Questions to consider when evaluating AI

 I am a slow adopter of new technologies. I have recently been playing around with AI at a very basic level on some research problems. Colleagues are also telling me about their experiences. I want to write something about its potential, both good and bad, for research and teaching. I want to hear from more people, particularly in condensed matter theory. However, before that, I think it is worth stepping back and asking some bigger questions than "Can AI help me publish more papers?" or "How do we stop students cheating on assessment?", as important as they are. I know the mathematics community is going through some angst and has issued a declaration about AI, and it is good to see that level of reflection.

Neil Postman (1931-2003) was a media theorist and cultural critic at New York University who spent a lifetime wrestling with questions about the broader implications of new technologies. In a talk given in 1998, he considered five things we need to know about technological change. Postman's enduring influence and relevance are marked by the fact that these five things featured in a column in The Washington Post, "Is the Internet Evil?" by Christine Emba, published in 2018.

Below, I summarise the five ideas from Postman's talk and provide questions (in italics) we should ask about any technology, particularly Artificial Intelligence (AI).

1. All technological change is a trade-off. 

"the greater the wonders of a technology, the greater will be its negative consequences" 

Don't just ask the question "What will a new technology do?" Also ask, "What will a new technology undo?"

"a sophisticated perspective on technological change includes one’s being skeptical of Utopian and Messianic visions drawn by those who have no sense of history or of the precarious balances on which culture depends."

Through adoption of the technology, what will we lose, individually and as a society?

2. The advantages and disadvantages of a new technology are never distributed evenly among the population.

Who will benefit? Who will be harmed? 

Winners will try to persuade losers that they will benefit as well.

Benefits and harms can relate to employment, finances, social status, health, and political power.

3.  Embedded in every technology are powerful ideas. 

"These ideas are often hidden from our view because they are of a somewhat abstract nature. But this should not be taken to mean that they do not have practical consequences."

"The telegraphic person values speed, not introspection. The television person values immediacy, not history... the computer person values information, not knowledge, certainly not wisdom." 

"The medium is the message."

What ideas are embedded in the technology?

How does it make us use our minds and bodies?

How does it affect our personal relationships and social cohesion?

4. Technological change is not additive; it is ecological. It changes everything.

"The consequences of technological change are always vast, often unpredictable and largely irreversible."

The entrepreneurs who started the television industry "did not mean to turn political discourse into a form of entertainment."

The consequences can be social, economic, political, environmental, religious, and health-related.

What are the unintended consequences of the technology?

5. When a technology becomes mythic, it is always dangerous because it is then accepted as it is, and is therefore not easily susceptible to modification or control.

"...our enthusiasm for technology can turn into a form of idolatry and our belief in its beneficence can be a false absolute. The best way to view technology is as a strange intruder, to remember that technology is not part of God’s plan but a product of human creativity and hubris, and that its capacity for good or evil rests entirely on human awareness of what it does for us and to us."

How does the technology lead to idolatry? Do some people worship it, its creators, or its owners?

Finally,

Do we use the technology or does the technology use us? 

In different words, will we shape our lives to fit the requirements of the technology, rather than have our values shape our use of the technology?

Topology matters in condensed matter physics

Topology is the field of mathematics describing the properties of geometric objects that do not change when they are smoothly deformed. These properties only change in steps by cutting or gluing. Concepts in topology can be illustrated with everyday objects such as balls, doughnuts, coffee cups, and pretzels. For example, a doughnut can be gradually and smoothly deformed into the shape of a coffee mug (Figure 33). No ripping or cutting is required. In contrast, it is impossible to turn a ball into a doughnut without cutting a hole. The number of holes in an object is referred as a topological invariant. For a ball, doughnut, and the simplest pretzel these numbers are zero, one, and two, respectively. Topology is about qualitative differences not quantitative details such as distances, angles, and sizes.

Figure 33. A doughnut can be smoothly deformed into a coffee cup. From the perspective of the mathematical field of topology all the objects above are identical.

In chapter 4 it was noted that in ordered states of matter, some properties are determined by topological defects, such as vortices in superconductors. These are topological objects in the following sense. In a superconductor, there is an electrical current circulating around a vortex and a magnetic field that passes through the centre of the vortex. The magnetic flux is equal to one unit or quantum of the magnetic flux. If the spatial distribution of the electrical current around the vortex is smoothly changed the total magnetic flux remains the same. Furthermore, it is not possible to smoothly deform the system in any way to make the vortex disappear. The magnetic flux associated with the vortex is a topological invariant.

Condensed matter physics is about qualitative difference: states of matter are qualitatively different from one another. Until the 1980s these differences were only associated with different types of symmetry, which in turn reflect the underlying ordering in the state. This chapter describes unanticipated discoveries of new states of matter that could not be described in terms of this traditional symmetry picture. But they can be described in terms of topology. These states exhibit macroscopic quantum effects, reminiscent of superconductors and superfluids. Understanding these states of matter involves venturing back into Flatland and also into some abstract mathematical spaces. Remarkably, these abstractions can be related to practical questions about international standards for electronic circuits.

An extract from Topology Matters, Chapter 8, Condensed Matter Physics: A Very Short Introduction 

Condensed matter physics in flatland

Adventures in Flatland

In everyday life we think of most objects as having three dimensions. But what would life be like in a two-dimensional world? For one thing, it would be harder to move around. We could no longer step over things but would have to move around them. In 1884 Edwin Abbott published Flatland: A Romance of Many Dimensions, under the pseudonym, A. Square, a satirical novella about social life in Victorian England. People are represented by geometrical objects. Men are represented by shapes such as triangles and hexagons. Women are represented by lines. The social status of men increases with the number sides that their shape has and how many of the sides are of the same length. Abbott’s book created limited interest and was largely forgotten by the 1920s. Interest revived when theoretical physicists started to think about worlds in different dimensions. This interest was stimulated by Albert Einstein’s theories of relativity, that proposed that we live in a four-dimensional world, not a three-dimensional one. Time is the fourth dimension, and there is an intimate and concrete connection between time and space. Attempts to unify gravity with other fundamental forces has led to physicists proposing and studying theories with more than four dimensions.

Changing the number of spatial dimensions leads to different physics because it changes what is mathematically possible. In three dimensions, there were only five highly symmetrical shapes known as Platonic solids (tetrahedron, cube, octahedron, icosahedron, and dodecahedron). In contrast, in two dimensions it is possible to make an infinite number of symmetrical shapes, known as regular polygons, shapes made of straight lines of equal length such as squares or hexagons. Similarly, the number of Bravais lattices differ in two and three dimensions. Changing the number of spatial dimensions changes both what is mathematically possible and what is physically possible.

What would condensed matter physics be like in Flatland? This question received limited attention before the 1970s. Occasionally, theoretical physicists would investigate mathematical models of crystals or magnets in one or two dimensions just because the mathematics was simpler and more tractable than in three dimensions. The goal was to obtain insight into physics in three dimensions. We will consider a famous example, the Ising model. 

In the 1970s, several surprising developments led to significant interest in condensed matter physics in spatial dimensions different from the usual three. First, it became possible to make a wide range of material systems that were two-dimensional. Secondly, theoretical work showed that states of matter, and phase transitions between them, can be qualitatively different in one, two, and three spatial dimensions. And thirdly, considering different numbers of spatial dimensions turned out to be very fruitful for theory, particularly for understanding phase transitions near critical points. 

An extract from "Adventures in Flatland," chapter 5 in Condensed Matter Physics: A Very Short Introduction

Rodney Baxter (1940-2025): Mathematical Physicist

I recently learnt that Rodney Baxter died earlier this year. He was adept at finding exact solutions to two-dimensional lattice models in statistical mechanics. He had a remarkably low public profile. But, during my lifetime, he was one of the Australian-based researchers who made the most significant and unique contributions to physics, broadly defined. Evidence of this is the list of international awards he received.

On Baxter's scientific achievements, see the obituary from the ANU, and earlier testimonials from Barry McCoy in 2000, and by Vladimir Bahzanov, on the award of the Henri Poincaré Prize to Baxter in 2021.

Exact solutions of "toy models" are important in understanding emergent phenomena. Before Onsager found an exact solution to the two-dimensional Ising model in 1944, there was debate about whether statistical mechanics could describe phase transitions and the associated discontinuities and singularities in thermodynamic quantities. 

Exact solutions provide benchmarks for approximation schemes and computational methods. They have also guided and elucidated key developments such as scaling, universality, the renormalisation group and conformal field theory.

Exact solutions guided Haldane's development of the Luttinger liquid and our understanding of the Kondo problem.

I mention the specific significance of a few of Baxter's solutions. His Exact solution of the eight-vertex model in 1972 gave continuously varying critical exponents that depended on the interaction strength in the model. This surprised many because it seemed to be against the hypothesis of the universality of critical exponents. This was later reconciled in terms of connections to the Berezinskii-Kosterlitz-Thouless transition (BKT) phase transition, which was discovered at the same time. I am not sure who explicitly resolved this.

It might be argued that Baxter independently discovered the BKT transition. For example, consider the abstract of a 1973 paper, Spontaneous staggered polarization of the F-model

"The “order parameter” of the two-dimensional F-model, namely the spontaneous staggered polarization P0, is derived exactly. At the critical temperature P0 has an essential singularity, both P0 and all its derivatives with respect to temperature vanishing."

Following earlier work by Lieb, Baxter explored the connection of two-dimensional classical models with one-dimensional quantum lattice models. For example, the solution of the XYZ quantum spin chain is related to the Eight-vertex model. Central to this is the Yang-Baxter equation. Alexander B. Zamolodchikov connected this to integrable quantum field theories in 1+1 dimensions. [Aside: the Yang is C.N. Yang, of Yang-Mills and Yang-Lee fame, who died last week.]

Baxter's work had completely unanticipated consequences beyond physics. Mathematicians discovered profound connections between his exact solutions and the theory of knots, number theory, and elliptic functions. It also stimulated the development of quantum groups.

I give two personal anecdotes on my own interactions with Baxter. I was an undergraduate at the ANU from 1979 to 1982. This meant I was completely separated from the half of the university known as the Institute for Advanced Studies (IAS), where Baxter worked. Faculty in the IAS there did no teaching, did not have to apply for external grants, and had considerable academic freedom. Most Ph.D. students were in the IAS. By today's standards, the IAS was a cushy deal, particularly if faculty did not get involved in internal politics. As an undergraduate, I really enjoyed my courses on thermodynamics, statistical mechanics, and pure mathematics. My honours supervisor, Hans Buchdahl, suggested that I talk to Baxter about possibly doing a Ph.D. with him. I found him quiet, unassuming, and unambitious. He had only supervised a few students. He wisely cautioned me that Ph.D. students might not be involved in finding exact solutions but might just be comparing exact results to series expansions.

In 1987, when I was a graduate student at Princeton, Baxter visited, hosted by Elliot Lieb, and gave a Mathematical Physics Seminar. This visit was just after he received the Dannie Heinemann Prize for Mathematical Physics from the American Physical Society. These seminars generally had a small audience, mostly people in the Mathematical Physics group. However, for Baxter, many string theorists (Witten, Callen, Gross, Harvey, ...) attended. They had a lot of questions for Baxter. But, from my vague recollection, he struggled to answer them, partly because he wasn't familiar with the language of quantum field theory. 

I was told that he got nice job offers from the USA. He could have earned more money and achieved a higher status. For personal reasons, he turned down the offer of a Royal Society Research Professorship at Cambridge.  But he seemed content puttering away in Australia. He just loved solving models and enjoyed family life down under.

Baxter wrote a short autobiography, An Accidental Academic. He began his career and made his big discoveries in a different era in Australian universities. The ANU had generous and guaranteed funding. Staff had the freedom to pursue curiosity-driven research on difficult problems that might take years to solve. There was little concern with the obsessions of today: money, metrics, management, and marketing. It is wonderful that Baxter was able to do what he did. It is striking that he says he retired early so he would not have to start making grant applications!

Can AI solve quantum-many body problems?

I find it difficult to wade through all the hype about AI, along with the anecdotes about its failings to reliably answer basic questions.

Gerard Milburn kindly brought to my attention a nice paper that systematically addresses whether AI is useful as an aid (research assistant) for solving basic (but difficult) problems that researchers in condensed matter theorists care about.

CMT-Benchmark: A Benchmark for Condensed Matter Theory Built by Expert Researchers

The abstract is below.

My only comment is one of perspective. Is the cup half full or half empty? Do we emphasise the failures or the successes?

The optimists among us will claim that the success in solving a smaller number of these difficult problems shows the power and potential of AI. It is just a matter of time before LLMs can solve most of these problems, and we will see dramatic increases in research productivity (defined as the amount of time taken to complete a project).

The pessimists and skeptically oriented will claim that the failures highlight the limitations of AI, particularly when training data sets are small. We are still a long way from replacing graduate students with AI bots (or at least using AI to train students in the first year of their PhD).

What do you think? Should this study lead to optimism, pessimism, or just wait and see?

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Large language models (LLMs) have shown remarkable progress in coding and math problem-solving, but evaluation on advanced research-level problems in hard sciences remains scarce. To fill this gap, we present CMT-Benchmark, a dataset of 50 problems covering condensed matter theory (CMT) at the level of an expert researcher. Topics span analytical and computational approaches in quantum many-body, and classical statistical mechanics. The dataset was designed and verified by a panel of expert researchers from around the world. We built the dataset through a collaborative environment that challenges the panel to write and refine problems they would want a research assistant to solve, including Hartree-Fock, exact diagonalization, quantum/variational Monte Carlo, density matrix renormalization group (DMRG), quantum/classical statistical mechanics, and model building. We evaluate LLMs by programmatically checking solutions against expert-supplied ground truth. We developed machine-grading, including symbolic handling of non-commuting operators via normal ordering. They generalize across tasks too. Our evaluations show that frontier models struggle with all of the problems in the dataset, highlighting a gap in the physical reasoning skills of current LLMs. Notably, experts identified strategies for creating increasingly difficult problems by interacting with the LLMs and exploiting common failure modes. The best model, GPT5, solves 30\% of the problems; average across 17 models (GPT, Gemini, Claude, DeepSeek, Llama) is 11.4±2.1\%. Moreover, 18 problems are solved by none of the 17 models, and 26 by at most one. These unsolved problems span Quantum Monte Carlo, Variational Monte Carlo, and DMRG. Answers sometimes violate fundamental symmetries or have unphysical scaling dimensions. We believe this benchmark will guide development toward capable AI research assistants and tutors.

One hundred years of Ising

In 1925, Ising published his paper on the solution of the model in one dimension. An English translation is here.https://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html

Coincidentally, next week I am giving a lecture on the Ising model to an undergraduate class in statistical mechanics. To flesh out the significance and relevance of the model, here are some of the interesting articles I have been looking at:

The Ising model celebrates a century of interdisciplinary contributions, Michael W. Macy, Boleslaw K. Szymanski and Janusz A. Hołyst

This mostly discusses the relevance of the model to understanding basic problems in sociology, including its relation to the classic Schelling model for social segregation.

The Ising model: highlights and perspectives, Christof Külske

This mostly discusses how the model is central to some work in mathematical physics and probability theory.

The Fate of Ernst Ising and the Fate of his Model, Thomas Ising, Reinhard Folk, Ralph Kennac, Bertrand Berche, Yurij Holovatche.

This includes some nice memories of Ising from his son, Thomas.

Aside: I wanted a plot of the specific heat for the one-dimensional model. According to Google AI "In a 1D Ising model with no external magnetic field, the specific heat is zero at all temperatures."

Weather, chaos, and emergence

Weather involves many scales of distance, time, and energy. Describing weather means making decisions about what range of scales to focus on. 

BTW. Did you know that there is a cyclone heading for Brisbane right now!

Now, back to physics :)

Key physics involves thermal convection which reflects an interplay of gravity, thermal expansion, viscosity and thermal conduction. This can lead to Rayleigh-Bénard convection and convection cells.

The multiple scales are associated with multiple entities:

-the molecules that make up the fluid

-small volumes of fluid that are in local thermodynamic equilibrium with a well-defined temperature, density, and velocity

-individual convection cells (rolls)

-collections of cells.

At each scale, the corresponding entities can be viewed as emerging from the interacting entities at the next smallest scale. Hence, they are collective degrees of freedom.

In principle, a complete description, including the transition to turbulence, is given by the equations of fluid dynamics, including the Navier-Stokes equation. Despite the apparent simplicity of these equations, making definitive predictions from them remains elusive.

A famous toy model was studied by the meteorologist Edward Lorenz in 1963, in a seminal paper, "Deterministic Nonperiodic Flow." Under the restrictive conditions of considering the dynamics of a single convection roll the model can be derived from the full hydrodynamic equations.

Lorenz's study stimulated the field of chaos theory, and is beautifully described in James Gleick's book Chaos: The Making of a New Science.

Here, I discuss Lorenz's model in the context of emergence.

 

The model consists of (just) three coupled ODEs (ordinary differential equations):

The variables x(t), y(t), and z(t) describe, respectively, the amplitude of the velocity mode, the temperature mode, and the mode measuring the heat flux Nu, the Nusselt number. x and y characterize the roll pattern.

The model has three dimensionless parameters: r, sigma, and b.

r is the ratio of the temperature difference between the hot and cold plate, to its critical value for the onset of convection. It can also be viewed as the ratio of the Rayleigh number to its critical value.

sigma is the Prandtl number, the ratio of the kinematic viscosity to the thermal diffusivity. Sigma is about 0.7 in air and 7 in water. Lorenz used sigma = 10.

b is of order unity and conventionally taken to have the value 8/3. It arises from the nonlinear coupling of the fluid velocity and temperature gradient in the Boussinesq approximation.

The model is a toy model because for values of r larger than r_c (defined below) "the three mode approximation for the PDEs describing thermal convection... ceased to be physically

realistic, but mathematically the model now starts to show its most fascinating properties,.."

Novelty

The model has several distinct types of long-time dynamics: stable fixed points (no convection), limit cycles (convective rolls), and most strikingly a chaotic strange attractor (represented below). The chaos is reflected in the sensitive dependence on initial conditions.

Briefly, a strange attractor is a curve of infinite length that never crosses itself and is contained in a finite volume. This means it has a fractal structure and a non-trivial Hausdorff dimension [calculated in this paper to be 2.0627160].

Discontinuities

Quantitative changes lead to qualitative changes. For r < 1, no convection occurs. For r > 1, convective rolls develop, but these become unstable for 

and a strange attractor develops.

Phase diagram

Lorenz only considered one set of parameter values [r =28, sigma=10, and b=8/3]. This was rather fortunate, because then strange attractor was waiting to be discovered. 

The phase diagram maps out the qualitatively different behaviours that occur as a function of sigma (vertical axis) and r (horizontal axis). 

Different phases are the fixed points P± associated with convective rolls (black), orbits of period 2 (red), period 4 (green), period 8 (blue), and chaotic attractors (white).

EXTENDED PHASE DIAGRAM OF THE LORENZ MODEL

H. R. DULLIN, S. SCHMIDT, P. H. RICHTER, and S. K. GROSSMANN

Universality

The details of the molecular composition of the fluid and the intermolecular interactions are irrelevant beyond how they determine the three parameters in the model. Hence, qualitatively similar behaviour can occur in systems with a wide range of chemical compositions and physical properties.

Unpredictability

Although the system of three ODEs is simple, discovery of the strange attractor and the chaotic dynamics was unanticipated. Furthermore, the dynamics in the chaotic regime are unpredictable, given the sensitivity to initial conditions.

Top-down causation

The properties and behaviour of the system are not just determined by the properties of the molecules and their interactions. The external boundary conditions, the applied temperature gradient and the spatial separation L of the hot and cold plates, are just as important in determining the dynamics of the system, including motion as much smaller length scales.

Emergence in classical optics: caustics and rainbows

                                                                Photo by Chris Lawton on Unsplash

I love seeing patterns such as those above in bodies of water. I did not know that they are an example of emergence, according to Michael Berry, who states:

“A caustic is a collective phenomena, a property of a family of rays that is not present in any individual ray. Probably the most familiar example is the rainbow.”

Caustics are envelopes of families of rays on which the intensity diverges. They occur in media where the refractive index is inhomogeneous. In the image above, there is an interplay of the uneven air-water interface and the difference in the refractive index between air and water. For rainbows, key parameters are the refractive index of the water droplets and the size of the droplets. The caustic is not the "rainbow", i.e., the spectrum of colours, but rather the large light intensity associated with the bow. The spectrum of colours arises because of dispersion (i.e., the refractive index of water depends on the wavelength of the light).

Caustics illustrate several characteristics of emergence properties: novelty, singularities, hierarchies, new scales, effective theories, and universality. 

Novelty. The whole system (a family of light rays) has a property (infinity intensity) that individual light rays do not.

Discontinuities. A caustic defines a spatial boundary across which there are discontinuities in properties.  

Irreducibility and singular limits. Caustics only occur in the theory of geometrical optics which corresponds to the limit where the wavelength of light goes to zero in a wave theory of light. Caustics (singularities) are not present in the wave theory.

Hierarchies. 

a. Light can be treated at the level of rays, scalar waves, and vector waves. At each level, there are qualitatively different singularities: caustics, phase singularities (vortices, wavefront dislocations, nodal lines), and polarisation singularities. 

b. Treating caustics at the level of wave theory, as pioneered by George Bidell Airy, reveals a hierarchy of non-analyticities, and an interference pattern, reflected in the supernumerary part of a rainbow.

New (emergent) scales. An example, is the universal angle of 42 degrees subtended by the rainbow, that was first calculated by Rene Descartes. Airy's wave theory showed that the spacing of the interference fringes shrinks as lambda^2/3.

Effective theories. At each level of the hierarchy, one can define and investigate effective theories. For ray theory, the effective theory is defined by the spatially dependent refractive index n(R)  and the ray action.

Universality. Caustics exist for any kind of waters: light, sound, and matter. They exhibit "structural stability". They fall into equivalence (universality) classes that are defined by the elementary catastrophes enumerated by Rene Thom and Vladimir Arnold and listed in the Table below. Any two members of a class can be smoothly deformed into one another.

The first column in the Table below is the name of the class given by Thom, and the second is the symbol used by Arnold. K is the number of parameters needed to define the class and the associated polynomial, which is given in the last column. 

For this post, I have drawn on several beautiful articles by Michael Berry.  A good place to start may be 

Nature's optics and our understanding of light (2015), which contains the figure I used above of the rainbow.

There is a beautiful description of some of the history and basic physics of the rainbow in Rainbows, Snowflakes, and Quarks: Physics and the World Around Us by Hans Christian Von Baeyer.

The pictures of the rainbow above were taken on a recent visit to Port Moresby in Papua New Guinea. 

Clarity about the relationship of emergence, complexity, predictability, and universality

Emergence means different things to different people. Except, that practically everyone likes it! Or at least, likes using the word. Terms associated with emergence include novelty, unpredictability, universality, stratification, and self-organisation. We need to be clearer about what we mean by each of these terms and how they are related or unrelated. Significant progress is reported in a recent preprint.

Software in the natural world: A computational approach to hierarchical emergence

Fernando E. Rosas, Bernhard C. Geiger, Andrea I Luppi, Anil K. Seth, Daniel Polani, Michael Gastpar, Pedro A.M. Mediano

This preprint is the subject of a nice article in Quanta Magazine.

The New Math of How Large-Scale Order Emerges by Philip Ball

Ball defines emergence in terms of unpredictability. He states: 

"Loosely, the behavior of a complex system might be considered emergent if it can’t be predicted from the properties of the parts alone."

He describes the work of Rosas et al. as follows, 

"A complex system exhibits emergence, according to the new framework, by organizing itself into a hierarchy of levels that each operate independently of the details of the lower levels."

This is defining emergence in terms of universality. Rosas et al. use an analogy with software, which runs independently of the details of the hardware of the computer and does not depend on microscopic details such as electron dynamics.

There are three types of closure associated with emergence: informational, causal, and computational.

Informational closure means that to predict the dynamics of the system at the macroscale one does not need any additional  information from the microscale.

Equilibrium thermodynamics is a nice example. 

Causal closure means that the system can be controlled at the macroscale without any knowledge of lower-level information.

"Interventions we make at the macro level, such as changing the software code by typing on the keyboard, are not made more reliable by trying to alter individual electron trajectories."

"...we can use macroscopic variables like pressure and viscosity to talk about (and control) fluid flow, and knowing the positions and trajectories of individual molecules doesn’t add useful information for those purposes. And we can describe the market economy by considering companies as single entities, ignoring any details about the individuals that constitute them."

Computational closure is a more technical concept. 

"a conceptual device called the ε-(epsilon) machine. This device can exist in some finite set of states and can predict its own future state on the basis of its current one. It’s a bit like an elevator, said Rosas; an input to the machine, like pressing a button, will cause the machine to transition to a different state (floor) in a deterministic way that depends on its past history — namely, its current floor, whether it’s going up or down and which other buttons were pressed already. Of course an elevator has myriad component parts, but you don’t need to think about them. Likewise, an ε-machine is an optimal way to represent how unspecified interactions between component parts “compute” — or, one might say, cause — the machine’s future state."

Aside: epsilon-machines featured significantly in my previous post about What is a complex system? 

"Computational mechanics allows the web of interactions between a complex system’s components to be reduced to the simplest description, called its causal state."

"...for an emergent system that is computationally closed, the machines at each level can be constructed by coarse-graining the components on just the level below: They are, in the researchers’ terminology, “strongly lumpable.”"

In some sense, this may be related to the notion of quasiparticles and effective interactions in many-body physics. 

Aside: In 1962, Herbert Simon identified hierarchies as an essential feature of complex systems, both natural and artificial. A key property of a level in the hierarchy is that it is nearly decomposable into smaller units, i.e., it can be viewed as a collection of weakly interacting units. The time required for the evolution of the whole system is significantly decreased due to the hierarchical character. The construction of an artificial complex system, such as a clock, is faster and more reliable if different units are first assembled separately and then the units are brought together into the whole. Simon argues that the reduction in time scales due to modularity is why biological evolution can occur on realistic time scales.  The 1962 article is reprinted in The Sciences of the Artificial.

The paper by Rosas et al. is one of the most important ones I have encountered in the past few years. I am slowly digesting it.

The beauty of the paper that it is mathematically rigorous. All the concepts are precisely defined and the central results are actually theorems. This replaces the vagueness of most discussions of emergence, including by myself.

The paper has helpful figures and considers concrete examples including Ehrenfest's Urn, an Ising model with Glauber dynamics, and a Hopfield neural network model.

I thank Gerard Milburn for bringing the Quanta article to my attention.

Certain benefits of Bayes

Best wishes for the New Year! One thing I hope to achieve this year is an actual understanding of things "Bayesian".

I am particularly interested because it gives a way to be more quantitative and precise about some of the intuitions that I use in science. For example, I tend to be skeptical of new experimental results (often hyped) that claim to go against well-established theories, regardless of how good the "statistics" of the touted result.

In this vein, Phil Anderson argued that Bayesian methods should have been used to rule out the significance of "discoveries" such as the 10 keV neutrino and the fifth force. In 1992 he wrote a Physics Today column on the subject.

An interesting metric for mathematical formula is the ratio of profound and wide implications to the simplicity of the formula and its derivation. I suspect that Bayes' formula for conditional probabilities would win first place!

P(A|B) denotes the probability of A given B. 

The proof takes about two lines. If you multiply both sides of the equation about by P(B) the identity holds because both sides of the equation are just different ways of writing P(A and B).

My first attempt to understand the applications and implications of Bayes was reading the relevant sections in Phil Nelson's beautiful book, Physical Models of Living Systems. There is a helpful section entitled, "Bayes formula provides a consistent approach to upgrading our degree of belief in light of new data."

More recently, I found this wonderful and short video very helpful, as it clearly defines terms, uses graphical representations, and gives some concrete examples.

 

A Bayesian perspective highlights the importance of reporting negative results and is the basis of a seminal paper

Why Most Published Research Findings Are False by John P. A. Ioannidis

A measure of the profundity of Bayes is that the Stanford Encyclopedia of Philosophy has two articles on the topic

Bayes Theorem

Bayesian Epistemology

Science and the universe are awesome

Since we are surrounded by scientific knowledge. We are so used to it that we can take science for granted and not reflect on how amazing science truly is. And how amazing the universe is that science reveals. Things that we know, learn, and do today in science would have been inconceivable decades ago, let alone centuries ago.

What specific things do you think are particularly awesome? This question was stimulated by Frank Wilczek's recent book, Fundamentals: Ten Keys to Reality. In writing the book, he says "what began as an exposition grew into a contemplation."

 My answer to the question has some significant overlap with Wilczek's ten. 

Below I list some of the things that I find awesome. I consider two classes: what science can do and what we learn about the universe from science.

Science works! It is amazing what science can do.

We can understand the material world.

Einstein said, "The most incomprehensible thing about the world is that it is comprehensible." In a previous post, I explored some different dimensions of the fact that the universe is comprehensible. The mystery includes human capabilities, both intellectual and physical, and the malleability of the material world.

We can make precise measurements.

Scientists have created incredibly powerful and specialised instruments for making very precise measurements such as spectrometers, telescopes and microscopes. Scientists can measure the tension in a single strand of DNA, the magnetic moment of an electron to a precision of one part in one billion billion, the spectrum of light emitted by a galaxy that is ten billion light years away, ...

We can predict the outcome of new experiments.

Scientists construct theories in their minds, on pieces of paper, in mathematical equations, and in computers. One way to evaluate the possible validity of a theory is to propose new experiments and predict the outcome. Famous examples include the existence of the chemical element aluminium, the existence of the planet Neptune, radio waves, a specific excited quantum state of the atomic nucleus of carbon atoms, the pollinator moth for Darwin's orchid, the deflection of the path of light from a distant star by our sun, gravitational waves, the Cosmic Microwave Background, quarks, the Higgs boson, the Berezinskii-Kosterlitz-Thouless phase transition, the hexatic phase, edge states in integer spin antiferromagnetic chains, topological insulators, ... Predictions are particularly impressive when they are unexpected and controversial.

We can use mathematics. 

Eugene Wigner received the Nobel Prize in Physics in 1963. In 1960 he published an essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences that concludes

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. 

We can manipulate and control nature.

Scientists and engineers can move single atoms, design drugs, make computers, build atom bombs, heart pacemakers, and mobile phones, manipulate genes, ......

We know so much but we know so little. 

On the one hand, the achievements of science are amazing. Yet, in spite of this, there are still significant mysteries and challenges. Examples include the nature of dark matter or human consciousness, a quantum theory of gravity, fine-tuning of fundamental constants, the quantum-classical boundary, protein folding, the nature of glasses, and how to calculate the properties of complex systems.

It is awesome what science reveals to us about the universe.

The immense scales of the observable universe

Our sun is just one star among the more than two hundred billion that make up our galaxy, the Milky Way. And that is just one of one trillion galaxies in the whole universe. It takes light from the most distant galaxies tens of billions of years to travel to us.

Length, time, and energy scales over many many orders of magnitude

These go far beyond our everyday experience and what we can see with the naked eye (from a millimetre to a kilometre). On the large scale, the visible universe involves distances of billions of light years (10^25 metres). On the small scale, there is the sub-structure of nucleons, which is smaller than femtometres (10^-15 m).  This wide range of length scales is nicely illustrated in the wonderful movie Powers of Ten and its update, The Cosmic Eye. There are corresponding time, energy, and temperature scales varying over many many orders of magnitude. For example, as one goes from ultracold atomic gases to quark-gluon plasmas, the  relevant energy and temperature scales vary over more than 20 orders of magnitude! At every scale, there are distinct phenomena and structures. 

Universal laws that are simple to state

The universe exhibits a diversity of rich and complex behaviour. Yet it can understand much of it in terms of simple universal laws that are easy to state, e.g., Newton's laws of motion, the laws of thermodynamics, Maxwell's equations of electromagnetism, Schrodinger's equation of quantum mechanics, the genetic code, ....  And, these are just a few of these laws. One does not need a multitude of laws to describe a multitude of instances of a multitude of phenomena.

Just a few building blocks

There are just a few fundamental particles in the standard model (leptons, neutrinos, and gauge bosons). Everything is made of them. They are the building blocks of atoms. They each have just a few physical properties: charge, spin, mass, and colour. Every single particle of a particular type in the universe has exactly the same properties. Exactly. As far as we know, they have been exactly the same throughout time, going back to the beginning of the universe, and whether they are in your body, or in a star in a distant galaxy.

Atoms are the building blocks of chemical compounds. Every single atom of a particular chemical element (and nuclear isotope) is absolutely identical. This allows astronomers to determine the chemical composition of distant stars, galaxies, and dust clouds.

Humans, plants, and animals all have the same molecular building blocks and there are just a few of them. Any DNA molecule is composed of just four different base pairs (denoted A, G, T, C) and proteins are composed of just twenty different amino acids.

There are two amazing things here. First, there are just so few building blocks. Second, every one of these building blocks is absolutely identical.

Emergence: simple rules produce complex behaviour

Humans, cells, and crystals can be viewed as systems composed of many interacting components. The components and their interactions can often be understood and described in simple terms. Nevertheless, from these interactions complex structures and properties can emerge.

Nature appears to be fine-tuned for life

This covers not just the values of fundamental physical constants that lead to the notion of fine-tuning and the anthropic principle. Water has unique physical and chemical properties that allow it to play a crucial role in life, such as the surface of lakes freezing before the bottom and aiding protein folding.

The intricate and subtle "machinery" of biomolecules

Proteins have very unique structures that are intimately connected to their specific functions, whether as catalysts or light sensors.

What do you think are the most amazing things about science and what we learn from it?

Is condensed matter physics too abstract?

Condensed matter physics is about the properties of real materials. Real stuff that you can see and touch and that you can use to make very practical things like TV screens and mobile phones. Yet, I find it fascinating and somewhat ironic that in condensed matter theory very abstract ideas and mathematical techniques keep cropping up (and being extremely useful): variable spatial dimensions, imaginary frequencies, topology, Chern numbers, conformal invariance, ...

Yet, there is a danger with abstraction. Theoretical condensed matter is not pure mathematics. Perhaps, too often fancy and beautiful mathematics is prized over physical intuition and insight. Theory may take precedence over experiment. How does one find the appropriate balance?

This is part of broader issues about the role of abstraction and formality in education.

Pierre de Gennes (1932-2007) was arguably the founder of soft matter as a research field, as recognized by the Nobel Prize in Physics in 1991. He began his career working on superconductivity and went on to develop a unified framework to understand soft matter (liquid crystals, polymers, foams, colloids...), introducing ideas such as order parameters, scaling, renormalisation, and universality.

After his Nobel, de Gennes gave many lectures in French high schools, which were then published as a book, Fragile Objects: Soft Matter, Hard Science, and the Thrill of Discovery. I highly recommend it, both as a popular introduction to soft matter, but also to hear the perspective of a great scientist on education and research.

de Gennes spent almost his whole life living and working in France. In the book he rants about the French system, particularly its obsession with entrance exams, mathematics, formality, and the abstract.

“Manual skills, visual acumen, the sense of observation, an interest for the physical world which surrounds us, are all qualities that are neglected or downgraded.”

“To work in a garage seems to me the best initiation to a professional life.”

“Ignorance of the real world causes grave distortions.”

“the positivist prejudice”

I found this fascinating because one thing de Gennes is famous for is showing how some properties of a polymer can be understood by considering a theory involving a vector of dimension n, where n was a continuous variable, in the limit where n approaches zero! That is pretty abstract! But, I guess the point is that he is not against abstraction, exams, and mathematics, per se. Rather, he is against them taking on a life of their own.

de Gennes concerns are also shared by Henri Alloul (well known for beautiful NMR experiments on strongly correlated electron materials) author of Introduction to the Physics of Electrons in Solids. In the Preface, he writes, 

In many countries, teaching traditions have always given pride of place to a formal, and essentially deductive, presentation of the physics, i.e., starting from formal hypotheses and leading up to observable consequences. This deductive approach leaves a purely a posteriori verificational role to observation, and hides the thinking that has gone into building up the models in the first place. Here we shall adopt the opposite approach, which begins with the fact that in science in general, and in solid state physics in particular, the qualitative understanding of a phenomenon is an important step which precedes the formulation of any theoretical development. We thus urge the reader to carry out a careful examination of the deeper significance of experimental observations, in order to understand the need for specific models and carry out realistic approximations.

The debate about abstract mathematics is also central to contrasting views about the Institute for Advanced Study at Princeton.

de Gennes's views would have also resonated with Harry Kroto who shared The Nobel Prize in Chemistry for the discovery of buckyballs. He credited playing with Meccano as a child as very important in his scientific development.

The physics of the SARS-CoV-2 virion

 Some progress is being made in understanding the structure and dynamics of the SARS-CoV-2 virions (virus particles) that are responsible for the pandemic. A nice starting point for the non-expert is a recent article in The New York Times.

A fundamental question is what is the structure and symmetry of the virion? In particular, does it have the icosahedral symmetry possessed by many virions, as discussed in a talk I gave earlier this year and in a recent review (with lots of nice pictures). As far as I am aware, there are still no definitive results on the overall structure and symmetry. 

This preprint has some really nice images and videos such as the video below. 

SARS-CoV-2 structure and replication characterized by in situ cryo-electron tomography

 View ORCID ProfileSteffen Klein,  View ORCID ProfileMirko Cortese,  View ORCID ProfileSophie L. Winter,  View ORCID ProfileMoritz Wachsmuth-Melm,  View ORCID ProfileChristopher J. Neufeldt, Berati Cerikan,  View ORCID ProfileMegan L. Stanifer,  View ORCID ProfileSteeve Boulant,  View ORCID ProfileRalf Bartenschlager,  Petr Chlanda

View ORCID Profile

Mathematical aside: the authors note that the geometric problem of how to place the spike protein (S) trimers on the surface of the virion is related to the "Tammes Problem" or the seventh unsolved mathematical problem listed by Steve Smale: how do you arrange a specific number of points on a sphere with the largest possible minimum distance between the points.

The paper below shows that the nucleocapsid protein (N) is similar to that for SARS-CoV and MERS. The protein can form dimers and tetramers, steps in the self-assembly of the whole virion.

Specific viral RNA drives the SARS CoV-2 nucleocapsid to phase separate

Christiane Iserman, Christine Roden, Mark Boerneke, Rachel Sealfon, Grace McLaughlin, Irwin Jungreis,  View ORCID ProfileChris Park, Avinash Boppana, Ethan Fritch, Yixuan J. Hou, Chandra Theesfeld, Olga G Troyanskaya, Ralph S. Baric,  View ORCID ProfileTimothy P. Sheahan, Kevin Weeks, Amy S. Gladfelter

Some nice soft matter physics is in the preprint below. It argues that the N protein can undergo liquid-liquid phase separation with the viral genome. Aside: even before covid, liquid-liquid phase separation was quite a hot topic in cell biology, as recently discussed by Tom McLeish. 

Architecture and self‐assembly of the SARS‐CoV‐2 nucleocapsid protein 

Qiaozhen Ye, Alan M. V. West, Steve Silletti, Kevin D. Corbett

Finally, the paper below combines molecular dynamics simulations with experiments to argue that the stalk of the spike protein has three hinges giving the head of the spike unexpected orientational freedom so it can scan the host cell surface.

In situ structural analysis of SARS-CoV-2 spike reveals flexibility mediated by three hinges

1. View ORCID ProfileBeata Turoňová1,2,*, 
2. View ORCID ProfileMateusz Sikora3,*, 
3. View ORCID ProfileChristoph Schürmann4,*, 
4. View ORCID ProfileWim J. H. Hagen1, 
5. View ORCID ProfileSonja Welsch5, 
6. View ORCID ProfileFlorian E. C. Blanc3, 
7. View ORCID ProfileSören von Bülow3, 
8. View ORCID ProfileMichael Gecht3, 
9. View ORCID ProfileKatrin Bagola6, 
10. View ORCID ProfileCindy Hörner4,7, 
11. View ORCID ProfileGer van Zandbergen6,8,9, 
12. View ORCID ProfileJonathan Landry10, 
13. View ORCID ProfileNayara Trevisan Doimo de Azevedo10, 
14. View ORCID ProfileShyamal Mosalaganti1,2, 
15. View ORCID ProfileAndre Schwarz1, 
16. View ORCID ProfileRoberto Covino3,11, 
17. View ORCID ProfileMichael D. Mühlebach4,7, 
18. View ORCID ProfileGerhard Hummer3,12,†, 
19. View ORCID ProfileJacomine Krijnse Locker13,†, 
20. View ORCID ProfileMartin Beck1,2,†
21.