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Towards Navier-Stokes from noncommutative geometry

edited November 2018 in - - Mathematical methods

Note: I played a bit with the spacetime continuum, i.e. I significantly modified this post after first submitting it.

In this post, I will work towards deriving the Navier-Stokes equation from noncommutative geometry. For some references and some mathematical background, see my previous post:

First, recall that our differential calculus is determined by the commutative relations

$$[d t, t] = [d x^i, t] = [d t, x^i] = 0\quad\text{and}\quad [d x^i, x^j] = \delta^{i,j} \eta d t,$$ for all \(i,j\in{1,2,3}\). These relations imply

$$d t d x^i = -d x^i d t\quad\text{and}\quad d x^i d x^j = -d x^j d x^i.$$ Due to the noncommutativity of 0-forms and 1-forms, in (3+1)-dimensions, we can either write a 1-form using left or right components, i.e.

$$\alpha = \sum_{i=1}^3\stackrel{\leftarrow}{\alpha}_i d x^i + \stackrel{\leftarrow}{\alpha}_t d t$$ or

$$\alpha = \sum_{i=1}^3 d x^i\stackrel{\rightarrow}{\alpha}_i + d t\stackrel{\rightarrow}{\alpha}_t$$ Using the commutative relations, it can be shown that

$$\stackrel{\leftarrow}{\alpha}_i = \stackrel{\rightarrow}{\alpha}_i\quad\text{and}\quad \stackrel{\rightarrow}{\alpha}_t = \stackrel{\leftarrow}{\alpha}_t - \eta\sum_i\partial_i\stackrel{\leftarrow}{\alpha}_i.$$ In particular, this means that

$$d\phi = \left(\partial_t \phi + \frac{\eta}{2}\sum_{i=1}^3 \partial_i^2\phi\right) d t + \sum_{i=1}^3 \left(\partial_i \phi\right) d x^i = d t \left(\partial_t \phi - \frac{\eta}{2}\sum_{i=1}^3 \partial_i^2\phi\right) + \sum_{i=1}^3 d x^i\left(\partial_i \phi\right).$$ To derive the Navier-Stokes equation, we simply begin by writing a connection \(A\) in right component form

$$A = -d t p + \sum_{i=1}^3 d x^i u_i.$$ The curvature of this connection is given by

$$F = d A + A A.$$ Straightforward (yet somewhat tedious) calculations show that

$$d A = d t \sum_{i=1}^3 d x^i\left(\partial_t u_i - \frac{\eta}{2}\sum_{j=1}^3 \partial_j^2 u_i + \partial_i p\right) + \sum_{i,j=1}^3 d x^i d x^j \left(\partial_i u_j\right)$$ and

$$A A = \eta d t \sum_{j=1}^3 d x^i \left(\partial_j u_i\right) u_j.$$ Putting these together, we have

$$F = d t \sum_{i=1}^3 d x^i \left[\partial_t u_i - \frac{\eta}{2}\sum_{j=1}^3 \partial_j^2 u_i + \eta \sum_{j=1}^3 \left(\partial_j u_i\right) u_j + \partial_i p\right] + \sum_{i,j=1}^3 d x^i d x^j \left(\partial_i u_j\right).$$ With only a thinly veiled suggestive notation, we'll write the curvature as

$$F = d t E + B,$$ where \(E\) is a noncommutative purely spatial 1-form and \(B\) is a noncommutative purely spatial 2-form.

Matching coefficients, we have the

Navier-Stokes Equation

$$\partial_t u_i - \frac{\eta}{2}\sum_{j=1}^3 \partial_j^2 u_i + \eta \sum_{j=1}^3 \left(\partial_j u_i\right) u_j + \partial_i p = E_i\quad\text{and}\quad \partial_i u_j - \partial_j u_i = B_{i,j}$$ for \(i,j\in{1,2,3}\).

Comments

  • 1.
    edited November 2018

    In complete analogy to the rescaling described here:

    if we rescale the connection according to

    $$A = \frac{1}{\eta}\left(-d t p + \sum_{i=1}^3 d x^i u_i\right),$$ we get the Navier-Stokes equation with the familiar coefficients, i.e.

    Navier-Stokes Equation

    $$\partial_t u_i - \frac{\eta}{2}\sum_{j=1}^3 \partial_j^2 u_i + \sum_{j=1}^3 \left(\partial_j u_i\right) u_j + \partial_i p = \eta E_i\quad\text{and}\quad \partial_i u_j - \partial_j u_i = \eta B_{i,j}$$ for \(i,j\in{1,2,3}\).

    When the curvature vanishes, there is a Cole-Hopf transformation for the curl-free Navier-Stokes equation. Using the trick observed here we can write

    $$A = \frac{1}{\eta}\left(-d t p + \sum_{i=1}^3 d x^i u_i\right) = -(d\phi)\phi^{-1}.$$ Expanding the right-hand side and matching coefficients gives us the

    Cole-Hopf Transformation for the Curl-Free Navier-Stokes Equation

    $$u_i = -\eta(\partial_i \phi)\phi^{-1}$$ and $$\partial_t \phi - \frac{\eta}{2}\sum_{i=1}^3 \partial_i^2 \phi - p\phi = 0.$$
    Comment Source:In complete analogy to the rescaling described here: * [Burgers equation revisited](https://forum.azimuthproject.org/discussion/comment/4184/#Comment_4184) if we rescale the connection according to $$A = \frac{1}{\eta}\left(-d t p + \sum_{i=1}^3 d x^i u_i\right),$$ we get the Navier-Stokes equation with the familiar coefficients, i.e. >**Navier-Stokes Equation** > >$$\partial_t u_i - \frac{\eta}{2}\sum_{j=1}^3 \partial_j^2 u_i + \sum_{j=1}^3 \left(\partial_j u_i\right) u_j + \partial_i p = \eta E_i\quad\text{and}\quad \partial_i u_j - \partial_j u_i = \eta B_{i,j}$$ > >for \\(i,j\in\{1,2,3\}\\). When the curvature vanishes, there is a Cole-Hopf transformation for the **curl-free** Navier-Stokes equation. Using the trick observed [here](https://forum.azimuthproject.org/discussion/comment/4183/#Comment_4183) we can write $$A = \frac{1}{\eta}\left(-d t p + \sum_{i=1}^3 d x^i u_i\right) = -(d\phi)\phi^{-1}.$$ Expanding the right-hand side and matching coefficients gives us the >**Cole-Hopf Transformation for the _Curl-Free_ Navier-Stokes Equation** > > $$u_i = -\eta(\partial_i \phi)\phi^{-1}$$ > >and > >$$\partial_t \phi - \frac{\eta}{2}\sum_{i=1}^3 \partial_i^2 \phi - p\phi = 0.$$
  • 2.

    This is a million miles over my head. However let me try to make an improbable connection.

    Paul Krugman (who claims expertise in resource economics) recently opined that peak oil would not be too serious, because oil only takes a small part of our budgets. But money doesn't go to oil (and to workers and other input), it flows through them. The money goes for the oil, then gets spent on oil rigs and other things, and so on.

    So understanding economics is about understanding flows. And even though I don't understand noncommutative geometry at all, still it would be great if it shed light on economics. A Nobel Prize in Economics awaits!

    Comment Source:This is a million miles over my head. However let me try to make an improbable connection. Paul Krugman (who claims expertise in resource economics) recently opined that peak oil would not be too serious, because oil only takes a small part of our budgets. But money doesn't go to oil (and to workers and other input), it flows through them. The money goes for the oil, then gets spent on oil rigs and other things, and so on. So understanding economics is about understanding flows. And even though I don't understand noncommutative geometry at all, still it would be great if it shed light on economics. A Nobel Prize in Economics awaits!
  • 3.
    edited May 2011

    Eric has worked for a number of years in financial circles, so he's thought a lot about the economic/financial applications of this kind of stuff.

    Personally, I'm a bit equivocal about financial models of this form. I have a uneasy feeling that when quants in general (not Eric in particular) write down these kind of models the kind of assumptions that lead to "tractable" models are favoured for those tractability reasons (rather than because they're supported by analysis or observation of the actual world). One simple example is models incorporating "always applicable" conservation laws: in the financial world it seems like "sensible" conservation laws hold, except in extreme situations where they don't. (Eg, when an entity defaults on its debts without enough assets to cover them.)

    Even things like oil, whilst "strictly" conserved often effectively vanished (eg, the oil in the Gulf of Mexico from the BP blowout).

    Comment Source:Eric has worked for a number of years in financial circles, so he's thought a lot about the economic/financial applications of this kind of stuff. Personally, I'm a bit equivocal about financial models of this form. I have a uneasy feeling that when quants in general (not Eric in particular) write down these kind of models the kind of assumptions that lead to "tractable" models are favoured for those tractability reasons (rather than because they're supported by analysis or observation of the actual world). One simple example is models incorporating "always applicable" conservation laws: in the financial world it seems like "sensible" conservation laws hold, except in extreme situations where they don't. (Eg, when an entity defaults on its debts without enough assets to cover them.) Even things like oil, whilst "strictly" conserved often effectively vanished (eg, the oil in the Gulf of Mexico from the BP blowout).
  • 4.
    edited November 2018

    John politely pointed out via email that the zero curvature condition In earlier versions of the comments above had no right to be called "Navier-Stokes" because zero curvature also says the solution is curl-free.

    This is fine. Instead of the original

    $$F = 0$$ we need

    $$F = d t E + B$$ where \(E\) is a purely spatial noncommutative 1-form and \(B\) is a purely spatial 2-form.

    This makes it no less fascinating. I've made these changes above. We are still not quite at the Navier-Stokes equation because we need equations for \(E\) and \(B\). I have an idea about how to proceed and the suggestive notation is a hint, but I'm running out of steam. Not sure if I'll finish this evening.

    Comment Source:John politely pointed out via email that the zero curvature condition In earlier versions of the comments above had no right to be called "Navier-Stokes" because zero curvature also says the solution is **curl-free**. This is fine. Instead of the original $$F = 0$$ we need $$F = d t E + B$$ where \\(E\\) is a purely spatial noncommutative 1-form and \\(B\\) is a purely spatial 2-form. This makes it no less fascinating. I've made these changes above. We are still not quite at the Navier-Stokes equation because we need equations for \\(E\\) and \\(B\\). I have an idea about how to proceed and the suggestive notation is a hint, but I'm running out of steam. Not sure if I'll finish this evening.
  • 5.

    Hi everyone,

    I disappeared for a while (more than 7 years!), but I recently found myself interested in these old results that are scattered across multiple discussions here on this forum so just published a concise version on my (old) blog:

    https://phorgyphynance.wordpress.com/2018/11/19/noncommutative-geometry-and-navier-stokes-equation/

    All of this was intended to lead toward some numerical implementations, so hopefully I get around to doing that this time :)

    Comment Source:Hi everyone, I disappeared for a while (more than 7 years!), but I recently found myself interested in these old results that are scattered across multiple discussions here on this forum so just published a concise version on my (old) blog: https://phorgyphynance.wordpress.com/2018/11/19/noncommutative-geometry-and-navier-stokes-equation/ All of this was intended to lead toward some numerical implementations, so hopefully I get around to doing that this time :)
  • 6.

    An applied application of a reduced form of Navier-Stokes is referred to in the long-running ENSO/QBO discussion thread. https://forum.azimuthproject.org/discussion/1471/qbo-and-enso#latest

    Comment Source:An applied application of a reduced form of Navier-Stokes is referred to in the long-running ENSO/QBO discussion thread. https://forum.azimuthproject.org/discussion/1471/qbo-and-enso#latest
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