2-form and dissipative systems

[homology]

Suppose you have a dissipative system where

$$\dot{q}=p/m$$

$$\dot{p}=-\gamma p -k\sin(q)$$

So there isn't a Hamiltonian for this system and Louiville's theorem doesn't hold. But the equations of motion still give us a vector field on phase space and we can still take the Lie derivative of $$\omega = dp\wedge dq$$ along it.

If I do that I get $$L_X \omega = -\gamma dp\wedge dq$$ (let me know if you want to see the details, I'm omitting them here to keep things brief) where X is the dynamical vector field. I'm still new to the geometric understanding of mechanics and so the above 'looks' like a differential equation, but I'm not sure what to make of this? I suppose I'd like to see how to use this to interpret $$\omega$$ as a function of time, but I'm stuck.

Any ideas are welcome.

 

[C. H. Fleming]

For this simple dissipation your Liouville operator is:

$$\boldsymbol{\mathcal{L}} = -\frac{p}{m} \frac{\partial}{\partial q} +k \sin(q) \frac{\partial}{\partial p} + \gamma \frac{\partial}{\partial p} p$$

and so in addition to the dissipative force $$\gamma p \frac{\partial}{\partial p}$$, you have the probability preserving term $$\gamma$$. Instead of Liouville's theorem you can apply its generalization: the method of characteristic curves.

I am not sure (off the top of my head) what it means to do what you are doing as dissipative mechanics is not symplectic and you are working with a symplectic form. I would also be interested if anyone knows the dissipative generalization of symplectic evolution in classical mechanics. In quantum mechanics, the group structure is much simpler and this is known.

 

[homology]

Okay, well your Louiville operator corresponds to vector field I'm differentiating $$\omega$$ along. I think I understand your objection to possibly using $$\omega$$ however I can't see why there still wouldn't be such a two form on the phase space, Hamiltonian or no Hamiltonian? clearly dq and dp are one forms, why wouldn't their wedge exist?

 

[C. H. Fleming]

What you have constructed exists - I don't question that. It's the interpretation I wonder about. The fact that it doesn't vanish along your dynamical vector field is probably telling you that no symplectic transformation is volume preserving here.

Moreover, if you omit the probability preserving $$\gamma$$ term from the correct Liouville operator, as to obtain a purely differential operator, then that differential part of the Liouville operator would actually be shrinking phase-space volume. That is probably what corresponds to the Lie-differential equation you have obtained $$L_X \omega = -\gamma \omega$$, which looks like exponential decay.

 

[homology]

So do you know if we can 'do' anything with such a 'differential equation'?

I think we both agree that omega exists and gives area, but that area is time dependent. I'd like to compute that. So give some time t and vectors v and w, I'd like to know how omega evaluates on v and w at time t. Then also, how that area changes with time.

My first thought is to evaluate the 'differential' equation on some vectors and perhaps obtain a differential equation for the area?

 

[C. H. Fleming]

I believe you can use the flow map to convert it to an ODE in time.

 

[homology]

Do you have a favorite reference for learning about flow maps? I'm not necessarily familiar with this, or do you mean the flow obtained by equations of motion?

By the way, I appreciate the time you've taken to engage in this thread, thank you.

 

[C. H. Fleming]

I come from a dissipative quantum mechanics background, so my only good reference to what you are doing is a fine book called "The Geometry of Physics" by Theodore Frankel. But, in brief the flow map is simply the map generated by your dynamical vector field. The dynamical vector field belongs to the Lie algebra, whereas the flow map belongs to the Lie group. Perhaps you know the flow map by another name. In QM I would simply call it the propagator or transition matrix (for the density matrix).

And no problem about engagement, I am weak on exterior forms, so this is making me think about things I know in a different context.

 

[homology]

I have that book! I'll have to check it out this weekend. But it does sound like yes, the flow you're talking about is the 'solution' to the dynamical vector field.

 

[A. Neumaier]

I would also be interested if anyone knows the dissipative generalization of symplectic evolution in classical mechanics. In quantum mechanics, the group structure is much simpler and this is known.

What about the entry ''Dissipative dynamics and Lagrangians'' of Chapter A1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#dissslag ?

 

[C. H. Fleming]

What about the entry ''Dissipative dynamics and Lagrangians'' of Chapter A1 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#dissslag ?

I am thinking of a much deeper question that I did not sufficiently specify.

In quantum mechanics, closed system evolution has unitary evolution with anti-hermitian generators ($$-\imath \mathbf{H}$$) whereas dissipative evolution has completely-positive evolution with Lindblad-GKS generators ($$\mathcal{L}$$). (And here I am specifically talking about the algebraic generators. The time-translation generators (a.k.a. the master equation) are only Lindblad in the Markovian regime. (For unitary dynamics this distinction is irrelevant even though the generators are also not always equivalent).).

For classical physics the closed system evolution is symplectic, but what characterizes the dissipative evolution? Is there, or can there be, something like the Lindblad-Gorini-Kossakowski-Sudarshan theorem for the generators of classical dissipative evolution? I don't do much classical physics, so I have never attempted to derive such a thing, but I should probably sit down and try one day.

 

[A. Neumaier]

I am thinking of a much deeper question that I did not sufficiently specify.

In quantum mechanics, closed system evolution has unitary evolution with anti-hermitian generators ($$-\imath \mathbf{H}$$) whereas dissipative evolution has completely-positive evolution with Lindblad-GKS generators ($$\mathcal{L}$$). (And here I am specifically talking about the algebraic generators. The time-translation generators (a.k.a. the master equation) are only Lindblad in the Markovian regime. (For unitary dynamics this distinction is irrelevant even though the generators are also not always equivalent).).

For classical physics the closed system evolution is symplectic, but what characterizes the dissipative evolution? Is there, or can there be, something like the Lindblad-Gorini-Kossakowski-Sudarshan theorem for the generators of classical dissipative evolution?

Yes. The classical analogue (and in many cases, a suitable classical limit) of a quantum dynamical semigroup is a Markov process.

The most general Markov processes are a combination of diffusion processes and jump processes. See the differential Chapman-Kolmogorov equation in Gardiner's Handbook of stochastic methods - (3.4.22) in the second edition. The derivation there is on the level of rigor of theoretical physics; but it is very likely that there is a fully rigorous version of this result in terms of measure-theoretic stochastic processes.

Should you or someone else find a reference to such a mathematical presentation, I'd be interested.

 

[C. H. Fleming]

Yes. The classical analogue (and in many cases, a suitable classical limit) of a quantum dynamical semigroup is a Markov process.

I don't want to limit scope to the dynamical semigroup, so the stochastic process may be non-Markovian and the quantum dynamical generator might not be of Lindblad-GKS form. The algebraic generator (which is not equivalent to the dynamical generator given time dependence) should still satisfy the Lindblad-GKS form. (I believe Kussakovski recently had a paper where he studied some of this in the Laplace domain. I also have a paper, coming out soon, looking at this perturbatively.) The reason this happens is because Lindblad-GKS derives directly from Choi's (group) theorem on CP maps, and so the Lindblad-GKS theorem is more fundamentally a theorem about algebra. The Markovan dynamics result (which is more widely appreciated) then derives from that.

So really, I am asking about the underlying Lie group and algebraic generators of dissipative classical mechanics. Choi and Lindblad-GKS characterize both for quantum mechanics. That is what I compare to the Hamiltonian and its unitary evolution. What characterizes dissipative classical mechanics? That is what I would compare to the symplectic evolution if I knew it. It would be something that is not (phase-space) volume preserving, but still probability preserving. When the question was posed about applying symplectic thinking to a dissipative system, my reaction was one of hesitance, because the evolution is not symplectic. That was why I made the comment that started this. I wish I knew the better algebra to think about.

The quantum correspondence in application between the algebraic and Markovian dynamics of the Lindblad-GKS theorem could be useful classically... if it still exists. Even then, I don't remember a Markovian classical theorem like Lindblad-GKS which says "the Liouvillian can only take the form ... in terms of $$x$$, $$\frac{\partial}{\partial x}$$, ...". Lindblad-GKS is extremely robust because it doesn't care what the model is, how you introduced the stochastic process, etc. I will look back over Gardiner though. Thanks for the recommendation.

 

[A. Neumaier]

I don't want to limit scope to the dynamical semigroup, so the stochastic process may be non-Markovian and the quantum dynamical generator might not be of Lindblad-GKS form. The algebraic generator (which is not equivalent to the dynamical generator given time dependence) should still satisfy the Lindblad-GKS form. (I believe Kussakovski recently had a paper where he studied some of this in the Laplace domain. I also have a paper, coming out soon, looking at this perturbatively.) The reason this happens is because Lindblad-GKS derives directly from Choi's (group) theorem on CP maps, and so the Lindblad-GKS theorem is more fundamentally a theorem about algebra. The Markovan dynamics result (which is more widely appreciated) then derives from that.

So really, I am asking about the underlying Lie group and algebraic generators of dissipative classical mechanics. Choi and Lindblad-GKS characterize both for quantum mechanics.

Could you please refer to an online source or journal paper on ''Choi's (group) theorem on CP maps'', so that I can see what you mean by ''the underlying Lie group'' in the quantum case?

I don't remember a Markovian classical theorem like Lindblad-GKS which says "the Liouvillian can only take the form ... in terms of $$x$$, $$\frac{\partial}{\partial x}$$, ...". Lindblad-GKS is extremely robust because it doesn't care what the model is, how you introduced the stochastic process, etc. I will look back over Gardiner though.

Gardiner does precisely that, though only for Markov processes. I need to see the non-semigroup quantum version to be able to connect it (perhaps) to some classical statement.

 

[C. H. Fleming]

Could you please refer to an online source or journal paper on ''Choi's (group) theorem on CP maps'', so that I can see what you mean by ''the underlying Lie group'' in the quantum case?

Consider the dynamical map:

$$\boldsymbol{\rho}(t) = \boldsymbol{\mathcal{G}}(t,0) \boldsymbol{\rho}(0)$$

where $$\boldsymbol{\rho}(t)$$ is the density matrix of the reduced system at time $$t$$ and we have a non-unitary theory (likely with a traced out environment) such that we can consider any initial state $$\boldsymbol{\rho}(0)$$ and also we might later include ancillary degrees of freedom (e.g. external entanglement).

Then with very few assumptions, $$\boldsymbol{\mathcal{G}}(t,0)$$ are completely-positive (CP) maps or semi-group elements. Then consider Choi's theorem on CP maps which immediately characterizes them:

http://en.wikipedia.org/wiki/Choi%27s_theorem_on_completely_positive_maps"

The (algebraic) generators of these semi-group elements are given by the Lindblad-GKS theorem, though it is usually only useful in the Markovian regime where the algebraic and dynamical generators are equivalent. (Otherwise, one can not extract very much from Lindblad-GKS.) If you refer back to the original papers of Lindblad and Gorini, Kossakowski and Sudarshan, you will see that they refer back to Choi. Choi's theorem describes the semi-group, Lindblad-GKS then describes the algebra which generates it. It's all a very beautiful structure. (I almost finished writing a review paper wherein I try to explain these lesser discussed (and applied) details.)

I think classically, one would have to think akin to non-symplectic flow maps and their generators. (Flow would then be a misnomer.) I am unfamiliar with what kind of structure these non-symplectic maps would be constrained to have. Maybe it is something simple that every classical physicist knows. My knowledge of symplectic manifolds is very weak.

 

[A. Neumaier]

Then consider Choi's theorem on CP maps which immediately characterizes them:
http://en.wikipedia.org/wiki/Choi%27s_theorem_on_completely_positive_maps"
[...]
I think classically, one would have to think akin to non-symplectic flow maps and their generators. (Flow would then be a misnomer.) I am unfamiliar with what kind of structure these non-symplectic maps would be constrained to have. Maybe it is something simple that every classical physicist knows. My knowledge of symplectic manifolds is very weak.

The classical version of Choi's theorem says that a linear mapping from C^n to C^m that maps real nonnegative vectors to real nonnegative vectors is given by a m x n matrix with nonnegative entries. (The proof is straightforward.) These form a semigroup. Their infinitesimal generators are the matrices that are off-diagonally nonnegative.

The associated dynamical semigroups that preserve the trace (i.e., the sum of the entries) are the Markov chains, while the more general version you are after seem to be Markov chains with arbitrarily long memory. In infinite dimensions, and assuming appropriate topologies, you get in place of a Markov chain a combined jump&diffusion process, and presumably the more general version is an arbitrary stochastic process. But since you did not specify the quantum version precisely enough, I can't tell.

Symplecticity never enters. The latter is present only when one specifies a Heisenberg algebra of distinguished operators, which provide a symplectic phase space structure.

 

[C. H. Fleming]

You will have to excuse my ignorance. Why are we mapping between nonnegative vectors? Naively I would imagine the starting point to be norm-preserving positive linear maps between positive functions of the phase-space coordinates (i.e. density functions instead of density matrices). Is there some representation that I am missing?

I also need to think about classical correlations to ancillary degrees of freedom and whatever the analog to complete positivity would be, if any. With the Markov chain that doesn't seem to matter.

 

[A. Neumaier]

Why are we mapping between nonnegative vectors?

Because Choi is mapping between positive semidefinite matrices. This corresponds to n-level quantum systems (whose classical analogue is an n-state probability space), not to particles moving in space. In the latter case, you'd have in place of positive semidefinite matrices integral operators on L^2(R^3) with nonnegative kernels.
I don't know whether the analogue of Choi's theorem has been proved rigorously. Certainly the corresponding Lindblad operators are used in quantum optics.

Naively I would imagine the starting point to be norm-preserving positive linear maps between positive functions of the phase-space coordinates (i.e. density functions instead of density matrices). Is there some representation that I am missing?

You need to select a quantum system to start with. This requires the choice of an algebra of operators. In Choi's case the algebra is C^{n x n}. The classical version corresponds to restricting to a maximal commuting subalgebra - wlog the algebra of diagonal matrices. This is equivalent with taking th algebra C^n with pointwise operations. Whence the setting I was using.

If you want to consider classical phase space, you'd have to think of it as the classical analogue of the quantum algebra of linear operators on L^2(R^3).

I also need to think about classical correlations to ancillary degrees of freedom and whatever the analog to complete positivity would be, if any. With the Markov chain that doesn't seem to matter.[/QUOTE]
This is too cryptic to make sense to me.

 

[C. H. Fleming]

Because Choi is mapping between positive semidefinite matrices. This corresponds to n-level quantum systems (whose classical analogue is an n-state probability space), not to particles moving in space. In the latter case, you'd have in place of positive semidefinite matrices integral operators on L^2(R^3) with nonnegative kernels.
I don't know whether the analogue of Choi's theorem has been proved rigorously. Certainly the corresponding Lindblad operators are used in quantum optics.

I believe the generalization of Choi's theorem is Stinespring's theorem (although I think it is a touch too general). As you already seem to know, people do apply Lindblad's theorem to systems with unbounded operators and the end result looks the same. This is typically safe. I think Davies first worked this out in Rep Math Phys '77, but Sciencedirect is down for maintenance so I can't pull up the paper. I believe the unbounded proof was incrementally fine tuned by several other papers that I don't know off the top of my head.

This is too cryptic to make sense to me.

Yes, its ubiquitous in QM but I've never seen talk of it in Classical physics, so perhaps it is irrelevant.

Say you have some positive maps $$\mathcal{G}$$ between density functions $$\rho$$ on $$2n$$-dimensional phase-space with coordinates $$z$$, and parametrized by time:

$$\rho(z,t) = \mathcal{G}(t,0) \rho(z,0)$$

Then say you add an additional $$2m$$ degrees of freedom to phase space and consider the trivially extended maps

$$\mathcal{G}(t,0) \otimes 1$$

between arbitrary density functions in the $$2(n+m)$$-dimensional phase-space. Then will this map also be positive on the higher dimensional phase space? In QM the answer is not necessarily, and that's why you have to invoke Choi's theorem in the first place. Now that I think about it more, the classical answer is more trivial: all positive maps are completely positive. You just have to invoke the fact that the density function is every where non-negative and that the maps are norm preserving. So indeed, this was irrelevant for me to think about.

Also I would add that in the quantum CP generators, as you likely know, you can see the unitary part. It would seem strange to me that in the classical CP generators, you could not see the symplectic part.

 

[A. Neumaier]

the classical answer is more trivial: all positive maps are completely positive.

Yes.

Also I would add that in the quantum CP generators, as you likely know, you can see the unitary part. It would seem strange to me that in the classical CP generators, you could not see the symplectic part.

No. The classical equivalent of a unitary operator is a bijection of the state space.

Symplecticity is classically expressed by the CCR for p and q in the Poisson bracket, and hence quantum mechanically by the Heisenberg CCR.

In general, the structure of a classical or quantum theory is determined by a distinguished Lie algebra of operators. (See my book http://lanl.arxiv.org/abs/0810.1019 for a deeper discussion.) Without that, Hilbert spaces are far too structureless - just one space for each cardinality of the basis. And most physical systems live in a separable, infinite-dimensional Hilbert space, of which there is only a single one.

 

[C. H. Fleming]

No. The classical equivalent of a unitary operator is a bijection of the state space.

So there are volume preserving flows which cannot be described by a Hamiltonian? I have been thinking about Hamiltonian motion instead of maps between pure states like I should. In QM they are equivalent, but I've never thought about it classically.

 

[A. Neumaier]

So there are volume preserving flows which cannot be described by a Hamiltonian?

Yes. Hamiltonian flow conserves much more than phase space volume.

The analogy is:
unitary mapping -- bijection
commutator bracket -- Poisson bracket

The Hamiltonian defines the flow only given the bracket. But the bracket is not an intrinsic part of the space. Indeed, one can have even in the quantum case [x,y]=xuy-yux, which satisfies for any u the Leibniz and Jacobi identities. in some systems, taking u distinct from the canonical choice i/hbar simplifies the presentation...

 

[C. H. Fleming]

Great progress. I see now, it is obvious that Hamiltonian motion is not necessary to satisfy the continuity of phase-space volume in the evolution of $$\rho$$. It is only sufficient.

I have been thinking about this with the wrong framework: phase space instead of the cotangent space of the symplectic manifold. First I think I should consider symplectomorphisms between symplectic manifolds (which automatically preserve probability) as the analogy I desire to relate to unitary transformations.

Then maybe I should consider relaxing the bijective condition of the diffeomorphism (while retaining probability preservation) as the analogy to Lindblad-GKS. I need to think about this much more.

 

[C. H. Fleming]

I have thought about this some more. Unitary maps are analogous to Symplectomorphic maps. Neither groups are simply bijections, but they are moreover isomorphisms. There are bijections between state vectors in Hilbert space which are not unitary, but they do not preserve the state overlap.

However, for dissipative classical mechanics, I believe you have steered me towards the correct path. One should consider stochastic matrices which act as the positive maps between state vectors, though translating that into the context of phase space, which would naturally involve some functional analysis, seems like it might be tricky. I will think about that some more.

 

[A. Neumaier]

I have thought about this some more. Unitary maps are analogous to Symplectomorphic maps.

But this is the wrong analogy. The classical symplectic structure is tied to the existence of conjugate observables satisfying canonical Poisson bracket relations. Thus any analogous quantum system must have corresponding conjugate observables satisfying CCR. Thus associated to any real Hilbert space V, one can associate canonically a symplectic phase space V x V with symplectic form omega(p,q) = p^*q-q^*p and a Hilbert space L^2(V); V=R^3 gives the 1-particle case. In this _particular_ situation, Unitary maps are analogous to Symplectomorphic maps.

But if your Hilbert space is C^n, there are no conjugate operators, and your analogy breaks down. The corresponding classical ''phase space'' (if one may call it that) is a discrete probability space with n elementary events.

 

[homology]

I hope its not too out of line to pop back in with something lower key. Its been great following the material you guys have been writing. I was hoping to put some more dots together with the calculation.

I was thinking about the flow $$\phi:T^*M\to T^*M$$ obtained from the vector field X (given in the original post). It would induce a pullback $$\phi^*:T^*(T^*M)\to T^*(T^*M)$$ and so if $$\omega=dp\wedge dq$$ and if, at a point we have a couple tangent vectors $$\vec{u},\vec{v}\in T(T^*M)$$ which span an area A then we could write:

$$(\phi^*\omega)(\vec{u},\vec{v})=\omega(\phi_*\vec{u},\phi_*\vec{v})$$

We also know that the Lie derivative of $$\omega$$ along X is given by:

$$L_X\omega=-\gamma \omega$$

I feel close and a little dumb at the moment. The Lie derivative is clearly giving us some measure of the time evolution of $$\omega$$. Just as the pullback is pushing the area, spanned by the tangent vectors, along the vector field, 'forward in time' Is it correct to say that

$$L_X\omega=\dot{\omega}$$

If so we'd get the expression $$\omega(t)=e^{-\gamma t}dp\wedge dq$$ then,

$$A(t+\Delta t)-A(t)=A(t)(e^{-\gamma\Delta t}-1)$$

divide both sides by $$\Delta t$$,


$$\frac{A(t+\Delta t)-A(t)}{\Delta t}=A(t)\frac{(e^{-\gamma\Delta t}-1)}{\Delta t}$$

In the limit $$\Delta t\to 0$$, the quotient on the right goes to $$-\gamma$$ so we end up with,

$$\frac{dA}{dt}=-\gamma A$$ which results in what we'd expect.

My goal is to do this without having to solve for the flow (which would involve the usual elliptic integral mish mash).

 

[A. Neumaier]

if $$\omega=dp\wedge dq$$ [...]
Is it correct to say that
$$L_X\omega=\dot{\omega}$$

Probably not. This doesn't follow from what you assumed so far, hence would be a new assumption.

It is not clear at all what you are doing and what you want to achieve in posts #1 and #26. If you work on th level of forms, you can't treat omega as time-dependent - neither are p and q. Omega just served to define the Poisson bracket, and then only a Hamiltonian (which you didn't specify at all) would define a a dynamics for p(t) and q(t) - which has almost nothing to do with the p and q of the forms.

But the standard recipe then gives a conservative system while your system isn't conservative. So you don't have a Hamiltonian.

Why do you want to force your system into a symplectic framework? If you want to do so, you first need to generalize the conservative dynamical equation fdot= {f,H} so that it has a chance to describe your system.

 

[homology]

True, sorry about the omega definition, that was sloppy, perhaps:$$\omega=w(t)dp\wedge dq$$ where $$w(0)=1$$?

I want to see how area/volume of phase space evolves for a dissipative system. Certainly there is a two form $$\omega=w(t)dp\wedge dq$$ though its not going to be the usual 2-form. I don't see why I need a Hamiltonian, there isn't one for this sytem, however we should still be able to talk about the area spanned by a set of tangent vectors. That area is going to change over time and so the area 2-form should be time dependent, no?

 

[A. Neumaier]

True, sorry about the omega definition, that was sloppy, perhaps:$$\omega=w(t)dp\wedge dq$$ where $$w(0)=1$$?

I want to see how area/volume of phase space evolves for a dissipative system. Certainly there is a two form $$\omega=w(t)dp\wedge dq$$ though its not going to be the usual 2-form. I don't see why I need a Hamiltonian, there isn't one for this sytem, however we should still be able to talk about the area spanned by a set of tangent vectors. That area is going to change over time and so the area 2-form should be time dependent, no?

No. Your omega is not the area between two tangent vectors in phase space. z= (p,q) denotes a single curve in phase space following your dissipative equation. It has only a single tangent vector at each particular time.

What you are looking for is how, given a set Omega in phase space, the volume (=area) of the set of all points z(t) with z(0) in Omega changes with time.

If your equation is dz/dt=F(z), the infinitesimal volume change factor is the trace of F'(z). So an infinitesimal volume A close to a trajectory z changes according to the differential equation dA/dt = tr F'(z) A

 

[homology]

(1) only a single tangent vector? Why aren't the tangent spaces 2D? I also don't see why omega is no longer the area when it was before. Or perhaps its more accurate to say I don't see why the absence or presence of a Hamiltonian changes the interpretation of omega in terms of area/volume.

(2) So then for my system:

$$<br /> \frac{d}{dt}\begin{pmatrix} p\\ q \end{pmatrix} =\begin{pmatrix} -\gamma & ??\\ 1/m & 0\end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix}<br />$$

The system is nonlinear, how should I represent F?

When you say F', what is it with respect to?

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.

Cheers (and thanks!)