Suppose you have a dissipative system where [tex] \dot{q}=p/m [/tex] [tex] \dot{p}=-\gamma p -k\sin(q) [/tex] 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 [tex] \omega = dp\wedge dq [/tex] along it. If I do that I get [tex] L_X \omega = -\gamma dp\wedge dq [/tex] (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 [tex] \omega [/tex] as a function of time, but I'm stuck. Any ideas are welcome.
For this simple dissipation your Liouville operator is: [tex]\boldsymbol{\mathcal{L}} = -\frac{p}{m} \frac{\partial}{\partial q} +k \sin(q) \frac{\partial}{\partial p} + \gamma \frac{\partial}{\partial p} p[/tex] and so in addition to the dissipative force [tex]\gamma p \frac{\partial}{\partial p}[/tex], you have the probability preserving term [tex]\gamma[/tex]. 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.
Okay, well your Louiville operator corresponds to vector field I'm differentiating [tex]\omega[/tex] along. I think I understand your objection to possibly using [tex]\omega[/tex] 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?
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 [tex]\gamma[/tex] 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 [tex]L_X \omega = -\gamma \omega[/tex], which looks like exponential decay.
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?
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.
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.
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.
What about the entry ''Dissipative dynamics and Lagrangians'' of Chapter A1 of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/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 ([tex]-\imath \mathbf{H}[/tex]) whereas dissipative evolution has completely-positive evolution with Lindblad-GKS generators ([tex]\mathcal{L}[/tex]). (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.
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.
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 [tex]x[/tex], [tex]\frac{\partial}{\partial x}[/tex], ...". 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.
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? 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.
Consider the dynamical map: [tex]\boldsymbol{\rho}(t) = \boldsymbol{\mathcal{G}}(t,0) \boldsymbol{\rho}(0)[/tex] where [tex]\boldsymbol{\rho}(t)[/tex] is the density matrix of the reduced system at time [tex]t[/tex] and we have a non-unitary theory (likely with a traced out environment) such that we can consider any initial state [tex]\boldsymbol{\rho}(0)[/tex] and also we might later include ancillary degrees of freedom (e.g. external entanglement). Then with very few assumptions, [tex]\boldsymbol{\mathcal{G}}(t,0)[/tex] 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.
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.
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.
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 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.
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. 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 [tex]\mathcal{G}[/tex] between density functions [tex]\rho[/tex] on [tex]2n[/tex]-dimensional phase-space with coordinates [tex]z[/tex], and parametrized by time: [tex]\rho(z,t) = \mathcal{G}(t,0) \rho(z,0)[/tex] Then say you add an additional [tex]2m[/tex] degrees of freedom to phase space and consider the trivially extended maps [tex]\mathcal{G}(t,0) \otimes 1[/tex] between arbitrary density functions in the [tex]2(n+m)[/tex]-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.
Yes. 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.