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)

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)

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)