User talk:Vasilii Tiorkin

Hello, Vasilii Tiorkin, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few links to pages you might find helpful:

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Please remember to sign your messages on talk pages by typing four tildes (~~~~); this will automatically insert your username and the date. If you need help, check out Wikipedia:Questions, ask me on my talk page, or ask for help on your talk page, and a volunteer should respond shortly. Again, welcome! RJFJR (talk) 15:19, 5 May 2017 (UTC)

Hi, it looks like you are sometimes active on wikipedia. I wanted to talk about Supersymmetric theory of stochastic dynamics, and ideally, have you explain it to me. FWIW, I do know supersymmetry, and once upon a time, I studied BRST but kind of forgotten it (I could learn it again.) Here's why I ask. That article makes some very dramatic and broad claims that I find surprising. I want to understand if they are as broad as they seem to be, or if they are narrower. And thus .. I'd like to talk to you directly, to straighten things out.

Let me start with the basics: so, for example, the article starts with a ${\displaystyle {\mathcal {W}}=...}$ but then doesn't "do anything" with it. It's rewritten in several different forms, but ... its not obvious how to use it to "solve" anything. Say, given some specific diffeq, its not clear how to obtain solutions using this, or how to actually calculate specific "useful" quantities/invariants/averages, etc. (perhaps I'm being stupid and impatient)

Right now, I'm not even sure what to ask or how to ask it; but the first step was to try to contact you. We can continue this conversation ... here, or on my talk page, or on the article talk page, or by private email. The conversation will probably be very very slow, because I have many other things to worry about, and the quantity of material here is so large, it will take me a while to absorb it. But I'd be thrilled if you could wave a magic wand and make me instantly smart. 67.198.37.16 (talk) 07:20, 23 May 2024 (UTC)

Hi, thank you for your message !

Regarding the Witten index, it is a subtle, fundamental and very common mistake in the Literature to use W as if it was a partition function of the SDE. In reality, it is a partition function of the noise (up to a topological factor) and it is not responsive to, say, the perturbation of the SDE. Accordingly, it cannot be used to study the response of the system to the perturbation - the response of W is identically zero. This is why I tried to stress it in the article that W cannot be used to "solve anything".

To address various interesting characteristics, one should turn to the partition function by switching from the periodic to antiperiodic boundary conditions for the ghosts. Unlike the Witten index, the partition function is not a topological invariant. This does not mean, however, that the topological supersymmetry is gone -- the topological supersymmetry is the property of the evolution operator and it is still there.

Let us use conventional email. My email is in this paper

https://www.sciencedirect.com/science/article/abs/pii/S0960077924001620

Vasilii Tiorkin (talk) 14:36, 23 May 2024 (UTC)

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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 (${\displaystyle \Delta \tau \to 0}$)

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

exists and the solution converges to its continuous time version:

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

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

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)