Talk:Supersymmetric theory of stochastic dynamics

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)

I gave the article another quick read-through, and firstly I must say it is orders of magnitude better than my first recollections of it, some fivish years ago. So; good job. I think a short "background" section with definitions of the basic mathematical objects would be extremely helpful for laypeople... as is customary in highly technical wiki pages. Since wikipedia does not assume the expertise, nor formal-notational fluency of its readers. Also as far as I understand the GTO is the fundamental theorem you put forward in this model? I think it may be prudent to highlight somehow either this; or the TFT operator-representation as the constitutive formulas for this page. As in put them first, or box them akin to math pages as in Navier-Stokes. To make people understand what is the focus of the theory.

Lastly I would like to add a short second paragraph to the lede - in order to make an oversimplified summary for laypeople to grasp the theory (at least as I understand it so far). You can later edit it to your liking if I miss the mark.

Added a second lede and image; use it as a base to edit as you see fit.

A

2001:861:44D0:9F20:C4E5:9CB4:F81D:885A (talk) 09:49, 4 June 2025 (UTC)

Hi there, I see you made quick work of this. Glad you accepted the second lede idea; however I'm afraid you made it once more too technical haha...! The point was to oversimplify it completely for a lay audience. Consider reverting, and making minor phrasing edits where you think I was inaccurate perhaps? Also the image caption is far too complicated too. Try to reduce it to bare minimum as was before. Cheers! 2001:861:44D0:9F20:70F6:27F3:456D:608 (talk) 19:07, 4 June 2025 (UTC)

The lede is better now; I still think something closer to the original phrasing is more warranted, from the perspective of some given reader with effectively zero physics knowledge. But this is fine. Also I think a picture could still lend more clarity with a simple one-line caption as before. Anyhow aside from the aforementioned "background" section for the notation and objects; I think I've contributed what I can to this article. I hope you continue the good work, and ever further improve! Buona fortuna.

A 2001:861:44D0:9F20:70F6:27F3:456D:608 (talk) 21:42, 4 June 2025 (UTC)

I've commented out the second lede since it mostly references original research and unverified claims. Hairer (talk) 06:18, 5 June 2025 (UTC)

Hello there. Consider looking at the original phrasing I made before edits and tell me if you think that is appropriate as a simple de-facto description of what this theory is trying to say for a general audience. As it stands it is too ambiguous...

A 2001:861:44D0:9F20:8E5D:DB1E:1389:FFE2 (talk) 08:45, 5 June 2025 (UTC)

I think the original phrasing was much better. It felt a bit too vague and I guess someone familiar with the theory could come up with a crisper formulation, but at least it stayed factual, away from grandiose unsupported claims.Hairer (talk) 11:54, 5 June 2025 (UTC)

Reinstate if you see fit, I think it was at least a good context for someone without a clue. Also I think the image I added later for lay people to understand what a topological order even is, was useful for a superficial impression. Up to you.

A 2001:861:44D0:9F20:8E5D:DB1E:1389:FFE2 (talk) 12:02, 5 June 2025 (UTC)

The current lede is categorically unencyclopedic. I think there is a misunderstanding which the author of this page does not comprehend; that per wikipedia's own guidelines, no one "owns" a wiki page. Even its creator. Seems like hardly any external revisions are accepted even where we try to help. I am doing my best to force myself into the perspective of an average reader whom has no knowledge of physics (which is precisely what wikipedia is for; not seasoned specialists). Frankly, even as someone who researched over a decade of mathematical physics, with stochastics specifically... I am mostly confused by the lofty claims and lack of simple mathematical clarity.

I think I've done everything I can here. As the fella once said: gentlemen, I wash my hands of this weirdness.

A 2001:861:44D0:9F20:102A:46C7:6CF8:3207 (talk) 17:32, 5 June 2025 (UTC)

Wait, please do not leave :) You probably did not like that new quote. Lets move it somewhere else and see if the paragraph gets better. Vasilii Tiorkin (talk) 18:55, 5 June 2025 (UTC)

Is the new version any better ? BTW, sorry if I changed it too much from the proposed version. I think it would take more than one iteration to collectively come up with a good version. Lets please take a few more turns editing this 2nd lede. What I tried to focus on mainly this time is making sure it is factual. Vasilii Tiorkin (talk) 19:12, 5 June 2025 (UTC)

I appreciate you want to keep me around haha... however at the same time I decided to stop editing wikipedia, after several years. Since, as you already know; it is a much more laboriously invloved -and at times pedantically toxic effort- (you should see some of the interactions I had in pure mathematics articles...) than we should care to spend our free time for. Especially to someone in your shoes: who has to create a page.

I think the base assumption here is that you take for granted the notion that this theory is already established. Therefore you use words like "reaveals", "identifies", or "provides". As opposed to the term 'provisional' which I used originally. Which rubs us all the wrong way. I am a bit of a crackpot myself in this manner, so I always try to cull my grandiosity. This theory has yet to be confirmed by third-sources in the scientific community - nor especially by experiment. Therefore all we can do is describe it "as-is" in [almost exclusively] your papers. So say what it *attempts* to do. Not what it already achieved. Especially since the great Lenny Susskind (though I'm more of a Penrose guy myself) said: we definitely do not live in a supersymmetric reality. I think it is important to remember again - wikipedia is a tertiary source that aggregates existing facts. Not a platform for novel or cutting-edge research.

I tried to give the most oversimplified, most generalized lede, which would provide a clear simple explanation with an image; to an innocent passerby who comes here from a popular page like chaos theory. And says oh - this is a thing? My perspective is not necessarily the correct one. Yet the entire reason I began this long and honestly fascinating correspondence; is that from the beginning when I first read it some five years ago... I could not answer the question: what *is* STS? Purely according to the contents this page. This question must be clear to anyone whom reads the whole thing through. And after many times doing so; I'm still not entirely sure. Is it a TFT? Is it a supersymmetric gauge theory? Is it the solution to all human suffering due to chaos?

Honestly I'd like to thank you fellows for this interaction. Where else would somebody have the opportunity to disucss such an interesting new theory with its creator; and a rigorously sharp fields-medalist as well... However without diving into the depths of these derivations I am afraid I won't be of much use. Some of my own research was done about tensors that can describe the ricci-flow of stochastic systems on manifolds. Which exhibited some supersymmetric duality. Therefore I came into this discussion with this background. Although I don't see much more I can contribute. I said what needs to be done: total oversimplification. That's the same standard I demand of my own work. As is oft said (most definitely not by Da Vinci nor Einstein though easily sounds like Feynman...): simplicity is the ultimate sophistication.

If you can achieve a point where a ten year old can explain what STS is, then you've done your job. That's what wikipedia and most encyclopedias are mostly for. Until then there's work to be done. And I don't want to badger on the same point over forever.

Hopefully one day we meet in some conference, and talk each others' ears off about how chaos, is not so chaotic in the end. And the universe has some foundational logic; all the way down to infinite scale-invariance. My money is on primitive roots of unity. 'Till then. Cheers!

A 2001:861:44D0:9F20:102A:46C7:6CF8:3207 (talk) 19:57, 5 June 2025 (UTC)

Thanks so much for the interaction and contributions — they really improved the page (and helped me better understand how Wikipedia works too). Sorry again if I came off as impolite at any point — it was never intentional. If you ever change your mind, you'd be more than welcome back — at least as far as I'm concerned. Cheers! Vasilii Tiorkin (talk) 21:15, 5 June 2025 (UTC)

It's been a pleasure. See you, out there. 2001:861:44D0:9F20:CF49:A6C2:F086:A323 (talk) 08:24, 6 June 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 [1]). 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)

Thank you! This is a good ref. I used it a few times long ago. Let me put it here too. I no longer have access to academia databases to find the exact Elworthy's ref. though.

On another subject, of primary importance is the understanding that diffeomorphisms (or trajectories defining them) are unique functionals of the noise-configuration (Ref.32 on this page by Slavik), provided that the configurations are within what we call the gaussian white noise and this is true. Therefore, the evolution operator defined as the noise average of the corresponding pullbacks is also unique. How rigorous is the derivation of the exact form of this unique operator is of secondary importance, even though I believe that under some general conditions the simple derivation given above is not any less rigorous than the one in the Kunita's book.Vasilii Tiorkin (talk) 23:33, 28 May 2025 (UTC)

After revisiting Kunita's book, it is clear to me that the traditional approach to SDEs is not aware of the mathematical meaning of the finite-time evolution operator. the easiest way to see it is that the most fundamental concept of "pullback" is not even in the Index of the book. In the traditional framework, stochastic evolution is described via a "generator of a semigroup ...", which is essentially another term for a stochastic evolution operator. The credit for GTO must do in its entirety to dynamical system theory. The traditional approach to SDEs can only be referenced in the context of the Stratonovich approach to SDEs, but not in the context of GTO.Vasilii Tiorkin (talk) 22:06, 15 July 2025 (UTC)

See https://arxiv.org/pdf/2105.15017 which is slightly more recent than Elworthy's book and refers to it, but is publicly available. In particular, see (8.8) for the definition of what you call the GTO (called $\delta P_{t}$ there) and Theorem 6.4.1 for a derivation of the generator in the case of 1-forms (the general case is identical). Hairer (talk) 16:42, 17 July 2025 (UTC)

Thanks for the ref. This paper addresses the class of SDEs with stochastic evolution operator from Witten's "Morse Theory and Supersymmetry" (1982). This is a narrow class of SDEs with gradient flows (many call it Langevin SDEs). These SDEs are never chaotic. Neither does this paper reveal the mathematical meaning of stochastic evolution operator -- it is still just a "generator". But let me change the corresponding part of the page a bit.Vasilii Tiorkin (talk) 14:27, 18 July 2025 (UTC)

Well, you can also look at Sections 2.3 and 2.4 of the book "On the Geometry of Diffusion Operators and Stochastic Flows" by Elworthy, LeJan, and Li which definitely contains the general case. I agree that none of these works address the link between spectral properties of the generator on forms and chaotic properties of the dynamic, but that wasn't my point. My point was just that this operator and its rigorous mathematical link to the corresponding stochastic dynamic has appeared in the mathematical literature a long time ago. Hairer (talk) 16:58, 18 July 2025 (UTC)

Thank you. I did not know it was Elworthy who first wrote it.Vasilii Tiorkin (talk) 18:49, 18 July 2025 (UTC)

By the way, what you call the "supersymmetric form of the GTO" was also remarked as Proposition 2.3.1 in that book. Hairer (talk) 18:08, 18 July 2025 (UTC)

Thanks. Indeed, Eqs. 2.3.1 and 2.3.3 are one step short of the statement that A is d-exact. Elworthy would not probably find the fact that A is d-exact valuable or useful in anyway -- TFTs were barely out of their formulation at that time and they were not yet generalized to pseudo-hermitian operators such as A.Vasilii Tiorkin (talk) 18:40, 18 July 2025 (UTC)

Applications

That whole sections seems quite problematic. The 'butterfly effect' part seems mostly redundant with the 'Stochastic chaos' section. The 1/f noise part has no citation whatsoever, so is either original research (which doesn't belong to Wikipedia) or needs citations.

The link to the kinenematic dynamo is speculative. It is based on one single article by the main editor of this page which has attracted a total of two citations in 10 years (excluding self-citations). The relation between STS and SOC is equally speculative. It is also based on one single article (again by the main editor of this page) that has only 2 citations on Google Scholar, one of which is a self-citation. Hairer (talk) 20:48, 4 June 2025 (UTC)

That is why we have this tag at the top, right ? But I agree, there are ways to improve this section.Vasilii Tiorkin (talk) 23:06, 4 June 2025 (UTC)

I've commented it out then. Hairer (talk) 06:15, 5 June 2025 (UTC)

Well, this is not exactly what I meant by improving this part. But yes, it has got to be made shorter. At this, we must keep the SOC part at all costs. This is for the betterment of humanity. It may save a few poor souls that are about to be press-ganged into the antiscientific cult of SOC. We can use another reference with a few dozen citations, which will make this subsection less "original research"". BTW, we are not too afraid of the original research because we have a tag on top. We will be more strict about it in the future after we removed the tag.Vasilii Tiorkin (talk) 12:26, 5 June 2025 (UTC)

Effective field theory

I have added a standard argument from quantum field theory that connects the evolution operator to the Goldstone theorem. I believe this is important, as it may help prevent a common, fundamental, and very subtle misunderstanding -- namely, the tendency to interpret the Witten index as a partition function. This mistake has appeared frequently in the literature (my work included), tracing back to the original paper by Parisi and Sourlas. At their time, however, the confusion was understandable -- the Witten index had not yet been introduced. I hope this brief addition offers a helpful clarification and helps others avoid falling into this conceptual trap.