2-form and dissipative systems

[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,

$$v(t)=\int_{D(0)} det\frac{\partial g^t \vec{x}}{\partial \vec{x}}dx$$

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

$$g^t\vec{x}=\vec{x}+\vec{F}(\vec{x})t+O(t^2), (t\to 0)$$

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:

$$\frac{\partial g^t \vec{x}}{\partial \vec{x}}=1+tdiv(\vec{F})+O(t^2)$$

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

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

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

$$\frac{dv}{dt}|_{t=0}=\int_{D(0)}div(\vec{F})dx$$

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...