The n-Category Café

Do you really want Kaehler? It looks like you don’t need integrability of the complex structure, so almost Kaehler would do just as well if not better.

Overall my wild guess is that you’re looking for Courant algebroids, or something even more general than a Courant algebroid (not that I know what I am talking about here).

Eugene wrote:

Do you really want Kaehler? It looks like you don’t need integrability of the complex structure, so almost Kaehler would do just as well if not better.

Good point! I’ll be happy for the answer to any question along these lines:

$$symplectic: Poisson :: Kaehler : ???$$

or

$$symplectic: Poisson :: almost Kaehler : almost ???$$

And when I finally get an answer, I’ll be eager to know the physical difference between dissipative systems described by ‘??? manifolds’ and those described by ‘almost ??? manifolds’.

My student Chris Rogers is becoming an expert on Courant algebroids, so I should ask him about your guess…

The ‘dissipative bracket’:

dF/dt={H,F}+[S,F]

So how would this work for a simple example, like the damped harmonic oscillator?

Gerard

Gerard wrote:

So how would this work for a simple example, like the damped harmonic oscillator?

Are you trying to ask me how Beris and Edwards’ equation

$$d O/d t = \{H,O\} + [H,O]$$

works for the damped harmonic oscillator? You actually asked me how Öttinger’s equation

$$d O/d t = \{H,O\} + [S,O]$$

works for the damped harmonic oscillator.

Öttinger’s formalism won’t work for the damped harmonic oscillator if the only degrees of freedom we use are the position and momentum of the oscillator, $q$ and $p$. Why? Because energy is conserved in Öttinger’s formalism! He demands

$$[S,H] = 0$$

so that

$$d H/d t = \{H,H\} + [S,H] = 0$$

Since energy is not conserved for the damped harmonic oscillator, we’re stuck.

In Beris and Edwards’ formalism, you just take the usual Hamiltonian for the harmonic oscillator

$$H = \frac{1}{2} (p^2 + q^2)$$

and the usual Poisson bracket

$$\{F,G\} = \frac{\partial F}{\partial p} \frac{\partial G}{\partial q} - \frac{\partial F}{\partial q} \frac{\partial G}{\partial p}$$

but then define a dissipative bracket

$$[F,G] = -k \frac{\partial F}{\partial p} \frac{\partial G}{\partial p}$$

where $k$ is the damping constant.

I’m annoyed that Öttinger’s formalism doesn’t seem to apply to the damped harmonic oscillator. But I can imagine three ways around this.

First of all, switch to Beris and Edwards’ formalism.

Second of all, explicitly couple the harmonic oscillator to a ‘heat bath’, which has extra degrees of freedom, and make friction transfer energy from the harmonic oscillator to the heat bath. I don’t actually know if this works, but I know an analogous trick works for the quantum version of the damped harmonic oscillator — it’s called the Caldeira–Leggett model. In this model, the heat bath is treated as an infinite collection of harmonic oscillators. If you follow the link, you’ll see people have considered the classical limit of the Caldeira–Leggett model. But it would also be nice to formulate a purely classical version of this model, if possible.

Third, drop the assumption that

$$[S,H] = 0$$

since this seems to be a removable portion of Öttinger’s formalism. I recently noticed that if we do this, Öttinger’s formalism actually includes Beris and Edwards’ formalism as a special case! We can take Öttinger’s formalism and choose $S = H$. Then Öttinger’s evolution equation

$$d O/d t = \{H,O\} + [S,O]$$

reduces to

$$d O/d t = \{H,O\} + [H,O]$$

which is Beris and Edwards’ equation. And then with a suitable choice of the dissipative bracket $[\cdot, \cdot]$ we can treat the damped harmonic oscillator.

However, this raises other questions. It seems physically absurd to take $S = H$, since entropy and energy have different units. I would like to get the free energy

$$F = H - T S$$

into the game, where $T$ is an adjustable constant (not an observable) called the temperature. So, I would like to replace Öttinger’s evolution equation

$$d O/d t = \{H,O\} + [S,O]$$

(where $O$ is any observable) by something like this:

$$d O/d t = \{H,O\} - T [S,O]$$

This is easy to do by rescaling the dissipative bracket by a factor of $-1/T$. If I use Öttinger’s strong assumption

$$\{S, O \} = [H, O] = 0$$

for all observables $O$, then I can rewrite Öttinger’s evolution equation as follows:

$$d O / d t = \{F, O\} + [F , O]$$

Now time evolution is generated by a single observable — not the energy, but the free energy! I like this idea.

AN: But things are more complex than that! It is a very long story, distributed in small bits and pieces in the literature. It has taken me a long time to figure out how things are connected.

JB: Are you going to tell the world? A paper that explained this stuff well could be very important.

I have several partial papers on this but nothing finished. It is really a complex story, and I haven’t found so far the time to give a publishable account of it. It is related to everything dissipative discussed in physics, from the anharmonic oscillator to damped mechanical structures and nonlinear elasticity to electric circuits to the Navier-Stokes equations to the Boltzmann equation to resonances in quantum systems to retarded solutions in QED, and to deeper levels, probably even to quantum gravity. On the other hand, it is related to existence problems in symmetric hyperbolic systems, to the ergodic hypothesis, to photons on demands, and lots of other stuff. The stuff is everywhere, and a good presentation is not much short of a theory of everything – not the pale shadow string theory is after, which has difficulties being a theory of anything at all, but a theory that connects lots of different applications under a single umbrella.

JB: Do you know what’s going on here? I don’t want a lot of different formalisms…

The multiple and slightly incompatible versions are proof that the field still has some research potential for theoreticians wanting to bring clarity into the picture. There are many schools of nonequilibrium thermodynamics, and they all have their own version of the theory. I just pointed you to some of the best stuff. There is more: For example, you should also read

Mueller and Ruggeri, Rational extended thermodynamics, Springer 1998

who proceed without any Poisson bracket, but use variational principles - still another very useful side of the same coin. Neither theory is the final word, but each contains some aspect of the truth. As I said, the real thing is a complex story, and it takes time and effort to see the connections and implications.

Getting an account that fits everything in a uniform way and makes the connections transparent is a big job that will fill a book (maybe a second volume of my book ”Classical and Quantum Mechanics via Lie algebras”, which apart from a few general remarks only treats equilibrium and the conservative case). Worse, in trying to do it well I stumble over lots of unsolved or poorly solved side issues. Neither Oettinger nor Beris and Edwards have the best formulation of the theory - many things are there too ad hoc. The correct formulation, which provides the full structure, is via dynamics in Lie algebras, but Oettinger is almost silent about this aspects, while Beris and Edwards don’t see the trees in their forest. (Almost all their Poisson brackets are in fact Lie-Poisson algebras. I treat these from an abstract point of view in my book, but had no space for the dynamical implications.)

At the moment I am trying to make sense from this perspective of Kadanoff-Baym equations (kinetic equations generalizing the Boltzmann equation to particles with a finite life time). Getting a match in each application takes some time, but then it shows the application from a new, useful perspective.

Maybe I can write about it in a more informal manner; perhaps I’ll try in July, after our summer term is over. But should you have time for a visit in Vienna, I could show you many things – much quicker than when having to type it into a keyboard!

JB: If someone — for example, you —- has already done it, I’d rather just know what they did.

Well, I have a full time job as a math professor, and have to do do all the physics in my spare time; so I can’t give long explanations unless I have them ready in my head and few formulas are sufficient to tell the story. This is not the case here. So until I have more time (which means not before July), I can only make superficial comments.

Your trick with the free energy goes in the right direction. It is not quite right since people want to study temperature dependence in a thermodynamically consistent way, which is lost the way you proceeded. What is ”right” is determined by lots of ”boundary conditions” that I collected over the years – one must recover all the major forms of dissipative systems that have been studied by physicists. If one can, then one knows one is right.

JB: What do you mean by ‘Lie–Poisson algebra’? Some people use that to mean the Poisson algebra of functions on the dual of a Lie algebra.

Yes, almost. The notion is used in the literature (e.g., the mechanics book by Marsden and Ratiu) in the sense you describe. But in Section 9.5 of my book

A. Neumaier and D. Westra, Classical and Quantum Mechanics via Lie algebras. arXiv:0810.1019

I consider a slight generalization, which identifies a distinguished central element 1 of the Lie algebra with the constant function 1, which is needed in many physical applications. This identification is not made in the standard construction, but has the advantage of adding flexibility. For example, in this way, the standard Poisson bracket on phase space functions f(p,q) becomes the Lie-Poisson algebra over the Heisenberg Lie algebra. This is part of my unification program. One gets all 2-cocycle stuff for free (well, at least hidden under the carpet).

JB: is there a more or less canonical way of equipping it with a symmetric ‘dissipative bracket’ as well?

No. And there shouldn’t. The dissipative bracket is far from unique. The reason is that dissipation is due to interaction with an unmodelled environment (which may be an external environment or neglected internal degrees of freedom, or both). Therefore the details depend on how the system is embedded into the unmodelled part.

In the examples, the dissipative part depends on phenomenological ”constants” (actually thermodynamic functions) such as transport coefficients, diffusion constants, and the like. The dissipative bracket can become extremely complicated (e.g., for combustion processes), while the conservative Poisson bracket nearly always comes from a fairly nice Lie-Poisson algebra (in my generalized sense) – although the underlying Lie algebra can also become a bit complex if a really complex physical system is studied. I have met only a single example of a dissipative system with a Poisson bracket that was not Lie-Poisson, and I suspect this arises due to poor modeling rather than something of importance.

JB: Something about your phrase ‘dynamics in Lie algebras’ reminds me of this paper by Bloch, Krishnaprasad and Marsden.

Yes, there is a connection. The double bracket dynamics is one of the standard ways dissipation arises in systems with a Lie structure, actually the simplest instance. There is a significant literature on double bracket flow that you can easily retrieve using http://scholar.google.com

Variants of the double bracket flow arise in similarity renormalization (perhaps the most elegant way to renormalize quantum fields numerically); see, e.g.,

T. S. Walhout, Similarity renormalization, Hamiltonian flow equations, and Dyson’s intermediate representation, Phys. Rev. D 59, 065009 (1999)

Double bracket terms are also characteristic of dissipative Lindblad terms in quantum optics, where they govern the losses of virtually all quantum optical dynamical systems. A glimpse on how it relates to my unification vision is given in Section 7.2 of my book. Actually, in the quantum case, (7.4) is the most general linear dissipative dynamics of interest! The classical case allows a bit more variety, essentially capturing all Markov processes and their deterministic approximations. In both cases, the interesting dynamics often has the double bracket form given in (3.9) on p.63, and for good reasons – not spelled out there, though. However, the two references given there include the book by Breuer and Petruccione that I had recommended before!

Let me also remind you again that Section 7.1 of my book contains a table that is similar to your earlier tables of analogies between mechanics and thermodynamics! Moreover, these are not just analogies, but special cases of a unified picture.

Thus there are good reasons to emphasize the Lie algebra aspect of everything. Unfortunately (but for you perhaps fortunately), many of the Lie algebras of interest are infinite-dimensional, and the mathematical structure of the corresponding Lie groups is highly nontrivial and full of topological pitfalls.

JB: my student Chris Rogers will be visiting the Erwin Schrödinger Institut during September and October, for the workshop on Higher Structures in Mathematics and Physics. He knows a lot about symplectic and Poisson geometry and their higher analogues… but he’s less passionate about dissipative systems than I am.

He is welcome, too. He will be well prepared if he has read my book. And perhaps he catches the virus while reading it!

JB: I wonder if I can get him to absorb and then emit the desired information?

Maybe you can talk him into writing a set of Latex lecture notes about what I can tell him. This would motivate me to present things coherently and in some detail – in September I’d have fairly much time to prepare some lectures on the topic. Then you and others could read them afterwards.

I find it MUCH more useful if references included titles! why did physicists ever adopt the ‘no title needed’ choice?

Full reference for you Jim:

• Robert Jongschaap and Hans Christian Öttinger, The mathematical representation of driven thermodynamic systems, Journal of Non-Newtonian Fluid Mechanics Volume 120, Issues 1-3, 1 July 2004, Pages 3-9. [Free pdf.]

What you say about stochastic non-Markovian equations is a rather extended opinion, but it is not right. In any case, the main question here really was that the GENERIC equation is built over several approximations, whereas other formalisms cited are more general.

Dynamics where S(t) does not increase monotonically are fundamental. It is only for large times over the scale (tau_C), that molecular details of the real initial state are ‘erased’ by the destruction fragment of the evolutor and one recovers the usual exponential relaxation which gives the law dS/dt >= 0.

There is some well-known effects that arise when people ignore all this. Balescu reports the early times of kinetic theory, when people took the Markovian equation obtained by Boltzman as fundamental and tried to seek for generalized Boltzmann equations applied to arbitrary densities in the supposition that relaxation was exponential and that dS/dt>=0 always hold. How wrong they were!

Some people claim to derive irreversible equations from reversible ones. As said by van Kampen “Obviously it is a logical impossibility to deduce from reversible equations an irreversible consequence” and emphasized that “One cannot escape from this fact by any amount of mathematical funambulism.”

I can understand that you dislike the label funambulism [1], but you just take another of his statements as “is a logical impossibility” or “your derivation cannot be right” instead doing ad hominem as your last:

“This should make one a bit cautious about their contributions.”

Sorry to remark here the obvious, but Nico van Kampen contributions to science and education are well-known. As D ter Haar has said he is one “of the most outstanding theoretical physicists of the second half of the 20th century” [2].

NOTES:

[1] However, it correctly labels the so called ‘derivations’.

[2] “Nico van Kampen: charlatans beware!” in Physics World, January 2001, p.46

My correct Spanish name is Juan R. González-Álvarez, but when posting it in the Name field gives an XML validation error “Line 10, column 131: non SGML character number 129”.

In the raw code, the error is associated to my name inside the class “comments-post”. I can see my name therein with strange invalid characters.

Note that the same characters give no error in the message body, only in the above class output from parsing the name field!