2-form and dissipative systems

[A. Neumaier]

(1) only a single tangent vector? Why aren't the tangent spaces 2D?

They are. But curves in an n-dimensional manifold have tangents that are vectors in the n-dimensional tangent spaces. Here n=2.

I also don't see why omega is no longer the area when it was before.

omega was never an area. It is a volume form, which means that (without your prefactor) omega(u,v) is the area of the parallelogram with vertices 0, u, v, and u+v.

So then for my system: [...] The system is nonlinear, how should I represent F?

In a dynamical system, F(z) is a nonlinear map. For your system, F(z) is the vector with components -gamma z_1 - k sin z_2 and z_1/m. People also write div F or nabla dot F for trace F'.

The particular stuff you're saying at the end of your last post (taking the trace of F' etc) is there a place I can find more on this? Otherwise I'm going to have to ask a number of questions which may become tiring for you.

I don't know where to find it; never look up these elementary things. Maybe others can help out.

 

[homology]

Curves in an n-dimensional manifold have tangents that are vectors in the n-dimensional tangent spaces. here n=2.

omega was never an area. It is a volume form, which means that (without your prefactor) omega(u,v) is the area of the parallelogram with vertices 0, u, v, and u+v.

Okay, I probably have been careless in associating omega with area/volume. But in one of my previous posts I did express the area as \omega(\vec{u},\vec{v}) where u,v are tangent vectors at some point (not tangent to the curve, just tangent to the phase space and so giving some notion of area at that point.

In a dynamical system, F(z) is a nonlinear map. For your system, F(z) is the vector with components -gamma z_1 - k sin z_2 and z_1/m. People also write div F or nabla dot F for trace F'.

I don't know where to find it; never look up these elementary things. Maybe others can help out.

Okay,

<br /> \frac{d}{dt}\begin{pmatrix} p \\ q\end{pmatrix} = \begin{pmatrix} -\gamma p - k\sin(q) \\ p/m\end{pmatrix}<br />

So the divergence, is this just (\partial/\partial p, \partial/\partial q)? If so then that would give me \nabla\cdot F = -\gamma

If its terribly elementary then perhaps I just know it in a different context? My department doesn't do anything geometrical, so I do this on the side, slowly, very slowly. Is there a name, or term for F? Or F', or div(F)?

I mean, div(F) is the divergence of the dynamical vector field, which, if the system was Hamiltonian, should be zero correct?

 

[A. Neumaier]

So the divergence, is this just (\partial/\partial p, \partial/\partial q)?

No. It is dF_1/dp + dF_2/dq. You seem to have used that but what you wrote is quite different. You need to take much more care in writing formulas.

If so then that would give me \nabla\cdot F = -\gamma

Yes. Thus dA/dt = -\gamma A

I mean, div(F) is the divergence of the dynamical vector field, which, if the system was Hamiltonian, should be zero correct?

Yes. Prove it for a general Hamiltonian system, as an exercise!

 

[homology]

Apologies for the careless verbage and notation and gratitude for your help, I'll work on this and then post something coherent :)

 

[C. H. Fleming]

Is it correct to say that

L_X\omega=\dot{\omega}

You want to say this

\phi^* L_X \omega = \frac{d}{dt} \phi^* \omega

The flow has time dependence, not the symplectic form, as Arnold already mentioned.

From this relation you probably immediately see

\frac{d}{dt} \phi^* \omega = - \gamma \phi^* \omega

which is the ODE you were looking for. I figured this out a while back, but I thought I would give you a chance to work it out yourself.

 

[C. H. Fleming]

Also I believe I have determined that the positivity constraint for Fokker-Plank equations defined on a continuous phase space with only local operations (coordinates and derivatives) is that it be constructed from an exterior derivative (e.g. in this case, all derivatives to the left of all coordinates). Without admitting nonlocal operations (phase-space integrals) there does not appear to be as much interesting structure as with stochastic matrices.

 

[A. Neumaier]

Also I believe I have determined that the positivity constraint for Fokker-Plank equations defined on a continuous phase space with only local operations (coordinates and derivatives) is that it be constructed from an exterior derivative (e.g. in this case, all derivatives to the left of all coordinates). Without admitting nonlocal operations (phase-space integrals) there does not appear to be as much interesting structure as with stochastic matrices.

There are two kinds of dissipative stochastic processes forming a dynamical semigroup on measures: Diffusion processes (given by Fokker-Planck equations) and jump processes (given by master equations) - and their combinations. For diffusion processes, the drift can be arbitrary and the diffusion must be positive semidefinite. The master equations are essentially infinite-dimensional versions of Markov chains.

 

[homology]

I take it that A. Neumanaier's first name is Arnold? In any event it reminded me of a section of V.I. Arnold that I hadn't quite grasped (and still don't thus the post).

While it makes sense to me that the divergence of the vector field should give some measure of how the volume changes I wanted to see a derivation. So I looked in V.I.Arnold's text (Section 16)

He starts with a vector field \vec{F}(\vec{x}) with local coordinates \vec{x} and the associated flow g^t. My questions have to do mostly with the appearance of t\to 0 which will turn up shortly.

Let D(0) be a region and v(0) its volume. v(t)=vol(D(t)) where D(t)=g^tD(0). We can also express v(t) as,

<br /> v(t)=\int_{D(0)} det\frac{\partial g^t \vec{x}}{\partial \vec{x}}dx<br />

We can evaluate the derivative of g^t by first expanding it,

<br /> g^t\vec{x}=\vec{x}+\vec{F}(\vec{x})t+O(t^2), (t\to 0)<br />

which is mostly okay. The flow at t=0 is the identity and its time derivative gives the vector field and there would be higher order terms. Why is 't' going to zero?

Then he uses a neat little formula that for any matrix A, det(I+At)=1+t\cdot tr(A)+O(t^2) which I'd like to figure out eventually (but that's for another time). This gives us:

<br /> \frac{\partial g^t \vec{x}}{\partial \vec{x}}=1+tdiv(\vec{F})+O(t^2)<br />

Jamming this into the integral for v(t) and differentiating with respect to time we have:

<br /> <br /> \frac{dv}{dt}=\int_{D(0)}div(\vec{F})dx+\int_{D(0)}\frac{\partial O(t^2)}{\partial t}dx<br /> <br />

V.I.Arnold takes the limit as t\to 0 which gives the convenient result:

<br /> \frac{dv}{dt}|_{t=0}=\int_{D(0)}div(\vec{F})dx<br />

But I don't want the restriction of t=0?

 

[A. Neumaier]

I take it that A. Neumanaier's first name is Arnold? In any event it reminded me of a section of V.I. Arnold that I hadn't quite grasped (and still don't thus the post).

Yes, this is the derivation of the formula I used. And yes, my first name is his family name. But my family name is Neumaier, not Neumanaier.

The flow at t=0 is the identity and its time derivative gives the vector field and there would be higher order terms. Why is 't' going to zero?

But I don't want the restriction of t=0?

Then do a similar expansion around t=t_0 in place of t=0.

 

[homology]

doh...