In week289 of This Week’s Finds, hear the latest news about . Then, continue exploring the grand analogy between different kinds of physics. We’ll get into a bit of thermodynamics — and chemistry, too! Finally, learn more about rational homotopy theory, this time entering the world of “differential graded Lie algebras”, which lets us use Lie theory to study topological spaces.
And guess what’s going on here:
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I’m suspicious of any attempt to make economics seem like physics. Unlike elementary particles or rocks, people don’t seem to be very well modelled by simple differential equations. However, some economists have used the above analogy to model economic systems. And I can’t help but find that interesting - even if intellectually dubious when taken too seriously.
Differential equations for particles and rocks are themselves somewhat intellectually dubious when taken too seriously, but at a higher level of mathematical sophistication :)
Having said that, I also have fun occasionally relating physics and finance (or phynance):
I’m suspicious of any attempt to make economics seem like physics.
Perhaps then More heat than light: economics as social physics, physics as nature’s economics by Philip Mirowski is for you:
More Heat Than Light is a history of how physics has drawn some inspiration from economics and also how economics has sought to emulate physics, especially with regard to the theory of value. It traces the development of the energy concept in Western physics and its subsequent effect upon the invention and promulgation of neoclassical economics. Any discussion of the standing of economics as a science must include the historical symbiosis between the two disciplines. Starting with the philosopher Emile Meyerson’s discussion of the relationship between notions of invariance and causality in the history of science, the book surveys the history of conservation principles in the Western discussion of motion. Recourse to the metaphors of the economy are frequent in physics, and the concepts of value, motion, and body reinforced each other throughout the development of both disciplines, especially with regard to practices of mathematical formalisation. However, in economics subsequent misuse of conservation principles led to serious blunders in the mathematical formalisation of economic theory. The book attempts to provide the reader with sufficient background in the history of physics in order to appreciate its theses. The discussion is technically detailed and complex, and familiarity with calculus is required.
John A. wrote:
It seems to me that the difference isn’t so much that people are so much more complicated as that the simplifying assumptions become significant so much sooner.
Good point. I’ve tried to improve the wording a bit.
Hi John,
I remember seeing in some book what you wanted to know about that lattice with e8 symmetry. I am still looking for it, but I guess these articles should be interesting for the subject. I googled for “E8” + “Ising” :
Paul A. Pearce, PHASE TRANSITIONS, CRITICAL PHENOMENA AND EXACTLY SOLVABLE LATTICE MODELS.
V. BAZHANOV, B. NIENHUIS AND S. O. WARNAAR, LATTICE ISING MODEL IN A FIELD: E8 SCATTERING THEORY, this one cites Zamolodchikov.
Perhaps the book you’re thinking of is G. Mussardo, Statistical Field Theory, Oxford, 2010?
2010? Hmmm, I don’t think that is the one. It is older. But take a look at this.
The A-D-E diagrams show up over and over again in conformal field theory and 2-D statistical mechanics. The critical Ising model is (A2 , A3) in the A-D-E classification described in the Scholarpedia article. That E8 shows up when the critical model is perturbed away from criticality is, I believe, an extra surprise.
I found it!
Philippe Di Francesco,Pierre Mathieu,David Sénéchal,”Conformal Field Theory”,Springer, 1997. The stuff around page 814.
If that still doesn’t help, I will use peg-leg magic to make the article appear.
The paper by Pearce is a very pleasant introduction to critical points in statistical mechanics and how an ADE classification shows up in this subject.
The paper by Bazhanov et al seems more relevant. It mentions the work of Zamolodchikov — precisely the work that I’d like to understand. It helps to know that the ferromagnetic system being studied is known mathematically as an ‘Ising model’. The paper begins:
Since the work by A.B. Zamolodchikov [1] it is known that certain perturbations of conformal field theories (CFT’s) lead to completely integrable models of massive quantum field theory (QFT). The existence of non-trivial higher integrals of motion and other dynamical symmetries [2-6] in such a QFT allows to compute the spectrum of the particles and their -matrix explicitly. At the same time, these QFT models can be obtained as the scaling limit of appropriate non-critical solvable lattice models in statistical mechanics (see [7] for an introduction and references on solvable lattice models). In the latter approach the spectrum and the S-matrices can be calculated from the Bethe Ansatz equations for the corresponding lattice model [8–10]. The natural problem arising in this connection is to find lattice counterparts for all known integrable perturbed CFT’s and vice versa. A description of known results of such correspondence lies outside the scope of this letter and we refer the interested reader to [1-10] and references therein. Here we consider one particularly important example of this correspondence associated with the Ising model at its critical temperature in a magnetic field, hereafter referred to as the magnetic Ising model.
A.B. Zamolodchikov [11] has shown that the CFT (corresponding to the critical Ising model) perturbed with the spin operator of dimension describes an exactly integrable QFT containing eight massive particles with a reflectionless factorised -matrix. Up to normalisation the masses of these particles coincide with the components of the Perron-Frobenius vector of the Cartan matrix of the Lie algebra E8.
[…]
The aim of this letter is to show that the above QFT describes the scaling limit of the dilute A3 model of Warnaar, Nienhuis and Seaton [12,13] in the appropriate regime.
I would love to understand how the magnetic Ising model gets to have 8 massive particles whose masses are related to the Cartan matrix — that’s what the new experiment is studying! But the above paper goes on to talk about the ‘dilute model’, whatever that is, rather than explaining Zamolodchikov’s work in more detail.
The paper by Pearce hints at the relation between the dilute model and , because in section 4.2 it says that the Rogers-Ramanujan identity can be proved by studying the dilute model!
Even the ‘ordinary’ Rogers-Ramanujan identities are pretty terrifying. The version, which you can see on page 10 of Pearce’s paper, is even scarier.
Clearly there’s quite a big web of deep mathematics at work here, and I probably don’t have the energy to penetrate it — at least not very quickly! But I’d love to hear from anyone who understands anything about this.
The following review article, as well as the book mentioned above, describe Zamolodchikov’s work:
Giuseppe Mussardo, Off-critical statistical models: Factorized scattering theories and bootstrap program. Physics Reports 218 (1992). Pages 215-379. Abstract.
An expansion on your remarks concerning the experimental situation: The E8 structure is not seen when the transverse magnetic field is scanned through its critical value - one must fix the transverse field at its critical value, and turn on a longitudinal field as well. The transverse-field-only situation is shown in Figure 2e of the Science paper, where only a single peak is seen. What happens when the longitudinal field is added is shown in figure 4, where the first two of the eight particles show up as peaks with the correct mass ratio.
The transverse magnetic field in the spin chain model corresponds to the temperature variable in the Ising model; the longitudinal field in the spin chain model corresponds to the external magnetic field in the Ising model. The critical point of the Ising model occurs when T=Tc, H=0, and corresponds to the M(3,4) minimal model of conformal field theory. It turns out that there are two integrable massive perturbations of this theory: One corresponds to moving T away from Tc; the other corresponds to turning on an external magnetic field.
That the magnetic perturbation results in an E8 structure is something that I believe is still not fully understood. The bootstrap approach described in the Mussardo article is one way to see how it emerges. It is presumably is also related to the fact M(3,4) can be obtained from a coset construction involving E8.
Finally, it should be noted that the integrable models referred to above are continuum models, related to the scaling theory of the 2-D Ising model. Neither the Ising model on the lattice nor the XX spin chain with a transverse field remains integrable when the longitudinal field is turned on. The dilute A3 model is an integrable lattice model in the same universality class as the Ising model, but which does contain a field that breaks the up-down symmetry of the spins.
Will: thanks a lot for your corrections and explanations! I’ve massively rewritten week289. Do you see more mistakes?
One basic thing I want to check: when you speak of a ‘longitudinal’ magnetic field, do you mean one pointing along the same axis the spins like to point along before any magnetic field at all is applied? I think so, from my reading of the Coldea paper.
As you can see, I’m engaged in one of my favorite hobbies: learning stuff by making mistakes in public and letting experts correct them!
John,
Looks good to me! In regard to your question about longitudinal field, it is indeed oriented along the direction in which the spins prefer to point before to the application of the transverse field.
The spin chain with only a transverse field has Hamiltonian . The critical value of the transverse field is . The model with structure, , is obtained by fixing and adding a small longitudinal field, which means adding the term to .
The unperturbed Hamiltonian defines an integrable model, in the sense that there is an infinite set of operators that commute with . The connection with the 2-D zero-field Ising model solved by Onsager is that commutes with the transfer matrices of that model.
Unfortunately, the perturbed Hamiltonian, is not integrable in the same sense. The perturbed conformal field theory to which it corresponds is, however, integrable. This means that the symmetry manifests itself only approximately in the lattice model, whereas it is exact in the perturbed conformal field theory. Luckily, there is a different lattice model, the dilute model, in which the symmetry is exact.
Hi John.
I haven’t had time to read your new post carefully, which I look forward to doing, but I did notice one mistake that I think I should mention right away: the normalized chains on a topological group is NOT a cocommutative dg Hopf algebra. Even with rational coefficients, the cocommutativity holds only up to homotopy, actually up to an infinite family of homotopies: the normalized chain complex of any space admits an E_\infty-coalgebra structure. This is the source of the Steenrod algebra action on the cohomology of the space.
One can also say that the comultiplication on the normalized chain complex is a strongly homotopy comultiplicative map, i.e., a morphism of chain coalgebras up to strong homotopy. This is somewhat less strong that saying that there is a full E_\infty-coalgebra structure, but is often sufficient for building interesting algebraic models of topological spaces.
I’ve enjoyed your introduction to rational homotopy theory!
Oh, right. I should have realized this. I’ll read some more and fix this part.
Thanks!
Back from Saturday shopping now, so I’ve had time to read more carefully what you wrote and to think a bit more.
The key references to understanding when the chains on a topological group are weakly equivalent to the universal enveloping algebra of a dg Lie algebra are:
–David Anick’s article on Hopf algebras up to homotopy, which was in the Journal of the AMS in 1989;
–Steve Halperin’s article on universal enveloping algebras in JPAA 83 (1992);
–Jonathan Scott’s article on Hopf algebras up to homotopy in AGT 5 (2005).
I’m wondering if something like this works:
Problem: The rational chains on a topological space form a dg coalgebra that’s not cocommutative — it’s only an -coalgebra.
Sullivan was able to fix the corresponding problem for rational cochains:
Problem: The rational cochains on a topological space form a dg algebra that’s not commutative — it’s only an -algebra.
Namely, he found a commutative dg algebra of ‘rational differential forms’ for a space , which is presumably equivalent to the -algebra of rational cochains on .
So, can we dualize and get a dg coalgebra that’s equivalent to the -coalgebra of rational chains on ?
Here we should heed the whispered warnings of the ancestors: never dualize unless you absolutely need to.
The dual of an infinite-dimensional Hopf algebra isn’t usually a Hopf algebra. is infinite-dimensional, so we probably don’t want to take its dual. But there’s a standard solution: the restricted dual of a Hopf algebra is again a Hopf algebra. The idea is to use the largest subspace of such that
$$m^* : A(X)^* \to (A(X) \otimes A(X))^*A(X)^* \otimes A(X)^*A(X)^\circXE_\infty$-coalgebras? Or does some other trick like this work?
I don’t know if the trick you suggest of considering the restricted dual of A(X) would work. I’m a bit skeptical, but that may just be my indoctrination by John Moore speaking.
If you want to see a clear and complete explanation of the relationship between the Sullivan and Quillen models in rational homotopy theory, I strongly suggest you take a look at “Rational homotopy models and uniqueness” by Martin Majewski, which was published as an AMS Memoir in 2000. His argument proceeds (very) roughly as follows.
Let X be a 1-connected space (resp., a 1-reduced simplicial set). By Anick’s theorem (from the paper I cited in my earlier comment), the rational cubical chain complex of the Moore loops on X (resp., the normalized rational chain complex of the Kan loop group of X) can be rigidified to a strictly cocommutative dg Hopf algebra, of which one can take the primitives. Majewski proves that the dg Lie algebra thus obtained is weakly equivalent both to Quillen’s dg Lie algebra model of X and to the dg Lie algebra canonically associated to Sullivan’s minimal model of X. (There’s a formula in Halperin’s paper, for example, for this associated dg Lie algebra.)
The key concept operating here is Koszul duality between the commutative operad and Lie operad, over the rationals…
Thanks again, Kathryn. I think an argument roughly like the one you sketch may also be lurking somewhere in the Félix–Halperin–Thomas book. For example, in Section 26 they say (I’ll paraphrase it):
The main objective of this section is to show that the isomorphism (26.1) is induced from a chain algebra quasi-isomorphism between a free Lie model for a rational homotopy type and (that is, the rational chains on the based loop space of ). This is a result of Majewski. Its significance is due to a theorem of Anick, who should directly the existence of a unique quasi-isomorphism class of free chain Lie algebras admitting such a quasi-isomorphism. However, these were potentially different from the Lie models constructed via Sullivan’s functor as described here. Majewski’s result shows that they coincide.
I’m willing to learn all this stuff, but it seems a bit elaborate and hard to explain, so I’m still hoping there’s a quick way to get ahold of cocommutative dg Hopf algebra that will substitute for .
If Sullivan could solve the ‘commutative cochain problem’ over the rationals, why can’t we solve the ‘commutative chain problem’ in an equally elegant way?
You’re right, John, that FHT are referring to the same result by Majewski that I sketched above.
The only direct–and, in my opinion, very elegant– solution of the “cocommutative chain problem” of which I am aware is in Quillen’s landmark 1969 Annals paper on rational homotopy theory. There he considers a sequence of six (!) pairs of adjoint functors that link the category of 1-connected spaces (where weak equivalences are rational homotopy equivalences) to the category of 1-connected, cocommutative rational dg coalgebras and shows that all of the pairs are Quillen equivalences.
When I was a grad student, this sequence of Quillen equivalences looked very intimidating, but I realize now that taken one by one, they’re not so bad. Given a 1-connected space X, you start by taking its singular simplicial set
S(X)
and throwing away all the simplices except the basepoint in degrees 0 and 1. You then apply the Kan loop group functor (the simplicial analogue of the based loop space functor) to S(X), obtaing an honest simplicial group
GS(X).
The next step is still somewhat mysterious to me: you take the group ring
Q[GS(X)]
and complete it with respect to powers of its augmentation ideal, obtaining a “reduced, complete simplicial Hopf algebra”,
\hat Q[GS(X)],
which happens to be cocommutative, since the group ring is cocommutative. Taking degreewise primitives, you then get a reduced simplicial Lie algebra
Prim(\hat Q[GS(X)]).
We’re getting close now! At the next stage, we finally apply the normalized chains functor, to get Quillen’s dg Lie model of X:
N(Prim(\hat Q[GS(X)])).
(So it seems that the key is to wait until the very end of the process to pass from the simplicial world to the chain world…)
Finally, to get a a cocommutative dg coalgebra model for X, we use a slight generalization of a functor first defined by Koszul for computing the homology of a Lie algebra, which always gives rise to a cocommutative dg coalgebra.
If Sullivan could solve the “commutative cochain problem” over the rationals, why can’t we solve the “commutative chain problem” in an equally elegant way?
Maybe he was well advised not to worry so much about
the whispered warnings of the ancestors: never dualize unless you absolutely need to.
Kathryn Hess kindly sketched the dual Quillen route. Thanks! I hadn’t been aware of that.
Despite the claim that this is very elegant, I get away with the impression that this route loses the conceptual transparency of what’s going on. Taken at face value, it looks like black magic.
John’s general strategy in the TWF is to go to dg-Lie algebras, and is advertized as a massive generalization of Lie theory. But in fact dg-Lie algebras are the strictification of general -Lie algebras aka -algebras. My feeling is that this extra strictification demand makes the construction more involved than it naturally is.
We may think of (in non-negative degree) with its standard model structure as being already a presentation for : the fibrant-cofibrant objects are precisely the -algebroids in that the fibrant-cofibrant objects with the point in degree 0 are precisely the -algebras (at least if everything is degreewise finite, otherwise one has to say say this more carefully). From this perspective Sullivan’s construction is immediate and elegant in that it directly produces an object in there. The further steps of fibrantly replacing that object and then further finding strictified equivalents may be done if desired (for computations), but is not what the gods asked us to do.
Maybe I’m missing the point of contention, but I think Quillen’s homological construction is completely intuitive: we are simply taking the Lie algebra of the “group” which is the loop space of X. In other words, starting from a pointed space X we take the corresponding group . From a group we pass to the enveloping algebra, ie distributions supported at the identity, completed. The topological analog of distributions is chains (dual to functions=cochains), so Quillen’s completed chains construction is exactly the completed enveloping algebra. From the (completed) enveloping algebra we recover the Lie algebra as its primitive elements. In the -context this is all we’re doing - ie we internalize the identifications of simplicial sets and spaces and Dold-Kan. Again maybe this latter part is the point of the discussion, in which case sorry to barge in. But the overall scheme of Quillen’s argument, seen from a modern POV, is simply classical Lie theory (hence its brilliance!)
Maybe I’m missing the point of contention, but I think Quillen’s homological construction is completely intuitive:
Thanks, David, I see, that’s helpful. So the strictification is all in the first step, of course, where is realized as a simplicial group.
The topological analog of distributions is chains (dual to functions=cochains), so Quillen’s completed chains construction is exactly the completed enveloping algebra.
I see, thanks.
Urs wrote:
John’s general strategy in the TWF is to go to dg-Lie algebras, and is advertized as a massive generalization of Lie theory. But in fact dg-Lie algebras are the strictification of general -Lie algebras aka -algebras. My feeling is that this extra strictification demand makes the construction more involved than it naturally is.
Of course I will talk about -algebras in future weeks. But my plan here is to explain things gently and slowly. My goal is to get everyone in the world to understand this stuff. And I don’t think most people can grok -algebras until they’re pretty comfortable with dg Lie algebras. So, I wanted to give an easy explanation of why the loop group of a space has a dg Lie algebra.
Ultimately of course you’re right: it’s good to ‘accept weakness’ — not try to get equations to hold on the nose. Trying to always work with strict algebraic structures is constantly flirting with danger, and ultimately it’s quite stupid. I’ve spent years trying to convince everyone in the world that equations are evil. So it may seem inconsistent to take the opposite approach. But it’s a fun expository challenge to see how far one can go with strict structures! There will come a point at which it’s either impossible, or so inconvenient that everyone can see the need for weakness.
Indeed, this question makes a nice test case: what’s the most natural way to get a dg Lie algebra from a rational homotopy type? Or is it just a bad idea?
My original attempt, currently still uncorrected in week289, ran afoul of strictness issues. I wanted rational chains on a topological group to form a dg cocommutative Hopf algebra. For this, I needed a model of rational chains on a space that forms a cocommutative coalgebra. But the rational simplicial chains only form a coalgebra. I still hope there’s a dual version of Sullivan’s ‘rational differential forms’ construction that does the job. I even think I see roughly how it goes. If someone knows it can’t work, I’d love to know why! But if it does work and someone already knows how, I’d love that even more!
If this approach doesn’t work, the approach that Kathryn sketched here also sounds nice. As she points out, the trick here is to pass from the simplicial world to the chain complex world at the very last minute, after you’ve turned your cocommutative Hopf algebra object into a Lie algebra object. Hopf algebras involve both operations and co-operations. Lie algebras only involves operations. So, the normalized chains functor treats Lie algebra objects better. At least that’s my impression as to why this works.
I briefly collected some of the material about the Quillen model mentioned above at Lab:rational homotopy theory.
heat it up or cool it down “adiabatically” - that is, while keeping it in thermal equilibrium all along.
That’s not “adiabatically”, that’s “reversibly”! “Adiabatically” means “without heat flow across the boundary between the system and its environment”.
Fixed.
Candy Hansen writes: “The channels carved by the escaping gas are often radially organized and are known informally as “spiders.”
When I saw the picture I thought the Martian south pole was inhabited by giant creatures similar to basket sea stars.
John wrote:
Okay, back down to earth. Last week I began to sketch an analogy between various kinds of physical systems, based on general concepts of “displacement” and “momentum”, and their time derivatives, called “flow” and “effort”:
I was so busy writing about global energy that I almost missed this thread about my one time favorite subject.
Try my website on this subject.
Anyway, what are some other examples of physical systems where we have a notion of “effort” and a notion of “flow”, such that effort times flow equals power?
Here are two:
Thermodynamics: entropy // temperature
A disadvantage of this approach is that entropy flow is not a conserved current, unlike electric current. Electric circuit equivalents for heat conduction are quite useful in engineering, I use them a lot myself. Like you say:
There are also weaker analogies to subjects where effort times flow doesn’t have dimensions of power. The two most popular are these:
Heat Flow: heat flow// temperature
An alternative, in which we have entropy production as the generating functional:
Thermodynamics: heat flow// Negcitemp
where “Negcitemp” = -1/kT. “Negcitemp”, a word invented by Zemansky, is just a transformed temperature, that in many contexts is actually nicer than temperature.
The equations for electrical circuits can be derived for the “Principle of Least Dissipation”. This is a bit like the Principle of Least Action. You minimise the sum of all dissipations in the resistors, with the voltages as variables. This solves the circuit. If your circuit is using the Principle of Least Dissipation, then the pairs of quantities all have dissipation, or power, as their product.
You can rewrite things so that the circuit is defined in terms of the Principle of Least Action, by making what I call a “space time circuit”. I explain this on my website. In a space time circuit, the flow/effort variables become displacement/momentum variables. Resistances in the time direction become negative. The pairs of quantities are now the canonical conjugates.
Another subtlety:
If you refine your discretization, the Voltages generally remain the same: They are “0-chunks”. Voltage differences across edges are “1-chunks”. Currents are “[D-1] chunks”, so that the dissipation in the resistor is a D-chunk: It scales as a D-volume, where D is the dimension of space.
As you say, there is a huge amount if fun connected with this. Looking forward to it!
Gerard
Hi, Gerard! It’s nice to hear from you again.
I was so busy writing about global energy that I almost missed this thread about my one time favorite subject.
What are you writing about, and why are you doing it? I can’t tell if you mean ‘the problem of finding a globally well-defined notion energy’, which is a subtle issue in general relativity, or ‘the problem of getting enough energy for our civilization without destroying the globe’.
An alternative, in which we have entropy production as the generating functional:
Thermodynamics: heat flow// Negcitemp
where “Negcitemp” = -1/kT.
The concept of negcitemp seems to show up in that article about the Legendre transform that I mentioned. This is why I wrote “But it seems to be using a slightly different analogy than the one I was just explaining… so my confusion is not eliminated.”
What’s going on with these different heat flow analogies? Why are there so many — and how are they related?
The equations for electrical circuits can be derived for the “Principle of Least Dissipation”. This is a bit like the Principle of Least Action.
I definitely plan to talk about this in future Weeks!
If you refine your discretization, the Voltages generally remain the same: They are “0-chunks”. Voltage differences across edges are “1-chunks”. Currents are “[D-1] chunks”, so that the dissipation in the resistor is a D-chunk: It scales as a D-volume, where D is the dimension of space.
I think what you’re calling a -chunk is what mathematicians would call a -cochain, or possibly a -chain. (These are two different but closely related concepts.)
Indeed, some of the pictures on your website suggest that you share an interest in -cochains with Eric Forgy and Robert Kotiuga, whose work on discrete versions of electromagnetism can be strongly seen in week288 and the ensuing blog discussion!
And, I definitely plan to talk about this side of things as well!
About World energy:
I was writing this.
The reason I made this was that I wanted to check certain things for myself.
Thinking about dQ = TdS - pdV:
In this case, S’ seems like a flow, just like V’. But like I wrote, if S is not conserved, Kirchhoff’s current law doesn’t work, so circuits don’t either. But in the reversible case, it is different: S is conserved. In acoustics, temperature as well as pressure fluctuate together, dQ = TdS - pdV = 0 applies. However, they are coupled by a state equation, like pV=RT. The state equation is used to eliminate S and T, so that the dynamic equations end up using just p and V.
I can’t figure out yet if there is some general principle behind this.
Legendre transform and thermodynamics:
Still thinking…
Chain complexes:
I haven’t yet caught up with the terminology yet, (cup product, graded commutative etc.) but I’ll assume for the moment that D-chunks are D-cochains.
I think an example of Eric’s “Cochain problem” is: How do you define the Poynting vector in a 2-complex? Or in the acoustic case, the Power flux. I’ve thought about this before, (I remember a correspondence with Eric about it some years ago) but not yet written anything about it yet. I call it “Cut functions”, but I guess a nicer name should be possible.
Start with the acoustic case.
Think of an acoustic circuit, that has a certain solution. Any solution has SUM((p_i-p_j)V_ij) = d/dt (SUM(p_i^2 + V_ij^2)) = 0. (Assume unit impedance for simplicity) Now, cut the circuit in 2 parts, along a certain line (I’ll use a 2D circuit here). The 2 half circuits now do not individually satisfy d/dt (SUM(p_i^2 + V_ij^2)) = 0. It would be nice if some quantity (W) integrated over the “cut” has the property:
d/dt (SUM(p_i^2 + V_ij^2)) = - SUM (W).
Let’s say an edge that is cut in 2 has an “inner” vertex, and an “outer” vertex. The vertices that are not connected to a cut are either “left” or “right” depending on which half of the original circuit they belong.
I believe you can prove:
SUM_left(SUM(p_i^2 + V_ij^2) + SUM_inner((p_i)V_ij) = 0
So (Assuming j is outer),
W_ij_inner = p_i V_ij
W_ij_outer = p_j V_ij
So the required power flux quantity is situated not on a vertex, not on an edge, but on a vertex-edge pair.
Perhaps confusing is that for each point in space, there are 2 times more components of the Power flux than components of flow. Which one you need depends on which side of a potential cut you are. Another subtlety is that W is not “Gauge invariant” : It depends on the absolute value of p. It is still OK to add a constant p to all vertices, as long as we do it to both sides of the cut.
In the electromagnetic case, you get a Poynting vector component on an edge-loop pair. Again, there are more of these than components of other vectors or n-forms. The one you need again depends on how you choose to cut the circuit.
hmm… It starts to make sense…
Gerard
Another thing:
I would really like it if I could understand angular momentum in circuits. Acoustic circuits describe a spin 0 particle (You can make an analog of the Klein Gordon equation with mass too, as I did on my web site). The Maxwell equations, with their 2 complex
“circuit” describe spin 1 particles.
I don’t see how this fits in the circuits.
So: Is there a meaningfull analog of angular momentum in circuits?
I don’t know the answer, but I’ll try to write down the direction I been thinking in so far.
Here is the acoustic circuit:
Because the voltage integrated along any closed loop is zero,
d/dt (SUM_loop (Momentum)) = 0
Also, iy you integrate along a line from a boundary to a boundary
d/dt (SUM_line (Momentum)) = p_start-p_finish
Intuitively, I feel momentum in a loop is related to angular momentum. In a circuit, we do no have an “r” to put in the (r × p) for angular momentum. But the “r” is not very nice anyway, it is frame dependent, and is very hard to generalise to curved space.
Think about the circuit analog of a pirouette:
You start with momentum along a certain loop. Although in acoustics, this momentum can never leave the loop, lets assume there is some mechanism that reversibly can transfer it to a loop that is “inside” the first loop. This second loop is smaller than the first, so you can imagine that the momentum per edge will be larger than along the first loop, so that the momentum integrated along it is equal to the momentum along the first loop. So perhaps the “r” gets hidden in the “loop length”.
But wait: In a pirouette the momentum *increases* when the skater folds in her arms and legs. Hmm… where does the energy come from?
Actually, I’m off to go skating myself. (Took a day off) I’ll think about pirouettes, but I will refrain from doing them myself.
In week289 I asked:
I’m also curious about lots of other things. For example: in classical mechanics it’s really important that we can define “Poisson brackets” of smooth real-valued functions on the cotangent bundle. So: how about in thermodynamics? Does anyone talk about the Poisson bracket of temperature and entropy, for example?
In response, someone suggested that maybe Berris and Edwards use something of this sort in Thermodynamics of Flowing Systems to get the Navier–Stokes equation.
I’ll try to remember to take a look — but I am still hoping someone can help me out some more here.
Since the overall universe cannot be in equilibrium, I googled the phrase “nonequilibrium Poisson” and found this for you:
Constraints in Nonequilibrium Thermodynamics by Hans C. Ottinger in J. Chem. Phys. 130, 114904 (2009):
“We elaborate how holonomic constraints can be incorporated into the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics. Dirac’s ideas for constructing constrained Poisson brackets are extended to dissipative brackets. The construction is presented such that it can be put into practice most readily. We illustrate the procedure by developing a symmetric thermodynamic description of diffusion in multicomponent systems and, as a further example, we impose an incompressibility constraint. As a consequence of its more elaborate and restrictive structure, GENERIC removes the ambiguities occurring in the classical thermodynamics of irreversible processes when one works with redundant variables.”
“We elaborate how holonomic constraints can be incorporated into the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics.”
All this is collected into an interesting and comprehensive, poisson-bracket-based theory in Oettinger’s book
Hans C. Oettinger, Beyond Equilibrium Thermodynamics, Wiley 2005
In week289 John Baez asked:
“So: how about in thermodynamics? Does anyone talk about the Poisson bracket of temperature and entropy, for example?”
Entropy is a canonical variable in the Hamiltonian description of fluid dynamics, see, e.g., R. Salmon, “Hamiltonian Fluid Dynamics”, Ann. Rev. Fluid Mech. 20 (1998) 225.
“But if anyone knows a clear, detailed treatment of the analogy between classical mechanics and thermodynamics, focusing on the Legendre transform, please let me know!”
A recent article, that you might find interesting (although perhaps not exactly what you are seeking for in terms of Legendre transform), is S.G. Rajeev, “Quantization of Contact Manifolds and Thermodynamics”, Annals Phys. 323 (2008) 768-782, http://arxiv.org/abs/math-ph/0703061.
Thanks for the references, Klaus. The paper by Rajeev definitely proves I’m not the only one who had the crazy idea of trying to ‘quantize’ the conjugate variables in thermodynamics, or who wondered what the relevant uncertainty principle might then be. Quoting:
In classical thermodynamics, as in classical mechanics, observables come in canonically conjugate pairs: pressure is conjugate to volume, temperature to entropy, magnetic field to magnetization, chemical potential to the number of particles etc. An important difference is that the thermodynamic state space is odd dimensional. Instead of the phase space forming a symplectic manifold (necessarily even dimensional) the thermodynamic state space is a contact manifold, its odd dimensional analogue. Upon passing to the quantum theory, observables of mechanics become operators; canonically conjugate observables cannot be simultaneously measured and satisfy the uncertainty principle
Is there is an analogue to this uncertainty principle1 for thermodynamically conjugate variables? Is there such a thing as ‘quantum thermodynamics’ where pressure or volume are represented as operators? The product of thermodynamic conjugates such as has the units of energy rather than action. So if there is an uncertainty relation , it is clear that cannot be Planck’s constant as in quantum mechanics.
And he notes: “There is already an uncertainty relation for statistical rather then quantum fluctuations of thermodynamical quantities, where the analogue of is ”.
But there are a couple of strange things here, even before I read the rest of the paper! First, the variables and are of type that in general systems theory are called ‘effort’ and ‘displacement’, while and are of the type ‘momentum’ and ‘displacement’. It’s the latter sort where the product has units of action, and where in the quantum theory. The analogous quantities would not be pressure and volume, but rather ‘pressure momentum’ and volume — as I explained in week288.
Secondly, it’s not only thermodynamics where contact geometry becomes important. It’s also important in classical mechanics!
Anyway, this paper looks very interesting — I’m just mentioning two places where I think it might be interesting to follow the analogies more religiously than Rajeev seems to be doing.
There’s a lot more to say. But I should read the paper.
Another piece for your chart can be found in algorithmic information theory.
Let be the first bits of .
Let be the length of the smallest program whose output is .
Let if the program halts, 0 otherwise.
Let .
Then the expectation value of is
Chaitin’s Omega number is .
It’s also true that
This is like the chemical potential idea: is the number of bits of information it takes to produce more bits of the expansion of .
Replacing ‘energy’ by ‘money’ in your economic example is bold. So far there is no empirical evidence that money is invariant over time. On the contrary, as long as money can simply be printed, it is not invariant.
However, I think your intuition still remains true.
Christian Schwarz from Duisburg university argues that ‘displacement’ should be ‘price’ and from the empirically justified ‘demand invariance under price-scaling’ one gets that the ‘momentum’ (as the space shift invariant) becomes ‘demand’ (as the invariant for the above symmetry. One can even derive commutation relations and uncertainty equations with this approach.
More can be found in my mathematics-blog filed under miroeconomics.
I wonder if the number ought really to be the purchasing power of the Global economy or something; after all, money isn’t worth the paper it’s printed on (or the gold it’s pressed into) it’s only as good as what it will buy. It’s still a tricky idea — cash is coupled to many fields not directly interchangeable, in ways that vary across space and time.
Of course, in thermodynamics, Gibbs’ free energy is often a more useful number than the raw total energy of a system; and that’s not an invariant either, but it tells you what is reversible and what isn’t.
One approach to comparing economies across nations
is that used by the Economist mag - it’s called the Big Mac index - no kidding!
Hmmm… I am always excited when I see cool analogies like these developed! But I feel like there should be more columns in the table - to include explicitly dissipative terms.
I realize that non-conservative terms may not fit so well into a program directed toward symplectic geometry, but, at least in the thermal and chemical systems, they seem unavoidable to me because in these systems dissipative effects predominate over “inertial” effects.
In all but the most exotic chemical reactions, the approach to equilibrium is characterized by concentrations whose differences from their equilibrium values decay (often exponentially) without oscillating (overshooting equilibrium like a damped oscillator would).
In fact, the damped mechanical harmonic oscillator (with displacement coordinate x, mass m, spring constant k, and a frictional force F = -b x’ proportional to velocity) serves as a good reference point for what I mean. (In the spirit of analogy, you could use a circuit having inductance, capacitance, and resistance, of course!)
As you know, a classical equation of motion for this system, when unforced, is
m x” + b x’ + k x = 0.
When m k < (b/2)^2, the system undergoes no oscillations, and we say it is overdamped. In fact, when this inequality is pronounced, the restoring force F = -k x is proportional to a VELOCITY rather than an ACCELERATION, and the mass plays essentially no role in the solution except when the initial velocity is very large. Same thing goes for electrical circuits, where it is often appropriate to ignore inductances as negligible.
I feel like this is more the case in thermal systems (with temperature gradients) or chemical systems (with concentration or chemical potential gradients). In fact, it is hard for me to see how a “temperature momentum” or “chemical momentum” would arise, as these would give rise to an overshooting of the natural equilibrium.
There are “oscillating reactions” in which the chemical potential of a substance undergoes oscillations, but as I said, these are very rare and often the result of something crazy like autocatalysis, where a product of the reaction actually catalyses the reaction itself and speeds it up in proportion to the product’s concentration.
Having said all that, there are really cool analogies to be made with dissipative systems!
There’s
Ohm’s Law: J = - G dV/dx,
in which a current density is proportional to a gradient of electric potential (an E field),
Fick’s Law: J = - D dC/dx,
in which a material flux is proportional to a concentration gradient,
Fourier’s Law: 1/A dQ/dt = - k dT/dx,
which relates heat flux through an area to a temperature gradient (and defines the thermal conductivity),
and a law (Newton’s?) for viscosity, which relates a frictional force to a transverse velocity gradient:
1/A Fx = - η dvx/dy.
Here, the “flux” is a flux of transverse momentum through the area A.
I’m sure there are others, but these are the only ones I can think of right now.
All of these laws relate a type of “flux,” a type of “gradient,” and a type of “conductivity.”
Hi, Garett! Remember telling me about the Y- transformation for electrical circuits?
I still don’t understand how this is related to the Yang–Baxter equation. But I plan to say a lot about the category-theoretic approach to electrical circuits and their analogues in other branches of physics and engineering. And the Y- transformation is part of the story I’ll tell. So, thanks for telling me about it!
I realize that non-conservative terms may not fit so well into a program directed toward symplectic geometry…
Not so! I know what you mean: Hamiltonian mechanics and symplectic geometry are typically aimed at understanding systems without dissipation. But I’ll definitely be talking a lot about dissipative systems in the Weeks to come — indeed I already brought in RLC circuits and the damped harmonic oscillator back in week288. And it turns out that purely dissipative systems have their own different relation to symplectic geometry!
So, thanks for listing a bunch of purely dissipative systems! As you may know, circuits built from resistors and capacitors are described by systems of first-order ordinary differential equations… but if you build a big grid of resistors and capacitors, its behavior can approximate that of the heat equation.
So far, I mainly see symplectic geometry raising its pretty head in an even simpler situation: DC circuits made entirely of resistors — or in the continuum limit, Laplace’s equation. Systems like this can be described by a variational principle — the ‘principle of least power dissipation’. And that’s how symplectic geometry gets into the game.
There’s a lot more to say… and I’ll try to say it all in future Weeks!
I look forward to hearing much more!
Yes, I remember discussing the Y-Delta transformation with you in the context of “duals.” There was something I forgot to mention to you that I noticed around that time that might be relevant. There’s a very common bit of circuitry around called a Wheatstone bridge. (I’m sure it is on Wikipedia). I’ve never seen it drawn this way, but the six circuit elements in this circuit are connected in the same way as the edges of a tetrahedron!
Also around that time I remember your introducing me to your idea of an “S-connector,” and I said “hey, that’s a transistor!” In an email, Peter Selinger raised some justified skepticism about this analogy. What I should have said was, “hey, that’s a passive linear current amplifier!” - which is one of the very important models of transistor behavior in common regimes of operation.
On the subject of analogies, I remember a picture I saw in an electronics book from my youth that draws a parallel between current flow in a transistor and water flowing in connected channels that might be of some interest. There’s a similar sort of picture here (under “How do Transistors work?”):
http://talkingelectronics.com/projects/ElectronicsGuide/ElectronicsGuide.html
That picture with the voltmeter and square grid of resistors reminds me of an interesting problem: What is the resistance between 2 points separated (n,m)?
You may think this is a function with a strong “rectangular” character, ie strongly correlated to (m+n). But it is surprisingly well correlated to sqrt(m^2+n^2), what you would expect for a continuous resistance sheet.
One way to see this is using the fact that random walks solve networks. Computationally highly inefficient, but nice conceptually.
A random walk of N steps in the n-direction, gives a binomial distribution N!/(n! (N-n!)). But this tends to a Gaussian for high N. If you include a second direction, you get approximately a product of 2 Gaussians,
P ~ exp(-(n/N)^2)*exp(-(m/N)^2) = exp(-(n^2+ m^2)/N^2)
So combinatorics induce isotropy!
Gerard
Garret wrote:
But I feel like there should be more columns in the table - to include explicitly dissipative terms.
By the way, we’ll see we don’t need more columns to describe dissipative terms: even equations that include dissipation, e.g. those for an RLC circuit, can be described using in terms of the variables and .
There’s a very common bit of circuitry around called a Wheatstone bridge. (I’m sure it is on Wikipedia). I’ve never seen it drawn this way, but the six circuit elements in this circuit are connected in the same way as the edges of a tetrahedron!
Good point!
So, the Wheatstone bridge is a special case of the symbols, or ‘tet net’, in the monoidal category whose morphisms are circuits. I’ll describe that monoidal category in a while.
Also around that time I remember your introducing me to your idea of an “S-connector,” and I said “hey, that’s a transistor!” In an email, Peter Selinger raised some justified skepticism about this analogy. What I should have said was, “hey, that’s a passive linear current amplifier!”
Thanks! If I ever finish that paper with Peter Selinger, I’ll include that information.
Bad link. (And no, the desired URI is not hidden anywhere in the HTML source of the page.)
Fixed, thanks! This link should be a trip down memory lane for you, Toby…
By the way, I think I was wrong to claim the Wheatstone bridge is a special case of a tet net. The tet net is a tetrahedron with edges labelled by identity morphisms of objects. The Wheatstone bridge has edges labelled by nonidentity morphisms. But there still may be something going on here!
Kostant is giving a talk about the new appearance of in condensed matter physics. If anyone can listen to it, please give us a report!
Abstract: A recent experimental discovery involving the spin structure of electrons in a cold one dimensional magnet points to a model involving the exceptional Lie group . The model predicts 8 particles the ratio of whose masses are the same as the ratios of the radii of the circles in the famous Gossett diagram (going back to 1900) of what is now understood to be a 2 dimensional projection of the 240 roots of arranged in 8 concentric circles. The ratio of the radii of the two smallest circles (read 2 smallest masses) is the golden number. This beautifully has been found experimentally. The ratio of the radii of the other masses has been written down conjecturally by Zamolodchikov. This again agrees with the analogous statement for the radii of the Gossett circles.
Some time ago we found an operator (very easily defined and reexpressed by Vogan as an element of the group algebra of the Weyl group) on 8-space whose spectrum is exactly the squares of the radii of the Gossett circles.
The operator is written in terms of the coefficients of the highest root. In McKay theory the are the dimensions of the irreducible representations of the binary icosahedral group. Our result works for any simple Lie group not just .
Re: This Week’s Finds in Mathematical Physics (Week 289)
You say:
But do elementary particles really obey such simple differential equations? We come up with, say, the Klein-Gordon equation only by making all sorts of simplifying assumptions. We find that its predictions hold only when those assumptions aren’t violated too badly, like if interactions are “weak enough”.
It seems to me that the difference isn’t so much that people are so much more complicated as that the simplifying assumptions become significant so much sooner.