily even dimensional and the variables do not have the meaning of momenta. See, e.g., Lorenz system.

Wait, I just realized that the same object is called state space in dynamical systems. Do you propose to use this term instead of phase space? This is a good idea, actually. Thanks. Will make the corresponding corrections soon. Vasilii Tiorkin (talk) 01:08, 11 May 2025 (UTC)

I would really like to see is how to apply Parisi-Sourlas to the very simplest possible case: the Wiener process, or Langevin in one dimension, or Fokker–Planck in the simplest possible case of one dimension. Having this would then provide a basic cornerstone on which to build.

This may be a good ref: Chapter 3.3 in TOPOLOGICAL FIELD THEORY, BIRMINGHAM et.al., Physics Reports 209, Nos. 4 (1991) 129—340. Vasilii Tiorkin (talk) 14:05, 10 May 2025 (UTC)

A discussion like this may take too much space on the page. Well, let me think how to make it short enough.Vasilii Tiorkin (talk) 01:08, 11 May 2025 (UTC)

I have some urgent assignments due, so it may be weeks or longer before I can look at this again. 67.198.37.16 (talk) 08:07, 10 May 2025 (UTC)

Hierarchy of mathematical spaces. Inner product spaces are a subset of normed vector spaces, which are a subset of metric spaces, which in turn are a subset of topological spaces.

OK, Let me try a third time. Regarding dot products: please look at this article: normed vector space and in particular, pay attention to the diagram on the right hand side. I reproduce it here. Notice the blue circle labelled "inner product space": this is where dot products live. You've got "white noise" and I don't know what that is. Maybe that's a Banach space or maybe that's a Frechet space? I don't know because its not defined. Maybe there's a norm. Maybe there's a metric. Maybe there's a topology. I'm not a mind-reader, I literally do not know what "white noise" is, unless you stick it in some space, perhaps into $\mathbb {R} ^{\infty }$ or perhaps into $\mathbb {R} ^{I}$ for the unit interval $I$ and then perhaps explain that maybe you are interested only in the classical Wiener space that is a subset of $\mathbb {R} ^{I}$. Or perhaps you mean something larger, such as the Cameron–Martin space. I don't know what "white noise" is, or how to "multiply it", unless you actually explain what "multiplication" is.

If you want your noise to NOT be "white" or "Gaussian", then you define some measure, perhaps a cylinder set measure what makes it "non-white" or "non-gaussian". However, you also keep saying "supersymmetry" and "Fadeev-Popov", and I would like to keep in mind that the Berezin integral that is used in Fadeev-Popov has one of two forms (I quote):

• $\int \exp \left[-\theta ^{T}A\eta \right]\,d\theta \,d\eta =\det A$

with $A$ being a complex $n\times n$ matrix.

• $\int \exp \left[-{\tfrac {1}{2}}\theta ^{T}M\theta \right]\,d\theta ={\begin{cases}\mathrm {Pf} \,M&n{\mbox{ even}}\\0&n{\mbox{ odd}}\end{cases}}$

with $M$ being a complex skew-symmetric $n\times n$ matrix, and $\mathrm {Pf} \,M$ being the Pfaffian of $M$, which fulfills $(\mathrm {Pf} \,M)^{2}=\det M.$

Both of these use an algebraic notation that makes them look very distinctly "Gaussian", and if you want to find some kind of non-Gaussian measure, then you have to explicitly define it. At any rate, I don't understand how to connect the Berezin concepts with the textbook-conventional concepts of abstract Wiener space and this article, as written, is not providing that connection. 67.198.37.16 (talk) 19:26, 10 May 2025 (UTC)

It does say there that $\xi (t)\in \mathbb {R} ^{D}$. However, I did not speak much about the randomness aspect of this function, e.g., that its "realizations" are distributed according to some probability functional. Thanks for pointing this out. Will do it soon.

On the other issue, fermions have very little to do with the noise. They are just auxillary fields to track evolution of differentials and/or to correctly represent the functional Jacobian coming from the evolution-induced deformation of the state space aka phase space. Vasilii Tiorkin (talk) 01:08, 11 May 2025 (UTC)

The very first equation in this article is nearly identical to the very first equation in the article Stochastic differential equation and this article should point at that article for a general overview.

Indeed... I did not edit that page. Someone else did. Thanks for pointing this out. Let me think how to reflect that page on this one.Vasilii Tiorkin (talk) 16:51, 13 May 2025 (UTC)

Yes, I understand that $\xi (t)\in \mathbb {R} ^{D}$ and I'm trying to point out that for the $D=1$ case, the definition of Wiener space is infinite-dimensional. Both you, and the article on Stochastic differential equations are attempting to glue together $D$ copies of some infinite-dimensional spaces that have some rather nasty and complicated structure. And that gluing is hand-wavey, imprecise. But that's not the fault of this article; the ultimate problem is that there isn't any article on WP that describes the $D>1$ generalization of abstract Wiener space or how it works or how to do geometry on it.

The time-derivative of WP is the gaussian white noise. If you use one, using the other would be redundant. At the same time, guassian white noise is a more transparent and intuitive concept. This is why I personally avoid WP altogether. And I do not understand why some people keep using WP. Vasilii Tiorkin (talk) 16:51, 13 May 2025 (UTC)

Grassmann variables are not "fermions" because they are not spinors. This is a common misconception. Turns out that the correct way to build spinors with with the Clifford algebra -- elements of the Clifford algebra also anti-commute, and they do so "correctly" in that you can build Weyl spinors out of them (and also Dirac spinors in certain cases) and these transform "as expected" under the rotation group. (The associated graded algebra *is* naturally isomorphic to supernumbers, which is I guess the reason why physicists love to pretend that supersymmetry is "like spin but without the spinor". However, I think that being just slightly more precise makes things more clear and easier to understand.)

I agree. The term "fermions" is used here is a generalized sense. "Ghosts" would be a more accurate term here. Vasilii Tiorkin (talk) 16:51, 13 May 2025 (UTC)

Anyway ... what's interesting is that you can build spinors for more-or-less anything that has a tangent manifold that looks locally Euclidean. So you can build spinors on any Riemannian manifold in any dimension. Well, finite dimension. I have no clue about how "infinite dimensional geometry" works. I imagine that maybe you can build clifford algebras and spinors just fine on infinite-dimensional manifolds. I imagine that this all somehow ties together with a giant constellation of related ideas. I am very slowly trying to dig deeper, but this is very much a side-quest for me. 67.198.37.16 (talk) 05:58, 13 May 2025 (UTC)

Please keep digging — you're unearthing useful suggestions. Vasilii Tiorkin (talk) 16:51, 13 May 2025 (UTC)

Mathematics vs Physics

Everything up to the end of the 'Averaging over Noise' section is now reasonably clean from a mathematical perspective, at least if one ignores technicalities about boundaries, behaviour at infinity, etc.

For simplicity, the phase space is assumed closed so it has no infinities and boundaries.

Closed is not the same as compact... Hairer (talk) 17:08, 16 May 2025 (UTC)

Exactly. A compact manifold can have a boundary, e.g., a disk, while a closed manifold, e.g., a sphere, can not.Vasilii Tiorkin (talk) 17:14, 16 May 2025 (UTC)

Compact implies closed so that statement makes no sense, unless you're using made up terminology. Also, $\mathbb{R}^d$ itself is closed and most certainly has an "infinity". Hairer (talk) 08:12, 17 May 2025 (UTC)

You're mistaken. Take a look at the first sentence from closed manifold: In mathematics, a closed manifold is a manifold without boundary that is compact. From the same page: A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. So compact does not imply closed. Also, $\mathbb{R}^d$ is a textbook example of an open manifold, not closed one.

The way to remember: compact means "finite size", while closed adds no-boundary to it. Another hint: a Poincare dual of a closed manifold (embedded into a higher dimensional manifold) is ${\hat {d}}$-closed.Vasilii Tiorkin (talk) 10:59, 17 May 2025 (UTC)

Fair enough, I hadn't realised that this is standard terminology. (Still seems weird to me that the closed unit disk isn't considered a "closed manifold", but well...) Hairer (talk) 12:41, 17 May 2025 (UTC)

Well, noone can know everything. Please keep being friendly and critical.Vasilii Tiorkin (talk) 01:52, 19 May 2025 (UTC)

Everything that comes after seems to be at the level of handwaving physics arguments / conjectures. This should be made much clearer.

I would be happy to clarify the presentation further -- please let me know specifically which parts you find unclear. As for the rigor of the approach itself, it is mathematically solid. All the tools used come from the standard toolkit of, say, cohomological topological field theories, which is a well-established part of modern math.

I removed mentioning functional integral representation of the noise from Averaging over the noise. Did it make it clearer ? Vasilii Tiorkin (talk) 12:07, 18 May 2025 (UTC)

It is somewhat clearer but what is the reason for all these hats in the notation?

This is for phisicists. They may not recognize this is an operator acting on an infinite-dimensional Hilbert space. I understand mathematicians are out there thinking about things like this. But physicists do not. We are not as clever, you know.:) 2601:58B:C00:CCC0:3D87:9E08:D6B9:A8E0 (talk) 13:37, 19 May 2025 (UTC)

A few more remarks:

1. The "averaging" section could still be shortened by simply saying that the GTO is defined as the semigroup on differential forms obtained by averaging the pullback induced by the inverse flow and that ${\hat {H}}$ is its generator with the formula you provided. The derivation you give works as a motivation but is purely formal, not rigorous.

Wait, I did not manage to address the point raised in the previous response. I addressed something else. Let me try again and rewrite it. I think my goal here is to have a derivation. This shows how simple and elegant this approach is. As to being rigorous, I think everything in this derivation as rigorous as it gets in the world of math. If not, please let me know what exactly you think is not rigorous. Also, I will soon write a section on this talk page to discuss Ito-Stratonovich dilemma, in case you mean something related to it. Vasilii Tiorkin (talk) 18:14, 21 May 2025 (UTC)

2. There is no reason for the dimension of the noise to match that of the manifold. Even for one of the simplest non-trivial diffusion on a manifold, namely Brownian motion on the two-dimensional sphere, one needs at least three noises if one wants the $G$'s to be smooth.

Absolutely. I just did not want to introduce yet another parameter which would be the dimentionality of the noise. Let me see if I can change this. Vasilii Tiorkin (talk) 18:20, 19 May 2025 (UTC)

3. As I already pointed out several times, the remark about the GTO having a clear mathematical meaning as opposed to SDEs is inappropriate. "Everybody" knows that the Stratonovich interpretation of SDEs is the one that's natural in a geometric context; that doesn't mean that SDEs don't have a clear mathematical meaning.

I always knew that I am stepping on someoneelses' tuft. But this was not my real intention, My true intention was to bring attention of experts like yourself to this issue. Now that I succeeded,:) I agree that removing this sentence is a right thing to do.2601:58B:C00:CCC0:3D87:9E08:D6B9:A8E0 (talk) 14:06, 19 May 2025 (UTC)

4. In the part on "GTO eigensystem", you'll definitely need to assume enough non-degeneracy on your SDE to guarantee that ${\hat {H}}$ is (hypo)elliptic, otherwise there's no reason for its spectrum to consist solely of discrete eigenvalues. Also, the notion of an operator being "pseudo-Hermitian" depends on some involution. What is that involution here?

As long as the spectrum is complex congjugate -- and this is true for all real operators -- the operator is pseudo-Hermitian. Ali Mostafazadeh says it and I think this is correct. Also, I believe your comment about involution is absolutely correct. Perhaps, this is what we, physicists, call pseudo-time-reversal -- involution of the operator/Hilbert space combined with the reversal of time. I tried to do something in this direction, but I failed to get to the bottom of it. I still do not have a solid feeling about it. It certainly needs a professional mathematician's touch.Vasilii Tiorkin

That paper assumes from the start (top of p.3) that the operator is diagonalisable, namely that its spectrum consists only of isolated eigenvalues. This is not true in general for the generator of a diffusion, unless you impose strong non-degeneracy conditions.Hairer (talk) 18:30, 21 May 2025 (UTC)

I think that in our case the assumption that the spectrum is discrete (and bounded from below) is sufficient. First, the case with no degeneracies is the most general case. Second, the spectral degeneracies can either be accidental or due to some additional symmetries. In both cases, they can be removed by a continuous deformation of the model. Now, one can construct a one-parameter (say parameter x) family of models such that at x=0 it is the original model and at x=1 it is the model with all degeneracies removed. Statements that are of topological nature are homotopically invariant and what can be proved for x=1 is also true for x=0 by homotopy. In other words, the topological properties are true disregard whether or not the spectrum has degeneracies, because, particularly, in proving them one does not need the assumption that all eigenstates are different.

To check, take a torus with diffusion only. It has got as many degeneracies are one can get in this class of models. Everything is ok for this model.Vasilii Tiorkin (talk) 11:54, 22 May 2025 (UTC)

I think you misunderstood what I mean by "non-degeneracy". I wasn't thinking about eigenvalues having higher multiplicity but about the diffusion matrix being invertible (or some weaker form of non-degeneracy like Hörmander's bracket condition). The spectrum being discrete is kind of fine as an assumption, although that paper you reference also seems to require the system of eigenfunctions to be complete (i.e basically no Jordan blocks). Hairer (talk) 16:25, 22 May 2025 (UTC)

Sorry, by bad. You obviously meant the positive-definiteness of the noise-induced metric, $G_{a}^{i}G_{a}^{j}$. Yes, I addressed this point above in the talk with Dr. 67.198.37.16. This condition is needed in order to make sure that spec(H) is discrete and bounded form below. The exact conditions can be probably established by a professional mathematician, like yourself. The best I do on this page is to assume that the spectrum of H has these properties from from the start.

Also, many fundamental topological properties of the model do not need the exact form of H: e.g., the De Rham cohomology, and consequently the splitting of the eigenstates of H into the singlets and the doubles, is defined solely by the exterior derivative and this operator consists only of Jordan blocks. Vasilii Tiorkin (talk) 16:58, 22 May 2025 (UTC)

BTW, this issue has nothing to do with the material in the "Averaging over noise" section. Vasilii Tiorkin (talk) 17:28, 22 May 2025 (UTC)

5. In the "Operator representation" part (again, please remove mention of "Itô-Stratonovich dilemma"; just because some people get confused by an ambiguity arising from bad notation doesn't make this a thing), the remark that "Other interpretations differ only by the shifted flow vector field" doesn't really make sense since the Itô-Stratonovich correction term doesn't actually give rise to a vector field. Hairer (talk) 07:31, 19 May 2025 (UTC)

Removed this sentence.Vasilii Tiorkin (talk) 15:04, 19 May 2025 (UTC)

Also, the introduction should be more factual, it reads like a buzzwordy grant application rather than like an encyclopedia article. Hairer (talk) 08:44, 16 May 2025 (UTC)

Perhaps. Let me think how to improve this. Vasilii Tiorkin (talk) 11:54, 16 May 2025 (UTC)

Rewrote this sentence. Perhaps it reads more encyclopedic now. Vasilii Tiorkin (talk) 15:09, 19 May 2025 (UTC)

Ito-Stratonovich dilemma

The entire family of Runge-Kutta methods is based on the understanding that, under some general assumptions, for any given initial condition, the continuous-time limit of the solution of ($\Delta \tau \to 0$)

$(1)\;(x_{k}-x_{k-1})/\Delta \tau ={\mathcal {F}}(\alpha x_{k}+(1-\alpha )x_{k-1},\xi _{k}),$

exists and the solution converges to its continuous time version:

$(2)\;{\dot {x}}={\mathcal {F}}(x,\xi ).$.

Importantly, the solution is unique and independent of $\alpha$. The parameter $\alpha$ only controls how the error approaches zero: \emph{e.g.}, for the direct Euler method, where $\alpha =0$, and the midpoint method, with $\alpha =1/2$, the error $\sim \Delta \tau$ and $\Delta \tau ^{2}$, respectively. In fact, the error $\sim (\alpha -1/2)something$, just like the correction to the various interpretations of the stochastic dynamics -- this is the origin of it.

Therefore, if we choose to take the continuous-time limit before averaging over the noise, the temporal evolution of the differential forms is governed by what is written here, which has a very clear mathematical meaning and is independent of the parameter $\alpha$.

This point of view on stochastic dynamics can be described as first taking the continuous-time limit and then averaging over the noise. The pathintegral representation and comventional approach to SDEs reverses the order of these operations. In result, the SEO looses its mathematical meaning -- it is no longer a pullback averaged over the noise, but, rather, a result of formal manipulations with formulas. Moreover, the error in the convergence of Eq.(1) to Eq.(2) conspires with the noise to yield the $\alpha$-dependence of the SEO.

The so emerging ambiguity in the evolution operator is the general property of pathintegrals, because they are designed like this -- integration first, the continuous time limit second. But the people who work on SDEs could do better. After all, there is only one noise configuration in the real world and this unambiguously suggests that the averaging over the noise must be the last operation one does. Basically, we average only over our ignorance of the noise.

So, the fact that the averaging over the noise and the taking the continuous time limit do not "commute" was misunderstood. As far as I remember, Von Kampen was the one to propose that the Ito interpretation means essentially that the noise is "slow". How can a white noise be slow, gentlemen ? This is nonsense, of course. A slow noise must be colored at least.

And then came the "people" who solidified this misinterpretation in the concept of Martingale (probability theory) and things that come from it. This is a pseudo-math of course.

Or take the Wiener process. The Guassian white noise is the simplest random function that one can think of. And the meaning of Science is to make things as simple as they can be. Is it not ? So, why would anybody want to use a more complicated relative of the GWN instead of GWN? The answer is simple: when you write your solution like a function of WP, it brings the story to the trivialest mathematical concept -- the trajectory -- the only concept that most of the finance people "or coders" understand. The real substence is not a trajectory, but, rather, the probability of the system to jump from one local attractor to another. If finance people understood it, we probably would not suffer that much:)

So, my point is this. When Martin says that the stuff in Supersymmetric theory of stochastic dynamics#Dynamical systems theory perspective is not mathematically solid, he actually means to defend this peuudomathematics of matringale/wiener process and stuff. We, physicists, never liked it intuitively. We had to cope with it, actually. But no moreVasilii Tiorkin (talk) 22:04, 23 May 2025 (UTC)

When I say something I usually mean exactly this. So when I say that your derivation is not mathematically solid, I mean exactly that it's not mathematically solid, period, I would appreciate you not putting words in my mouth. As to the incoherent rant above, I don't know what to say except that accusing mathematicians of pseudo-mathematical nonsense after putting on several displays of mathematical ignorance is a bit rich. Hairer (talk) 07:19, 24 May 2025 (UTC)

I have already removed all references of the Ito-Stratonovich dilemma on the page, just as you requested. So there is not much to argue about in this respect.

What you call nonrigorous is the simple derivation of GTO based on the most fundamental fact that Lie derivative is an inifnitesimal pullback (see here). What follows next -- the integration over time and averaging over the noise -- is trivial. There is simply no room there for any approximation and/or assumption. And this concludes the derivation of the GTO, which is very much rigorous for anybody who has background in algebraic topology.

I understand that formulas involving Lie derivative may be confusing for someone who does not have a background in algebraic topology -- and the fact that you confuse closed and compact manifolds hints on exactly that. But this does not mean that the derivation is not rigorous. And there is nothing to be ashamed of, you are not an expert in this particular field.

As a non-expert in this field, please be critical but do not through around you opinion as if you believe that physicists are answerable to financial mathematicians. We know exactly what we do, and we do not need approval from anybody, especially from a field that has never managed to contribute anything useful to our subject of interest.

If you really think it is not rigorous, please publish a paper demonstrating it in a peer-review journal. (and I will certainly publish a comment) Before that, however, your opinion is not credible -- who knows, maybe your real motivations are political and go against the true goals of Wikipedia. Indeed, you already showed your interest in deleting this wikipage in the past so your opinion is certainly biased.

I spent enough time thinking about Ito-Stratonovich dilemma and my opinion on this is settled. Discussing it on the talk page is ok -- this is what the talk pages are designed for. Namely,

When time is continuous, there is only one interpretation -- Stratonovich

When time is discrete, it is not an SDE. It is a stochastic difference equation and different interpretations (or $\alpha$'s) do exist.

One may say that the continuous-time limit of an $\alpha$-dependent stochastic difference equation has got to be an SDE. Of course it is, but this limit has got a shifted flow vector field in terms of SDEs.

This is important, actually, because, unlike in finances, in physics one needs to know the exact form of the evolution operator and, say, in curved phase/state spaces the noise is not additive.

But I have no interest in arguing about this Ito-Stratonovich dilemma. You have my opinion, and I know that many have similar points of view. This is actually enough reason for this point of view to be on a wikipage, but lets swipe it under the carpet. I do not care. If someone truly wants to put effort into something that will soon fade away on its own, by all means -- who am I to say otherwise?

So, with all due respect, I have a different point of view and my reply can be summarized as follows:

I already removed all the references of Ito-Stratonovich dilemma from the page.

I forgot to mention that I genuinely believe the new insight into the Ito–Stratonovich dilemma, as provided by the unique and rigorous GTO, is an important byproduct of this theory. I’d love to discuss it here in particular, but out of sheer respect for you, I’ll refrain. Let’s try and keep things friendly.Vasilii Tiorkin (talk) 17:58, 25 May 2025 (UTC)

Your opinion that the derivation of GTO is not rigorous is an opinion of a non-expert in this particular field and it is certainly biased -- you already tried to delete this page.

If you publish your opinion in a peer-review journal, we will certainly reflect it on this page.

In journals where I publish my papers, the derivation of the GTO is considered very much rigorous. This is opinion of (at least some) editors and reviewers who are experts in related field and who are often unbiased.

I will extend, however, the discussion of the derivation of GTO on this page to make it clearer.Vasilii Tiorkin (talk) 17:58, 25 May 2025 (UTC)

Complete Simplification

Hello good people! I'm the random IP from the "Cleanup" discussion (for clarity and continuity I will hereon sign as A). I just wanted to drop in on this, dizzyingly intriguing and sprawlingly hyper-formal albeit straying discussion... To make a simple reminder to everyone involved: we do not seek proofs nor defense of theses on wikipedia. That's for the journals! We need clear, simple; somewhat minimalistic phrasings, of complex topics. Which are accessible to anyone as everyone. That's why I put the 'OR' tag on this page -which it still fully qualifies for alongside "too technical"- and now I want to make it clear from an external perspective (of someone with a background in mathematical physics). This page needs to have half the derivations; twice the third-party citations, and triple the simplified explanations for lay people to understand! This. Is. An encyclopedia. Not a peer-review board discussion page. We consolidate existing information here. Not, formulate novel bleeding-edge provisional theories. I commend once more the enormous effort Tiorkin has put into this page - But we are missing the point of what this website is...! I want to see this page continue to exist; and be useful to all.

Make it as oversimplified as possible. That's my advice. As said before, take the existing text; so reduce and reform it into a classic wiki math format, with a "Definition" section. Along the lines of: "Supersymmetric Stochastic Dynamics is A, B, C, {Equations}"... If this cannot be done, and a clear definition with solid governing equations [or principle of action...] is impossible to formulate; this theory is too young to be a wiki page.

Most Cordially,

A 2001:861:44D0:9F20:D3:407A:975:D37F (talk) 17:09, 24 May 2025 (UTC)

Well, I tried to make it simpler, but now it has been criticized for not being rigorous, which implies, particularly, that more derivation must be present. It is either simpler and less rigorous, or it is more rigorous and less simple. It is not easy to find a midpoint that is good for both goals.

As to the technicality of it current version, I may be wrong but the way I see it, it is not too much more technical than, say, this page Lie derivative#The Lie derivative of a tensor field.

Anyways, let me try this. I can give a detailed derivation of the GTO on the talk page, so those who are interested in this rigorous derivation can still discuss it. As to the page itself, we can only state the result and explain its meaning on the page itself in the A,B,C manner.Vasilii Tiorkin (talk) 18:41, 25 May 2025 (UTC)

Excellent. Exactly what I meant. I appreciate your willingness to adapt a great deal (I find this is exceedingly rare around here); and I already feel too ovebearing for pressing the same point repeatedly... You are in a rather tough spot and I sympathise a great deal with your position. I don't want to be pedantic whatsoever here – I feel there is a wealth of encyclopedic information hiding somewhere inside this page, and I would like to see it liberated.

As for the rigour, yes there are highly technical wiki pages and that's fine. I.e if it's a copy out of a textbook for something well-established, that's fine. I just think there's no point to "defend your thesis" on the page of a rather new theory. Just simply present as-is. Keep up the good work!

A 2A01:CB16:203F:24F0:0:45:349B:D601 (talk) 18:26, 26 May 2025 (UTC)

Thank you for encouragement! Let me keep simplifying. For example, there is no need for section "Averaging over noise", because averaging is a part of derivation.Vasilii Tiorkin (talk) 18:37, 26 May 2025 (UTC)

Generalized Transfer Operator

In the context of the question of the rigor of the derivation of GTO, the following derivation is at least 9 years old (I used it in Eqs. (83)-(87) in [2]). I believe it is actually older. I certainly remember seeing this exact form of GTO in one of D.Ruelle papers (I put a temporarily ref there but have to double check if it is an exact ref), and he must have derived it in a similar way.

There are most certainly a few others who used this form of evolution operator. In most publications, however, the evolution operator will be different because ${\hat {H}}$ below is correct only in the coordinate-free setting, where the probability distribution is viewed as a top differential form. Most people view the probability distribution as a simple function and in this case ${\hat {H}}$ below is not correct.

Anyways, in the real world, every dynamical system experiences only one noise configuration and for any noise configuration, even nondifferentiable with respect to time, ${\dot {x}}={\mathcal {F}}(t)$ defines diffeomorphisms or flows acting on differential forms as, $\partial _{t}\psi (t)=-{\hat {L}}_{{\mathcal {F}}(t)}\psi (t)$, where ${\hat {L}}$ is the Lie derivative, which is an infinisimal pullback, and the minus sign because it is the inverse diffeomorphisms that evolve the wavefunctions.

For small but finite $\Delta t$, the solution is $\psi (t+\Delta t)=({\hat {1}}-\int _{t}^{t+\Delta t}d\tau {\hat {L}}_{{\mathcal {F}}(\tau )}+\int _{t}^{t+\Delta t}d\tau _{1}{\hat {L}}_{{\mathcal {F}}(\tau _{1})}\int _{t}^{\tau _{1}}d\tau _{2}{\hat {L}}_{{\mathcal {F}}(\tau _{2})}...)\psi (t)$ The expression in the parenthesis is the pullback to the finite-time diffeomorphism between $t{\text{ and }}t+\Delta t$. It depends on the noise configuration, which we can average over to get the average evolution operator $\psi (t+\Delta t)=({\hat {1}}-\Delta t{\hat {L}}_{F}+(1/2)2\Theta \Delta t{\hat {L}}_{G_{a}}{\hat {L}}_{G_{a}}...)\psi (t)$ where we used $\langle \xi ^{a}(t)\rangle _{noise}=0,\langle \xi ^{a}(t)\xi ^{b}(t')\rangle _{noise}=\delta ^{ab}\delta (t-t'),\int _{t}^{\tau _{1}}d\tau _{2}\delta (\tau _{1}-\tau _{2})=1/2$ (because delta is a symmetric function of its argument). In the limit $\Delta t\to 0$, we get $\partial _{t}\psi (t)=-{\hat {H}}\psi (t),\;{\hat {H}}={\hat {L}}-\Theta {\hat {L}}_{G_{a}}{\hat {L}}_{G_{a}}$. This is it. No approximation, assumptions, nothing. The only potential criticism I can forsee is that this derivation should breakdown when the noise configuration is not integrable. Such configurations, however, are strongly suppressed in the gaussian white noise, whose functional probability distribution $\propto exp(-\int _{t}^{t+\Delta t}d\tau \xi (\tau )^{2})$.

The so-obtained evolution operator is unique and it is of the Stratonovich form. This is because we assumed that time is continuous from the start. And this is exactly what separates stochastic differential equations from stochastic difference equations, in which one can choose to average over noise before taking the continuous time limit rendering the dependence of the result on an artificial parameter typically called $\alpha$. This $\alpha$-dependent point of view is applicable only to dynamical systems with discrete time, but not to real life dynamical systems in which time is always continuous.

For a rigorous derivation of this formula in the mathematics literature, see for example Kunita's 1990 book "Stochastic flows and stochastic differential equations", Corollary 4.9.4 (in the more general context of acting on arbitrary tensor fields). It probably goes back much further, I guess Elworthy's 1978 paper contains some version of it, but I don't have access to it right now. Hairer (talk) 18:57, 28 May 2025 (UTC)