# Canonical Probabilities by Directly Quantizing Thermodynamics

I’ve had this derivation kicking around a while, and today seemed like as good a day as any to make a fuller write-up of it:

• B. C. Stacey, “Canonical probabilities by directly quantizing thermodynamics” (PDF).

The idea is that Boltzmann’s rule $p(E_n) \propto e^{-E_n / k_B T}$ pops up really naturally when you ask for a rule that plays nicely with the composing-together of uncorrelated systems. This, in turn, gives a convenient expression to the idea that classical physics is what you get when you handle quantum systems sloppily.

# What I Do

At the moment, I’m taking a quick break from reading some rather dense mathematical prose, and I spent yesterday plugging away at a draft of my research group’s next technical publication. This led me to reflect on a lesson that I think a lot of science education leaves out: Even in a technical article, you have to have a story to carry the progression through. “These are all the boffo weird roadside attractions we found while proving the theorems in our last paper” is honest, but not adequate.

Our research project is the reconstruction of the mathematical formalism of quantum theory from physical principles. We tease apart the theory, identify what is robustly strange about it — for many more quantum phenomena can be emulated with classical stochasticity than are often appreciated — and try to build a new representation that brings the most remarkable features of the physics to the forefront. In special relativity, we have Einstein’s postulates, and the dramatic tension between them: Inertial observers can come to agree upon the laws of physics, but they cannot agree upon a standard of rest. In thermodynamics, we have the Four Laws, which come with their own dramatic tension, in that energy is conserved while entropy is nondecreasing. Both of these theories are expressed in terms of what agents can and cannot do, yet they are more than “mere” engineering, because they apply to all agents. Or, to say it another way, it is to the benefit of any agent to pick up the theory and use it as a guide.

What, then, is the analogue for quantum theory? If the textbook presentation of quantum physics is like the formulae for the Lorentz transform, with all those square roots and whatnot, or the Maxwell relations in thermo, with all those intermingling partial derivatives that we invent hacks about determinants to remember, what is quantum theory’s version of Einstein’s postulates or the Four Laws?

That’s the grandiose version, anyway. The reason I got invited to speak at an American Mathematical Society meeting is that the geometric structures that arise in this work are vexingly fascinating. You want about Galois fields and Hilbert’s 12th problem? We’ve got ’em! How about sphere packing and unexpected octonions? We’ve got those, too! And the structure that leads down the latter path turns out, on top of that, to yield a new way of thinking about Mermin’s 3-qubit Bell inequality. It is all lovely, and it is all strange.

The SIC problem gives us the opportunity to travel all throughout mathematics, because, while the definition looks pretty small, the question is bigger on the inside.

# Multiscale Structure, Information Theory, Explosions

I’d like to talk a bit about using information theory to quantify the intuition that a complex system exhibits structure at multiple scales of organization. My friend and colleague Ben Allen wrote an introduction to this a while ago:

Ben’s blog post is a capsule introduction to this article that he and I wrote with Yaneer Bar-Yam:

I also cover this topic, as well as a fair bit of background on how to relate probability and information, in my PhD thesis:

In this post, I’ll carry the ideas laid out in these sources a little bit farther in a particular direction.
Continue reading Multiscale Structure, Information Theory, Explosions

# 17 Equations that Clogged My Social-Media Timeline

An image burbled up in my social-media feed the other day, purporting to be a list of “17 Equations that Changed the World.” It’s actually been circulating for a while (since early 2014), and purports to summarize the book by that name written by Ian Stewart. This list is typo-ridden, historically inaccurate and generally indicative of a lousy knowledge-distribution process that lets us down at every stage, from background research to fact-checking to copy-editing.
Continue reading 17 Equations that Clogged My Social-Media Timeline

# New Paper Dance

B. C. Stacey, “Geometric and Information-Theoretic Properties of the Hoggar Lines” (2016), arXiv:1609.03075 [quant-ph].

We take a tour of a set of equiangular lines in eight-dimensional Hilbert space. This structure defines an informationally complete measurement, that is, a way to represent all quantum states of three-qubit systems as probability distributions. Investigating the shape of this representation of state space yields a pattern of connections among a remarkable spread of mathematical constructions. In particular, studying the Shannon entropy of probabilistic representations of quantum states leads to an intriguing link between the questions of real and of complex equiangular lines. Furthermore, we will find relations between quantum information theory and mathematical topics like octonionic integers and the 28 bitangents to a quartic curve.

All things told, SIC-POVMs are just about the damnedest things I’ve ever studied in mathematics.

(Also listed on SciRate.)

# Less Heteronormative Homework

A few weeks ago, I found an old physics book on a colleague’s “miscellaneous” shelf: University of Chicago Graduate Problems in Physics, by Cronin, Greenberg and Telegdi (Addison-Wesley, 1967). It looked like fun, so I started working through some of it.

Physics problems age irregularly. Topics fall out of vogue as the frontier of knowledge moves on, and sometimes, the cultural milieu of the time when the problem was written pokes through. Take the first problem in the “statistical physics” chapter. It begins, “A young man, who lives at location $A$ of the city street plan shown in the figure, walks daily to the home of his fiancee…”

No, no, no, that just won’t do any more. Let us set up the problem properly:

Asami is meeting Korra for lunch downtown. Korra is $E$ blocks east and $N$ blocks north of Asami, on the rectangular street grid of downtown Republic City. Because Asami is eager to meet Korra, her path never doubles back. That is, each move Asami takes must bring her closer to Korra on the street grid. How many different routes can Asami take to meet Korra?

Solution below the fold.

# Multiscale Structure via Information Theory

We have scienced:

B. Allen, B. C. Stacey and Y. Bar-Yam, “An Information-Theoretic Formalism for Multiscale Structure in Complex Systems” [arXiv:1409.4708].

We develop a general formalism for representing and understanding structure in complex systems. In our view, structure is the totality of relationships among a system’s components, and these relationships can be quantified using information theory. In the interest of flexibility we allow information to be quantified using any function, including Shannon entropy and Kolmogorov complexity, that satisfies certain fundamental axioms. Using these axioms, we formalize the notion of a dependency among components, and show how a system’s structure is revealed in the amount of information assigned to each dependency. We explore quantitative indices that summarize system structure, providing a new formal basis for the complexity profile and introducing a new index, the “marginal utility of information”. Using simple examples, we show how these indices capture intuitive ideas about structure in a quantitative way. Our formalism also sheds light on a longstanding mystery: that the mutual information of three or more variables can be negative. We discuss applications to complex networks, gene regulation, the kinetic theory of fluids and multiscale cybernetic thermodynamics.

There’s much more to do, but for the moment, let this indicate my mood:

“You’ll get so preoccupied with equations that you forget to eat!” #BadWaysToPromoteScienceToYoungWomen

# Delayed Gratification

A post today by PZ Myers nicely expresses something which has been frustrating me about people who, in arguing over what can be a legitimate subject of “scientific” study, play the “untestable claim” card.

Their ideal is the experiment that, in one session, shoots down a claim cleanly and neatly. So let’s bring in dowsers who claim to be able to detect water flowing underground, set up control pipes and water-filled pipes, run them through their paces, and see if they meet reasonable statistical criteria. That’s science, it works, it effectively addresses an individual’s very specific claim, and I’m not saying that’s wrong; that’s a perfectly legitimate scientific experiment.

I’m saying that’s not the whole operating paradigm of all of science.

Plenty of scientific ideas are not immediately testable, or directly testable, or testable in isolation. For example: the planets in our solar system aren’t moving the way Newton’s laws say they should. Are Newton’s laws of gravity wrong, or are there other gravitational influences which satisfy the Newtonian equations but which we don’t know about? Once, it turned out to be the latter (the discovery of Neptune), and once, it turned out to be the former (the precession of Mercury’s orbit, which required Einstein’s general relativity to explain).

There are different mathematical formulations of the same subject which give the same predictions for the outcomes of experiments, but which suggest different new ideas for directions to explore. (E.g., Newtonian, Lagrangian and Hamiltonian mechanics; or density matrices and SIC-POVMs.) There are ideas which are proposed for good reason but hang around for decades awaiting a direct experimental test—perhaps one which could barely have been imagined when the idea first came up. Take directed percolation: a simple conceptual model for fluid flow through a randomized porous medium. It was first proposed in 1957. The mathematics necessary to treat it cleverly was invented (or, rather, adapted from a different area of physics) in the 1970s…and then forgotten…and then rediscovered by somebody else…connections with other subjects were made… Experiments were carried out on systems which almost behaved like the idealization, but always turned out to differ in some way… until 2007, when the behaviour was finally caught in the wild. And the experiment which finally observed a directed-percolation-class phase transition with quantitative exactness used a liquid crystal substance which wasn’t synthesized until 1969.

You don’t need to go dashing off to quantum gravity to find examples of ideas which are hard to test in the laboratory, or where mathematics long preceded experiment. (And if you do, don’t forget the other applications being developed for the mathematics invented in that search.) Just think very hard about the water dripping through coffee grounds to make your breakfast.

T. Biancalani, D. Fanelli and F. Di Patti (2010), “Stochastic Turing patterns in the Brusselator modelPhysical Review E 81, 4: 046215, arXiv:0910.4984 [cond-mat.stat-mech].

Abstract:

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. The mean-field equations are derived and shown to yield to organized Turing patterns within a specific parameters region. When determining the Turing condition for instability, we pay particular attention to the role of cross-diffusive terms, often neglected in the heuristic derivation of reaction-diffusion schemes. Stochastic fluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic of the mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameter yielding to the stochastic self-organization is wider than that determined via the conventional Turing approach, suggesting that the condition for spatial order to appear can be less stringent than customarily believed.

A. Franceschini et al. (2011), “Transverse Alignment of Fibers in a Periodically Sheared Suspension: An Absorbing Phase Transition with a Slowly Varying Control Parameter” Physical Review Letters 107, 25: 250603. DOI: 10.1103/PhysRevLett.107.250603.

Abstract: Shearing solutions of fibers or polymers tends to align fiber or polymers in the flow direction. Here, non-Brownian rods subjected to oscillatory shear align perpendicular to the flow while the system undergoes a nonequilibrium absorbing phase transition. The slow alignment of the fibers can drive the system through the critical point and thus promote the transition to an absorbing state. This picture is confirmed by a universal scaling relation that collapses the data with critical exponents that are consistent with conserved directed percolation.

Last October, a paper I co-authored hit the arXivotubes (1110.3845, to be specific). This was, on reflection, one of the better things which happened to me last October. (It was, as the song sez, a lonesome month in a rather immemorial year.) Since then, more relevant work from other people has appeared. I’m collecting pointers here, most of them to freely available articles.

I read this one a while ago in non-arXiv preprint form, but now it’s on the arXiv. M. Raghib et al. (2011), “A Multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics”, Journal of Mathematical Biology 62, 5: 605–53. arXiv:1202.6092 [q-bio].

Abstract: The pervasive presence spatial and size structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based on mean densities (local or global) only. Individual-based models (IBM’s) were introduced over the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite insightful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space-time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. Continuum approximations of these stochastic processes distill their fundamental properties, but they often result in infinite hierarchies of moment equations. We use the principle of constrained maximum entropy to derive a closure relationship for one such hierarchy truncated at second order using normalization and the product densities of first and second orders as constraints. The resulting `maxent’ closure is similar to the Kirkwood superposition approximation, but it is complemented with previously unknown correction terms that depend on on the area for which third order correlations are irreducible. This region also serves as a validation check, since it can only be found if the assumptions of the closure are met. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the maxent closure to predict equilibrium values for mildly aggregated spatial patterns.

# Of Two Time Indices

In the appendix to a paper I am currently co-authoring, I recently wrote the following within a parenthetical excursus:

When talking of dynamical systems, our probability assignments really carry two time indices: one for the time our betting odds are chosen, and the other for the time the bet concerns.

A parenthesis in an appendix is already a pretty superfluous thing. Treating this as the jumping-off point for further discussion merits the degree of obscurity which only a lengthy post on a low-traffic blog can afford.

# Fugacity

The question came up while discussing the grand canonical ensemble the other day of just where the word fugacity came from. Having a couple people in the room who received the “benefits of a classical education” (Gruber 1988), we guessed that the root was the Latin fugere, “to flee” — the same verb which appears in the saying tempus fugit. Turns out, the Oxford English Dictionary sides with us, stating that fugacity was formed from fugacious plus the common +ty suffix, and that fugacious (meaning “apt to flee away”) goes back to the Latin root we’d guessed.

Gilbert N. Lewis appears to have introduced the word in “The Law of Physico-Chemical Change”, which appeared in the Proceedings of the American Academy of Arts and Sciences 37 (received 6 April 1901).
On occasion, somebody voices the idea that in year $N$, physicists thought they had everything basically figured out, and that all they had to do was compute more decimal digits. I won’t pretend to know whether this is actually true for any values of $N$ — when did one old man’s grumpiness become the definitive statement about a scientific age? — but it’s interesting that not every physicist with an interest in history has supported the claim.