## Non-Equilibrium Thermodynamics in Biology

11 May, 2021

Together with William Cannon and Larry Li, I’m helping run a minisymposium as part of SMB2021, the annual meeting of the Society for Mathematical Biology:

• Non-equilibrium Thermodynamics in Biology: from Chemical Reaction Networks to Natural Selection, Monday June 14, 2021, beginning 9:30 am Pacific Time.

You can register for free here before May 31st, 11:59 pm Pacific Time. You need to register to watch the talks live on Zoom. I think the talks will be recorded.

Here’s the idea:

Abstract: Since Lotka, ﻿physical scientists have argued that living things belong to a class of complex and orderly systems that exist not despite the second law of thermodynamics, but because of it. Life and evolution, through natural selection of dissipative structures, are based on non-equilibrium thermodynamics. The challenge is to develop an understanding of what the respective physical laws can tell us about flows of energy and matter in living systems, and about growth, death and selection. This session will address current challenges including understanding emergence, regulation and control across scales, and entropy production, from metabolism in microbes to evolving ecosystems.

It’s exciting to me because I want to get back into work on thermodynamics and reaction networks, and we’ll have some excellent speakers on these topics. I think the talks will be in this order… later I will learn the exact schedule.

#### Christof Mast, Ludwig-Maximilians-Universität München

Coauthors: T. Matreux, K. LeVay, A. Schmid, P. Aikkila, L. Belohlavek, Z. Caliskanoglu, E. Salibi, A. Kühnlein, C. Springsklee, B. Scheu, D. B. Dingwell, D. Braun, H. Mutschler.

Title: Heat flows adjust local ion concentrations in favor of prebiotic chemistry

Abstract: Prebiotic reactions often require certain initial concentrations of ions. For example, the activity of RNA enzymes requires a lot of divalent magnesium salt, whereas too much monovalent sodium salt leads to a reduction in enzyme function. However, it is known from leaching experiments that prebiotically relevant geomaterial such as basalt releases mainly a lot of sodium and only little magnesium. A natural solution to this problem is heat fluxes through thin rock fractures, through which magnesium is actively enriched and sodium is depleted by thermogravitational convection and thermophoresis. This process establishes suitable conditions for ribozyme function from a basaltic leach. It can take place in a spatially distributed system of rock cracks and is therefore particularly stable to natural fluctuations and disturbances.

#### Supriya Krishnamurthy, Stockholm University

Coauthors: Eric Smith

Title: Stochastic chemical reaction networks

Abstract: The study of chemical reaction networks (CRNs) is a very active field. Earlier well-known results (Feinberg Chem. Enc. Sci. 42 2229 (1987), Anderson et al Bull. Math. Biol. 72 1947 (2010)) identify a topological quantity called deficiency, easy to compute for CRNs of any size, which, when exactly equal to zero, leads to a unique factorized (non-equilibrium) steady-state for these networks. No general results exist however for the steady states of non-zero-deficiency networks. In recent work, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments. Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for non-trivial examples, that in this manner we can predict any moment of interest, for CRNs with non-zero deficiency and non-factorizable steady states. It is however an open question how scalable these techniques are for large networks.

#### Pierre Gaspard, Université libre de Bruxelles

Title: Nonequilibrium biomolecular information processes

Abstract: Nearly 70 years have passed since the discovery of DNA structure and its role in coding genetic information. Yet, the kinetics and thermodynamics of genetic information processing in DNA replication, transcription, and translation remain poorly understood. These template-directed copolymerization processes are running away from equilibrium, being powered by extracellular energy sources. Recent advances show that their kinetic equations can be exactly solved in terms of so-called iterated function systems. Remarkably, iterated function systems can determine the effects of genome sequence on replication errors, up to a million times faster than kinetic Monte Carlo algorithms. With these new methods, fundamental links can be established between molecular information processing and the second law of thermodynamics, shedding a new light on genetic drift, mutations, and evolution.

#### Carsten Wiuf, University of Copenhagen

Coauthors: Elisenda Feliu, Sebastian Walcher, Meritxell Sáez

Title: Reduction and the Quasi-Steady State Approximation

Abstract: Chemical reactions often occur at different time-scales. In applications of chemical reaction network theory it is often desirable to reduce a reaction network to a smaller reaction network by elimination of fast species or fast reactions. There exist various techniques for doing so, e.g. the Quasi-Steady-State Approximation or the Rapid Equilibrium Approximation. However, these methods are not always mathematically justifiable. Here, a method is presented for which (so-called) non-interacting species are eliminated by means of QSSA. It is argued that this method is mathematically sound. Various examples are given (Michaelis-Menten mechanism, two-substrate mechanism, …) and older related techniques from the 50-60ies are briefly discussed.

#### Matteo Polettini, University of Luxembourg

Coauthor: Tobias Fishback

Title: Deficiency of chemical reaction networks and thermodynamics

Abstract: Deficiency is a topological property of a Chemical Reaction Network linked to important dynamical features, in particular of deterministic fixed points and of stochastic stationary states. Here we link it to thermodynamics: in particular we discuss the validity of a strong vs. weak zeroth law, the existence of time-reversed mass-action kinetics, and the possibility to formulate marginal fluctuation relations. Finally we illustrate some subtleties of the Python module we created for MCMC stochastic simulation of CRNs, soon to be made public.

#### Ken Dill, Stony Brook University

Title: The principle of maximum caliber of nonequilibria

Abstract: Maximum Caliber is a principle for inferring pathways and rate distributions of kinetic processes. The structure and foundations of MaxCal are much like those of Maximum Entropy for static distributions. We have explored how MaxCal may serve as a general variational principle for nonequilibrium statistical physics – giving well-known results, such as the Green-Kubo relations, Onsager’s reciprocal relations and Prigogine’s Minimum Entropy Production principle near equilibrium, but is also applicable far from equilibrium. I will also discuss some applications, such as finding reaction coordinates in molecular simulations non-linear dynamics in gene circuits, power-law-tail distributions in “social-physics” networks, and others.

#### Joseph Vallino, Marine Biological Laboratory, Woods Hole

Coauthors: Ioannis Tsakalakis, Julie A. Huber

Title: Using the maximum entropy production principle to understand and predict microbial biogeochemistry

Abstract: Natural microbial communities contain billions of individuals per liter and can exceed a trillion cells per liter in sediments, as well as harbor thousands of species in the same volume. The high species diversity contributes to extensive metabolic functional capabilities to extract chemical energy from the environment, such as methanogenesis, sulfate reduction, anaerobic photosynthesis, chemoautotrophy, and many others, most of which are only expressed by bacteria and archaea. Reductionist modeling of natural communities is problematic, as we lack knowledge on growth kinetics for most organisms and have even less understanding on the mechanisms governing predation, viral lysis, and predator avoidance in these systems. As a result, existing models that describe microbial communities contain dozens to hundreds of parameters, and state variables are extensively aggregated. Overall, the models are little more than non-linear parameter fitting exercises that have limited, to no, extrapolation potential, as there are few principles governing organization and function of complex self-assembling systems. Over the last decade, we have been developing a systems approach that models microbial communities as a distributed metabolic network that focuses on metabolic function rather than describing individuals or species. We use an optimization approach to determine which metabolic functions in the network should be up regulated versus those that should be down regulated based on the non-equilibrium thermodynamics principle of maximum entropy production (MEP). Derived from statistical mechanics, MEP proposes that steady state systems will likely organize to maximize free energy dissipation rate. We have extended this conjecture to apply to non-steady state systems and have proposed that living systems maximize entropy production integrated over time and space, while non-living systems maximize instantaneous entropy production. Our presentation will provide a brief overview of the theory and approach, as well as present several examples of applying MEP to describe the biogeochemistry of microbial systems in laboratory experiments and natural ecosystems.

#### Gheorge Craciun, University of Wisconsin-Madison

Title: Persistence, permanence, and global stability in reaction network models: some results inspired by thermodynamic principles

Abstract: The standard mathematical model for the dynamics of concentrations in biochemical networks is called mass-action kinetics. We describe mass-action kinetics and discuss the connection between special classes of mass-action systems (such as detailed balanced and complex balanced systems) and the Boltzmann equation. We also discuss the connection between the “global attractor conjecture” for complex balanced mass-action systems and Boltzmann’s H-theorem. We also describe some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

#### Hong Qian, University of Washington

Title: Large deviations theory and emergent landscapes in biological dynamics

Abstract: The mathematical theory of large deviations provides a nonequilibrium thermodynamic description of complex biological systems that consist of heterogeneous individuals. In terms of the notions of stochastic elementary reactions and pure kinetic species, the continuous-time, integer-valued Markov process dictates a thermodynamic structure that generalizes (i) Gibbs’ macroscopic chemical thermodynamics of equilibrium matters to nonequilibrium small systems such as living cells and tissues; and (ii) Gibbs’ potential function to the landscapes for biological dynamics, such as that of C. H. Waddington’s and S. Wright’s.

#### John Harte, University of Berkeley

Coauthors: Micah Brush, Kaito Umemura

Title: Nonequilibrium dynamics of disturbed ecosystems

Abstract: The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories, and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time-varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic, by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth, or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time-varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.

## Crimson Solar

4 May, 2021

A bit of good news near my home.

On Monday May 3rd, the U.S Department of the Interior approved a new solar project in southern California. It’ll occupy about 2000 acres in the desert near Blythe—a town near the border with Arizona, and the Colorado River.

It’s called the Crimson Solar Project. It’s a 350-megawatt solar photovoltaic facility that will cost about $550 million. They say it will be able to power about 87,500 homes. The choice of site for this solar project is part of California’s Desert Renewable Energy Conservation Plan. The publication The Hill says this project will “store up to 530 megawatts of energy”—but I have no idea what that means, since megawatts isn’t a unit of energy. ## Renaissance Polyphony: the Franco-Flemish School 29 April, 2021 Near the beginning of the pandemic I heard a conversation where Brian Eno said that he’d been listening to a radio station on the internet, based somewhere deep in rural Russia, that plays nothing but Eastern Orthodox chants 24 hours a day, with no announcements. He said that this sort of music appealed to him while locked down at home. Somehow this led to me listening to a lot of early music on YouTube. I started with some Baroque composers I’d never paid attention to before, like Corelli and Albinoni. Previously my interest in classical music started around Beethoven and focused on the 20th century, with Bach and Vivaldi as brilliant pearls surrounded by darkness. As I began listening to more Baroque music I became very happy, like someone who’d been locked in a room for years and suddenly found a key to another room. There was so much to explore. The Baroque period lasted from about 1580 to 1750: that’s almost two centuries of music! By getting to know it, I’ve been starting to understand the birth of common practice tonality—that is, the language of chord progressions that dominates classical music, and also, to some extent, lots of modern pop music. And I can finally start thinking clearly about what makes Bach great. Imagine listening to just one rock group, or one jazz band. Practically nobody does that, of course, but imagine it: it would be impossible to distinguish what’s special to the artist you love from what’s common to the style as a whole. So you’d think they were more creative than they really are—but you also wouldn’t know what was truly creative about their work. I’m still in the midst of this project, which also includes listening to Bach more thoroughly. But I got a bit distracted. I got pulled back further in time—back into Renaissance polyphony. This gave me access to another two centuries of rich, complicated music, roughly 1400 to 1600. And it let me understand the roots of Bach’s music. Bach’s music looks back to polyphony just as much as it looks forward into the future. For this he was considered a bit old-fashioned during his day; only later did people realize his greatness. The best of Renaissance polyphony is arguably just as complex and interesting as Bach’s counterpoint: it’s just less immediately gripping. The reason is that it doesn’t follow ‘common practice’, with its familiar strategies of building and releasing tension, and its repetitive rhythmic pulse. The vocal lines are often very smooth, flowing like water. So it’s less exciting, but it’s wonderfully entrancing. I went back even further, into the medieval—but at a certain point I ‘hit bottom’, at least when it comes to the art of harmony and juggling multiple independent melodic lines, which is what I like so much about Baroque counterpoint and Renaissance polyphony. Gregorian chant is great, but it has a single vocal line. The practice of using chords was first documented around 895: in a style called the organum, Gregorian chant was supplemented by either a supporting bass line, two parallel voices singing the melody, or both. But the ‘Big Bang’ of polyphony happened around 1170 when Léonin introduced two independent melody lines. Around 1200 his follower Pérotin went ahead and started using three or four! Imagine the dizzying sense of freedom these guys must have felt, with no precedents to guide them. Medieval polyphony should be fascinating, and again it’s a huge territory: roughly two centuries, from 1200 to 1400. But right now I seem to be focusing on Renaissance polyphony—and especially the so-called Franco-Flemish school, which is an incredibly rich vein of music. I’d like to say a lot about the Franco-Flemish school, but for now I just want to list a few of their best-known composers. In fact, I’ll only mention one of each ‘generation’, and give you a sample of their music. #### Guillaume Dufay (1397 – 1497) Dufay is the most famous of the first generation of the Franco-Flemish school. (This first generation is also called the Burgundian School.) He is often considered a transitional figure from the medieval to the Renaissance. His isorhythmic motets illustrate that—their tonality is dissonant and dramatic compared to typical Renaissance polyphony: #### Johannes Ockeghem (1410/1425 – 1495) Ockeghem is the most famous of the second generation of the Franco-Flemish school. His innovations firmly moved this school out of the medieval into the world of Renaissance polyphony. Some of his compositions are almost avant-garde in their structure—a mass that asks you to sing it in any of four modes, a mass where the different vocal lines are sung at different rates and drift out of synch—but they’re carried off so smoothly you might not notice. People tend to call Ockeghem “the Bach of the 1400s”, which is sort of ridiculous because there’s just one J. S. Bach, but it gives a hint of his importance. He was not a prolific composer, but he was also an honored singer, choirmaster, and teacher. #### Josquin de Prez (~1450 – 1521) Josquin, of the third generation, is the real superstar of the Franco-Flemish school. Luther wrote that “He is the master of the notes. They must do as he wills; as for the other composers, they have to do as the notes will.” He was influenced by Ockeghem, and composed a motet in commemoration of Ockeghem’s death. But he polished Ockeghem’s ideas and in some ways simplified them, making music that was easier to understand and more popular. Despite being prolific and very influential, nothing is known of his personality, and the only known writing that may be in his own hand is a piece of graffiti on the wall of the Sistine Chapel. #### Nicholas Gombert (1495 – 1460) Gombert is probably the most famous of the fourth generation of the Franco-Flemish school. He wrote polyphonic masses and motets with as many as 6 separate melody lines, and sometimes with one voice imitating another after a very short time interval. His work marks the height of complexity of the Franco-Flemish school. #### Orlande de Lassus (1530 – 1594) Lassus, of the fifth generation of the Franco-Flemish school, is one of the composers of a style known as musica reservata—roughly, sophisticated and highly chromatic music. But he also wrote drinking songs in German, and one of his motets satirizes poor singers. He wrote over 2,000 pieces of music, all vocal—none purely instrumental! In the “The Adventure of the Bruce-Partington Plans,” Sherlock Holmes said he was working on a monograph about the polyphonic motets of Lassus. So there you are: a microscopic overview of a musical tradition lasting over 200 years. Maybe listing one just composer of each generation was a bad idea: I feel I’m doing an injustice to Gilles Binchois of the first generation, often considered the finest melodist of the 1400s, and Jacob Obrecht of the third generation, who was the most famous composer of masses in the late 1400s before Josquin came along. Obrecht was very adventurous: he often played melodies backwards (in retrograde), and once he even took the notes from a melody and played them in order of duration, long to short, to get a new melody. There is also a lot going on in Renaissance polyphony outside the Franco-Flemish school! There’s the British tradition, including great composers such as William Byrd and Thomas Tallis. And I haven’t even mentioned the most famous of all polyphonists: Giovanni Pierluigi da Palestrina! Or the most dissonant and in some ways the most intriguing of the lot: Carlo Gesualdo. But it’s easy to get lost in unfamiliar territory, so I just wanted to give a quick outline of the Franco-Flemish school. At the very least, listening to their music should be fun. ## Net Zero Carbon Emissions—A Trap? 27 April, 2021 You’ve got to read this article: • James Dyke, Robert Watson and Wolfgang Knorr, Climate scientists: concept of net zero is a dangerous trap, The Conversation, 22 April 2021. By “net zero” they mean the idea that by cutting carbon emissions and introducing technologies that suck carbon dioxide from the air, we can reach net zero carbon emissions by around 2050 and stay below 1.5° warming. This idea is built into the Paris Agreement. They’re saying this idea, especially the 2050 deadline, is mainly a way to keep business as usual going for a few more decades, to get politicians would sign the Paris Agreement. And they’re saying it won’t really work, since we have no plausible scheme to suck enough carbon dioxide from the air. I wrote earlier about how we could suck a lot of CO2 from the air: The heavy lifting in these schemes is done with plants. Planting trees helps, and improving agriculture helps, but the National Academy of Sciences thinks creating biofuels and then capturing the CO2 they emit when burned could be even bigger. The acronym-loving experts call this BECCS: bioenergy with carbon capture and storage. Right now, people involved in the Paris Agreement tend to think our only hope is an “overshoot scenario” where we put shitloads of carbon dioxide into the air before 2050 and then suck it up later. The authors of the article I want you to read say this is folly. Who are these folks, anyway? James Dyke is a senior lecturer in Global Systems at the University of Exeter. Robert Watson is a professor emeritus in Environmental Sciences at the University of East Anglia. And Wolfgang Knorr is a senior research scientist in Physical Geography and Ecosystem Science at Lund University. Since you may not read the link, even though the fate of civilization hangs in the balance, let me quote a chunk of what they say. If we had acted on Hansen’s testimony [in 1988], we would have been able to decarbonise our societies at a rate of around 2% a year in order to give us about a two-in-three chance of limiting warming to no more than 1.5°C. It would have been a huge challenge, but the main task at that time would have been to simply stop the accelerating use of fossil fuels while fairly sharing out future emissions. Four years later, there were glimmers of hope that this would be possible. During the 1992 Earth Summit in Rio, all nations agreed to stabilise concentrations of greenhouse gases to ensure that they did not produce dangerous interference with the climate. The 1997 Kyoto Summit attempted to start to put that goal into practice. But as the years passed, the initial task of keeping us safe became increasingly harder given the continual increase in fossil fuel use. It was around that time that the first computer models linking greenhouse gas emissions to impacts on different sectors of the economy were developed. These hybrid climate-economic models are known as Integrated Assessment Models. They allowed modellers to link economic activity to the climate by, for example, exploring how changes in investments and technology could lead to changes in greenhouse gas emissions. They seemed like a miracle: you could try out policies on a computer screen before implementing them, saving humanity costly experimentation. They rapidly emerged to become key guidance for climate policy. A primacy they maintain to this day. Unfortunately, they also removed the need for deep critical thinking. Such models represent society as a web of idealised, emotionless buyers and sellers and thus ignore complex social and political realities, or even the impacts of climate change itself. Their implicit promise is that market-based approaches will always work. This meant that discussions about policies were limited to those most convenient to politicians: incremental changes to legislation and taxes. I’m not sure how much the problem is one of modeling. It could be that if enough powerful people want to keep on with business as usual going, it’ll happen no matter what. But we could argue that the modelers are complicit if they don’t speak out. And indeed that’s one of the main points of this article! Anyway, continuing: Around the time they were first developed, efforts were being made to secure US action on the climate by allowing it to count carbon sinks of the country’s forests. The US argued that if it managed its forests well, it would be able to store a large amount of carbon in trees and soil which should be subtracted from its obligations to limit the burning of coal, oil and gas. In the end, the US largely got its way. Ironically, the concessions were all in vain, since the US senate never ratified the agreement. Postulating a future with more trees could in effect offset the burning of coal, oil and gas now. As models could easily churn out numbers that saw atmospheric carbon dioxide go as low as one wanted, ever more sophisticated scenarios could be explored which reduced the perceived urgency to reduce fossil fuel use. By including carbon sinks in climate-economic models, a Pandora’s box had been opened. It’s here we find the genesis of today’s net zero policies. That said, most attention in the mid-1990s was focused on increasing energy efficiency and energy switching (such as the UK’s move from coal to gas) and the potential of nuclear energy to deliver large amounts of carbon-free electricity. The hope was that such innovations would quickly reverse increases in fossil fuel emissions. But by around the turn of the new millennium it was clear that such hopes were unfounded. Given their core assumption of incremental change, it was becoming more and more difficult for economic-climate models to find viable pathways to avoid dangerous climate change. In response, the models began to include more and more examples of carbon capture and storage, a technology that could remove the carbon dioxide from coal-fired power stations and then store the captured carbon deep underground indefinitely. This had been shown to be possible in principle: compressed carbon dioxide had been separated from fossil gas and then injected underground in a number of projects since the 1970s. These Enhanced Oil Recovery schemes were designed to force gases into oil wells in order to push oil towards drilling rigs and so allow more to be recovered—oil that would later be burnt, releasing even more carbon dioxide into the atmosphere. Carbon capture and storage offered the twist that instead of using the carbon dioxide to extract more oil, the gas would instead be left underground and removed from the atmosphere. This promised breakthrough technology would allow climate friendly coal and so the continued use of this fossil fuel. But long before the world would witness any such schemes, the hypothetical process had been included in climate-economic models. In the end, the mere prospect of carbon capture and storage gave policy makers a way out of making the much needed cuts to greenhouse gas emissions. When the international climate change community convened in Copenhagen in 2009 it was clear that carbon capture and storage was not going to be sufficient for two reasons. First, it still did not exist. There were no carbon capture and storage facilities in operation on any coal fired power station and no prospect the technology was going to have any impact on rising emissions from increased coal use in the foreseeable future. The biggest barrier to implementation was essentially cost. The motivation to burn vast amounts of coal is to generate relatively cheap electricity. Retrofitting carbon scrubbers on existing power stations, building the infrastructure to pipe captured carbon, and developing suitable geological storage sites required huge sums of money. Consequently the only application of carbon capture in actual operation then—and now—is to use the trapped gas in enhanced oil recovery schemes. Beyond a single demonstrator, there has never been any capture of carbon dioxide from a coal fired power station chimney with that captured carbon then being stored underground. Just as important, by 2009 it was becoming increasingly clear that it would not be possible to make even the gradual reductions that policy makers demanded. That was the case even if carbon capture and storage was up and running. The amount of carbon dioxide that was being pumped into the air each year meant humanity was rapidly running out of time. So then people turned to another method—in theory, that is, not practice. This was BECCS: bioenergy with carbon capture and storage. With hopes for a solution to the climate crisis fading again, another magic bullet was required. A technology was needed not only to slow down the increasing concentrations of carbon dioxide in the atmosphere, but actually reverse it. In response, the climate-economic modelling community – already able to include plant-based carbon sinks and geological carbon storage in their models – increasingly adopted the “solution” of combining the two. So it was that Bioenergy Carbon Capture and Storage, or BECCS, rapidly emerged as the new saviour technology. By burning “replaceable” biomass such as wood, crops, and agricultural waste instead of coal in power stations, and then capturing the carbon dioxide from the power station chimney and storing it underground, BECCS could produce electricity at the same time as removing carbon dioxide from the atmosphere. That’s because as biomass such as trees grow, they suck in carbon dioxide from the atmosphere. By planting trees and other bioenergy crops and storing carbon dioxide released when they are burnt, more carbon could be removed from the atmosphere. With this new solution in hand the international community regrouped from repeated failures to mount another attempt at reining in our dangerous interference with the climate. The scene was set for the crucial 2015 climate conference in Paris. As its general secretary brought the 21st United Nations conference on climate change to an end, a great roar issued from the crowd. People leaped to their feet, strangers embraced, tears welled up in eyes bloodshot from lack of sleep. The emotions on display on December 13, 2015 were not just for the cameras. After weeks of gruelling high-level negotiations in Paris a breakthrough had finally been achieved. Against all expectations, after decades of false starts and failures, the international community had finally agreed to do what it took to limit global warming to well below 2°C, preferably to 1.5°C, compared to pre-industrial levels. The Paris Agreement was a stunning victory for those most at risk from climate change. Rich industrialised nations will be increasingly impacted as global temperatures rise. But it’s the low lying island states such as the Maldives and the Marshall Islands that are at imminent existential risk. As a later UN special report made clear, if the Paris Agreement was unable to limit global warming to 1.5°C, the number of lives lost to more intense storms, fires, heatwaves, famines and floods would significantly increase. But dig a little deeper and you could find another emotion lurking within delegates on December 13. Doubt. We struggle to name any climate scientist who at that time thought the Paris Agreement was feasible. We have since been told by some scientists that the Paris Agreement was “of course important for climate justice but unworkable” and “a complete shock, no one thought limiting to 1.5°C was possible”. Rather than being able to limit warming to 1.5°C, a senior academic involved in the IPCC concluded we were heading beyond 3°C by the end of this century. Instead of confront our doubts, we scientists decided to construct ever more elaborate fantasy worlds in which we would be safe. The price to pay for our cowardice: having to keep our mouths shut about the ever growing absurdity of the required planetary-scale carbon dioxide removal. The article goes on, but you get the point if you’ve read this far. Even the so-called experts on climate change are complicit in painting a rosy scenario about what will happen if we do what the Paris Agreement asks. I’m interested to see if any climate scientists step up to argue against this article—or, for that matter, agree with it. ## Dufay’s Isorhythmic Motets 23 April, 2021 I’ve been reading about Renaissance music. People sometimes say that it began in the early 1400s when musicians rebelled against the dry, complicated mathematical structures of late medieval music and switched to a more emotionally expressive style. For example, the New Oxford History of Music writes: The isorhythmic motet, the highest achievement of medieval rationalism, reached its climax during Dufay’s prentice years (c. 1410-20), with works in which the quasi-mathematical construction arouses more admiration than pleasure. But since I’m a mathematician, this actually got me interested in isorhythmic motets! I found them hard to understand from written descriptions. Isorhythm involves a rhythmic pattern called a talea which is applied to different melodies. Often the talea lasts for a different amount of time than the melody, which leads to some interesting effects. For example, here is a melody that lasts for 28 measures divided into 7 talea that each last for 4 measures: This is from a composition written sometime around 1360. Isorhythm gets a lot more complicated than this! The music typically has several parts, in which talea get sped up or slowed down independently. But reading about these things didn’t give me much of a feel for what isorhythmic motets actually sound like. When I listened to some by Guillaume Dufay, they didn’t sound dry at all! For example: It’s quite thrilling and romantic, actually! Listen to how he uses the leading-tone, the note one half a step below the tonic, to build tension. This is a big medieval thing. If you don’t know what the heck I’m talking about, wait to the very end of the piece! Here Dufay clobbers us with a loooong leading-tone, the medieval equivalent of the wail of electric guitar at the end of a classic rock song. He’s really hamming it up. Now, Dufay is famous for being the first really major Renaissance composer and breaking from the medieval traditions, so maybe his isorhythmic motets are more exciting than average. But still, I hear they are intensely mathematical. In fact there’s an album of Dufay’s isorhythmic motets with the great title Quadrivium which features a booklet by a mathematical physicist named Guido Magnano, an expert on general relativity at the University of Turin. On a Russian website I read: This album by the Italian vocal-instrumental group Cantica Symphonia takes off from the proportional aspect of large Dufay works like the motet Nuper rosarum flores, long thought to have been based on the proportions of the great cathedral of Florence but recently discovered to have probably been modeled on another building. The album was actually sponsored by the mathematics department of the University of Turin, and an essay by professor Guido Magnano explores the mathematical bases of the musical system Dufay knew. For the average listener the musical manifestations of these principles are going to be hard to hear sitting in front of your stereo; the chief interpretive decision made by Cantica Symphonia is to strive for a transparent texture, judiciously using a small instrumental ensemble to bring out structural details. Save for the fact that the voice parts are sung solo, it’s a Renaissance performance in the classic “pure” mold. For the numerologically inclined or for the serious student of the Renaissance era, the disc is an interdisciplinary goldmine. Recorded in an Italian church, the disc matches its engineering to its aims, and the packaging by the Spanish label Glossa is uncommonly attractive. I decided I needed to get my hands on that booklet. The album is on YouTube: Unfortunately the actual CD costs$47.53 on Amazon. Luckily I was able to get it for much less online from Barnes and Noble… thereby procuring the booklet, which is what I really want. This should arrive in a week or so, so with luck I’ll tell you more later. I’m also quite fascinated by Dufay and the whole Franco-Flemish school of Renaissance music that he helped start, and—as usual when I’m just starting to learn about something—I have dreams of blogging about it.

In the meantime, I found out a bit from an interview with Guido Magnano, where he says this:

Your first disc for Glossa, Quadrivium, placed special emphasis on the question of mathematical proportions in Dufay’s motets. If these considerations apply also to the works recorded here, can you provide some examples of how this came through in practice?

Guido Magnano: Mathematical proportions do not occur in medieval and Renaissance music as occasional, accessory stylistic elements: the Pythagorean-Platonic paradigm states that music itself is nothing but “auditory perception of numbers”. The hypothetical relationship between the mensural proportions of the motet Nuper rosarum flores and the proportions of Brunelleschi’s Duomo, although fascinating and quite plausible, should not obscure that in other motets, particularly in the later isorythmic motets (Fulgens iubar and Moribus et genere), Dufay attains an even higher degree of formal complexity.

The motet Magnam me gentes, (12:4:2:3) also included in this CD, has a mensural structure very close to Nuper (6:4:2:3). Worth noting is that the 15th century humanist Marsilio Ficino introduced a “Platonic-Hermetic” movement, attributing occult significance to numerical relations. Did Dufay himself share these ideas? Do the numerical ratios in his motets hide a symbolic content? Some modern scholars have claimed so, even though the pieces for which numerological interpretations have been proposed were written some thirty years before Ficino’s works, and it is impossible to obtain a conclusive proof that such interpretations reflect Dufay’s intentions.

More concretely, one could ask to what extent mathematical proportions can be perceived by the listener. The “mensural proportions” (which are but one example of numerical ratios in this music) are merely changes of meter: in an isorhythmic motet, for instance, the basic sequence of note values (talea) is repeated with all durations multiplied by a fixed ratio (e.g. 2:1, 1:2 or 2:3). Whenever the change occurs simultaneously in all voices, it can be clearly heard; in other cases, it remains hidden in the polyphonic texture. The mensural proportions also determine the ratio of the lengths of the various sections of the piece, and the choice of appropriate proportions was considered to be essential to the overall structure of the piece, much as in the Pythagorean scale where such ratios (1:2, 2:3, …) determine the consonance of a chord. As Leibniz states three centuries later: “Music is the pleasure the human mind experiences from counting without being aware that it is counting.”

Here is Nuper rosarum flores as played by Cantica Symphonia:

This is the isorhythmic motet that Dufay wrote for the consecration of the cathedral in Florence in 1436, the one with Brunelleschi’s famous dome. He was 35 then, living in Florence and working for the Pope. Later he would return to Cambrai, in what is now Northern France.

I’ve visited this cathedral and taken the terrifying tour that lets you climb up to the dome, go above it into the rafters between the dome and roof, and then out onto the roof. It’s an amazing structure:

For sheer joy, so far my favorite performance of Dufay’s isorhythmic motets is the album O gemma lux by the Huelgas Ensemble:

Here’s the man himself—Guillaume Dufay:

To be honest, nobody is completely sure whether this is Guillaume Dufay or another famous composer of the early 1400s, Gilles Binchois.

## Compositional Robotics

20 April, 2021

A bunch of us are organizing a workshop on applications of category theory to robotics, as part of the IEEE International Conference on Robotics and Automation:

2021 Workshop on Compositional Robotics: Mathematics and Tools, online, 31 May 2021. Organized by Andrea Censi, Gioele Zardini, Jonathan Lorand, David Spivak, Brendan Fong, Nina Otter, Paolo Perrone, John Baez, Dylan Shell, Jason Kane, Alexandra Nilles, Andew Spielberg, and Emilio Frazzoli.

Submit your papers here by 21 May 2021!

Here’s the idea of the workshop:

In the last decade research on embodied intelligence has seen important developments. While the complexity of robotic systems has dramatically increased, both for single robots and interacting multi-robot systems (e.g., autonomous vehicles and mobility systems), the design methods have not kept up.

The standard answer to dealing with complexity is exploiting compositionality, but there are no well-established mathematical modeling and design tools that have the reach for compositional analysis and design at the level of a complex robotic system.

The goal of this workshop is to integrate mathematical principles and practical tools for compositional robotics, with a focus on applied category theory as a meta-language to talk about compositionality.

The workshop will happen on May 31st virtually. Details will follow.

#### Session I: Mathematics and Tools for Compositionality

In the morning, some of the world’s leading experts in Applied Category Theory (ACT) will provide tutorials to present an invitation to various aspects of compositionality, both at the theoretical and the practical level. In particular, Dr. Jonathan Lorand will teach Category Theory basics, Dr. David Spivak and Dr. Brendan Fong will introduce the audience to the concept of compositionality, Prof. John Baez will explain how the previously defined concepts can be used when modeling various types of systems, and Dr. Andrea Censi will present the theory of co-design, tailored to robotic applications.

#### Session II: Keynote Talks and Open Contributions

The afternoon session features two keynotes on the application of compositionality in robotics:

• Prof. Aaron Ames, Bren Professor of Mechanical and Civil Engineering and Control and Dynamical Systems, California Institute of Technology.

• Prof. Daniel Koditschek, Alfred Fitler Moore Professor of Electrical & Systems Engineering, School of Engineering & Applied Science, University of Pennsylvania. Prof. Koditschek will be assisted by Dr. Paul Gustafson (Wright State University) and Dr. Matthew Kvalheim (University of Pennsylvania).

Both speakers are leading experts in their fields and have succesfully applied category theory and compositionality to real challenges in robotics. Finally, we plan for eight talk-slots for open submissions. Submissions should focus on mathematical perspectives (not limited to ACT) and applications of compositionality.

## Applied Category Theory 2021 — Call for Papers

16 April, 2021

The deadline for submitting papers is coming up soon: May 12th.

Fourth Annual International Conference on Applied Category Theory (ACT 2021), July 12–16, 2021, online and at the Computer Laboratory of the University of Cambridge.

Plans to run ACT 2021 as one of the first physical conferences post-lockdown are progressing well. Consider going to Cambridge! Financial support is available for students and junior researchers.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying many different kinds of systems using category-theoretic tools. These systems are found across computer science, mathematics, and physics, as well as in social science, linguistics, cognition, and neuroscience. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, disseminate the latest results, and facilitate further development of the field.

We accept submissions of both original research papers, and work accepted/submitted/ published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume. The keynote addresses will be drawn from the best accepted papers. The conference will include an industry showcase event.

We hope to run the conference as a hybrid event, with physical attendees present in Cambridge, and other participants taking part online. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please do not book your travel or hotel accommodation yet.

### Financial support

We are able to offer financial support to PhD students and junior researchers. Full guidance is on the webpage.

### Important dates (all in 2021)

• Submission Deadline: Wednesday 12 May
• Author Notification: Monday 7 June
• Financial Support Application Deadline: Monday 7 June
• Financial Support Notification: Tuesday 8 June
• Priority Physical Registration Opens: Wednesday 9 June
• Ordinary Physical Registration Opens: Monday 13 June
• Reserved Accommodation Booking Deadline: Monday 13 June
• Adjoint School: Monday 5 to Friday 9 July
• Main Conference: Monday 12 to Friday 16 July

### Submissions

The following two types of submissions are accepted:

Proceedings Track. Original contributions of high-quality work consisting of an extended abstract, up to 12 pages, that provides evidence of results of genuine interest, and with enough detail to allow the program committee to assess the merits of the work. Submission of work-in-progress is encouraged, but it must be more substantial than a research proposal.

Non-Proceedings Track. Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure that these pages contain sufficient evidence of the quality and rigour of your work.

Papers in the two tracks will be reviewed against the same standards of quality. Since ACT is an interdisciplinary conference, we use two tracks to accommodate the publishing conventions of different disciplines. For example, those from a Computer Science background may prefer the Proceedings Track, while those from a Mathematics, Physics or other background may prefer the Non-Proceedings Track. However, authors from any background are free to choose the track that they prefer, and submissions may be moved from the Proceedings Track to the Non-Proceedings Track at any time at the request of the authors.

Contributions must be submitted in PDF format. Submissions to the Proceedings Track must be prepared with LaTeX, using the EPTCS style files available at http://style.eptcs.org.

The submission link will soon be available on the ACT2021 web page: https://www.cl.cam.ac.uk/events/act2021

### Program Committee

Chair:

• Kohei Kishida, University of Illinois, Urbana-Champaign

Members:

• Richard Blute, University of Ottawa
• Spencer Breiner, NIST
• Daniel Cicala, University of New Haven
• Robin Cockett, University of Calgary
• Bob Coecke, Cambridge Quantum Computing
• Geoffrey Cruttwell, Mount Allison University
• Valeria de Paiva, Samsung Research America and University of Birmingham
• Brendan Fong, Massachusetts Institute of Technology
• Jonas Frey, Carnegie Mellon University
• Tobias Fritz, Perimeter Institute for Theoretical Physics
• Fabrizio Romano Genovese, Statebox
• Helle Hvid Hansen, University of Groningen
• Jules Hedges, University of Strathclyde
• Chris Heunen, University of Edinburgh
• Alex Hoffnung, Bridgewater
• Martti Karvonen, University of Ottawa
• Kohei Kishida, University of Illinois, Urbana -Champaign (chair)
• Martha Lewis, University of Bristol
• Bert Lindenhovius, Johannes Kepler University Linz
• Ben MacAdam, University of Calgary
• Dan Marsden, University of Oxford
• Jade Master, University of California, Riverside
• Joe Moeller, NIST
• Koko Muroya, Kyoto University
• Simona Paoli, University of Leicester
• Daniela Petrisan, Université de Paris, IRIF
• Peter Selinger, Dalhousie University
• Michael Shulman, University of San Diego
• David Spivak, MIT and Topos Institute
• Joshua Tan, University of Oxford
• Dmitry Vagner
• Jamie Vicary, University of Cambridge
• John van de Wetering, Radboud University Nijmegen
• Vladimir Zamdzhiev, Inria, LORIA, Université de Lorraine
• Maaike Zwart

## Black Dwarf Supernovae

14 April, 2021

“Black dwarf supernovae”. They sound quite dramatic! And indeed, they may be the last really exciting events in the Universe.

It’s too early to be sure. There could be plenty of things about astrophysics we don’t understand yet—and intelligent life may throw up surprises even in the very far future. But there’s a nice scenario here:

• M. E. Caplan, Black dwarf supernova in the far future, Monthly Notices of the Royal Astronomical Society 497 (2020), 4357–4362.

First, let me set the stage. What happens in the short run: say, the first 1023 years or so?

For a while, galaxies will keep colliding. These collisions seem to destroy spiral galaxies: they fuse into bigger elliptical galaxies. We can already see this happening here and there—and our own Milky Way may have a near collision with Andromeda in only 3.85 billion years or so, well before the Sun becomes a red giant. If this happens, a bunch of new stars will be born from the shock waves due to colliding interstellar gas.

By 7 billion years we expect that Andromeda and the Milky Way will merge and form a large elliptical galaxy. Unfortunately, elliptical galaxies lack spiral arms, which seem to be a crucial part of the star formation process, so star formation may cease even before the raw materials run out.

Of course, no matter what happens, the birth of new stars must eventually cease, since there’s a limited amount of hydrogen, helium, and other stuff that can undergo fusion.

This means that all the stars will eventually burn out. The longest lived are the red dwarf stars, the smallest stars capable of supporting fusion today, with a mass about 0.08 times that of the Sun. These will run out of hydrogen about 10 trillion years from now, and not be able to burn heavier elements–so then they will slowly cool down.

(I’m deliberately ignoring what intelligent life may do. We can imagine civilizations that develop the ability to control stars, but it’s hard to predict what they’ll do so I’m leaving them out of this story.)

A star becomes a white dwarf—and eventually a black dwarf when it cools—if its core, made of highly compressed matter, has a mass less than 1.4 solar masses. In this case the core can be held up by the ‘electron degeneracy pressure’ caused by the Pauli exclusion principle, which works even at zero temperature. But if the core is heavier than this, it collapses! It becomes a neutron star if it’s between 1.4 and 2 solar masses, and a black hole if it’s more massive.

In about 100 trillion years, all normal star formation processes will have ceased, and the universe will have a population of stars consisting of about 55% white dwarfs, 45% brown dwarfs, and a smaller number of neutron stars and black holes. Star formation will continue at a very slow rate due to collisions between brown and/or white dwarfs.

The black holes will suck up some of the other stars they encounter. This is especially true for the big black holes at the galactic centers, which power radio galaxies if they swallow stars at a sufficiently rapid rate. But most of the stars, as well as interstellar gas and dust, will eventually be hurled into intergalactic space. This happens to a star whenever it accidentally reaches escape velocity through its random encounters with other stars. It’s a slow process, but computer simulations show that about 90% of the mass of the galaxies will eventually ‘boil off’ this way — while the rest becomes a big black hole.

How long will all this take? Well, the white dwarfs will cool to black dwarfs in about 100 quadrillion years, and the galaxies will boil away by about 10 quintillion years. Most planets will have already been knocked off their orbits by then, thanks to random disturbances which gradually take their toll over time. But any that are still orbiting stars will spiral in thanks to gravitational radiation in about 100 quintillion years.

I think the numbers are getting a bit silly. 100 quintillion is 1020, and let’s use scientific notation from now on.

Then what? Well, in about 1023 years the dead stars will actually boil off from the galactic clusters, not just the galaxies, so the clusters will disintegrate. At this point the cosmic background radiation will have cooled to about 10-13 Kelvin, and most things will be at about that temperature unless proton decay or some other such process keeps them warmer.

Okay: so now we have a bunch of isolated black holes, neutron stars, and black dwarfs together with lone planets, asteroids, rocks, dust grains, molecules and atoms of gas, photons and neutrinos, all very close to absolute zero.

I had a dream, which was not all a dream.
The bright sun was extinguishd, and the stars
Did wander darkling in the eternal space,
Rayless, and pathless, and the icy earth
Swung blind and blackening in the moonless air.

— Lord Byron

So what happens next?

We expect that black holes evaporate due to Hawking radiation: a solar-mass one should do so in 1067 years, and a really big one, comparable to the mass of a galaxy, should take about 1099 years. Small objects like planets and asteroids may eventually ‘sublimate’: that is, slowly dissipate by losing atoms due to random processes. I haven’t seen estimates on how long this will take. For larger objects, like neutron stars, this may take a very long time.

But I want to focus on stars lighter than 1.2 solar masses. As I mentioned, these will become white dwarfs held up by their electron degeneracy pressure, and by about 1017 years they will cool down to become very cold black dwarfs. Their cores will crystallize!

Then what? If a proton can decay into other particles, for example a positron and a neutral pion, black dwarfs may slowly shrink away to nothing due to this process, emitting particles as they fade away! Right now we know that the lifetime of the proton to decay via such processes is at least 1032 years. It could be much longer.

But suppose the proton is completely stable. Then what happens? In this scenario, a very slow process of nuclear fusion will slowly turn black dwarfs into iron! It’s called pycnonuclear fusion. The idea is that due to quantum tunneling, nuclei next to each other in the crystal lattice within a black dwarf will occasionally get ‘right on top of each other’ and fuse into heavier nucleus! Since iron-56 is the most stable nucleus, eventually iron will predominate.

Iron is more dense than lighter elements, so as this happens the black dwarf will shrink. It may eventually shrink down to being so dense that electron pressure will no longer hold it up. If this happens, the black dwarf will suddenly collapse, just like heavier stars. It will release a huge amount of energy and explode as gravitational potential energy gets converted into heat. This is a black dwarf supernova.

When will black dwarf supernovae first happen, assuming proton decay or some other unknown processes don’t destroy the black dwarfs first?

This is what Matt Caplan calculated:

We now consider the evolution of a white dwarf toward an iron black dwarf and the circumstances that result in collapse. Going beyond the simple order of magnitude estimates of Dyson (1979), we know pycnonuclear fusion rates are strongly dependent on density so they are greatest in the core of the black dwarf and slowest at the surface. Therefore, the internal structure of a black dwarf evolving toward collapse can be thought of as an astronomically slowly moving ‘burning’ front growing outward from the core toward the surface. This burning front grows outward much more slowly than any hydrodynamical or nuclear timescale, and the star remains at approximately zero temperature for this phase. Furthermore, in contrast to traditional thermonuclear stellar burning, the later reactions with higher Z parents take significantly longer due to the larger tunneling barriers for fusion.

Here “later reactions with higher Z parents” means fusion reactions involving heavier nuclei. The very last step, for example, is when two silicon nuclei fuse to form a nucleus of iron. In an ordinary star these later reactions happen much faster than those involving light nuclei, but for black dwarfs this pattern is reversed—and everything happens at ridiculously slow rate, at a temperature near absolute zero.

He estimates a black dwarf of 1.24 solar masses will collapse and go supernova after about 101600 years, when roughly half its mass has turned to iron.

Lighter ones will take much longer. A black dwarf of 1.16 solar masses could take 1032000 years to go supernova.

These black dwarf supernovae could be the last really energetic events in the Universe.

It’s downright scary to think how far apart these black dwarfs will be when they explode. As I mentioned, galaxies and clusters will have long since have boiled away, so every black dwarf will be completely alone in the depths of space. Distances between them will be doubling every 12 billion years according to the current standard model of cosmology, the ΛCDM model. But 12 billion years is peanuts compared to the time scales I’m talking about now!

So, by the time black dwarfs start to explode, the distances between these stars will be expanded by a factor of roughly

$\displaystyle{ e^{10^{1000}} }$

compared to their distances today. That’s a very rough estimate, but it means that each black dwarf supernova will be living in its own separate world.

## The Expansion of the Universe

9 April, 2021

We can wait a while to explore the Universe, but we shouldn’t wait too long. If the Universe continues its accelerating expansion as predicted by the usual model of cosmology, it will eventually expand by a factor of 2 every 12 billion years. So if we wait too long, we can’t ever reach a distant galaxy.

In fact, after 150 billion years, all galaxies outside our Local Group will become completely inaccessible, in principle by any form of transportation not faster than light!

• Toby Ord, The edges of our Universe.

This is where I got the table.

150 billion years sounds like a long time, but the smallest stars powered by fusion—the red dwarf stars, which are very plentiful—are expected to last much longer: about 10 trillion years!  So, we can imagine a technologically advanced civilization that has managed to spread over the Local Group and live near red dwarf stars, which eventually regrets that it has waited too long to expand through more of the Universe.

The Local Group is a collection of roughly 50 nearby galaxies containing about 2 trillion stars, so there’s certainly plenty to do here. It’s held together by gravity, so it won’t get stretched out by the expansion of the Universe—not, at least, until its stars slowly “boil off” due to some randomly picking up high speeds. But will happen much, much later: more than 10 quintillion years, that is, 1019 years.

## The Koide Formula

4 April, 2021

There are three charged leptons: the electron, the muon and the tau. Let $m_e, m_\mu$ and $m_\tau$ be their masses. Then the Koide formula says

$\displaystyle{ \frac{m_e + m_\mu + m_\tau}{\big(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\big)^2} = \frac{2}{3} }$

There’s no known reason for this formula to be true! But if you plug in the experimentally measured values of the electron, muon and tau masses, it’s accurate within the current experimental error bars:

$\displaystyle{ \frac{m_e + m_\mu + m_\tau}{\big(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\big)^2} = 0.666661 \pm 0.000007 }$

Is this significant or just a coincidence? Will it fall apart when we measure the masses more accurately? Nobody knows.

Here’s something fun, though:

Puzzle. Show that no matter what the electron, muon and tau masses might be—that is, any positive numbers whatsoever—we must have

$\displaystyle{ \frac{1}{3} \le \frac{m_e + m_\mu + m_\tau}{\big(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau}\big)^2} \le 1}$

For some reason this ratio turns out to be almost exactly halfway between the lower bound and upper bound!

Koide came up with his formula in 1982 before the tau’s mass was measured very accurately.  At the time, using the observed electron and muon masses, his formula predicted the tau’s mass was

$m_\tau$ = 1776.97 MeV/c2

while the observed mass was

$m_\tau$ = 1784.2 ± 3.2 MeV/c2

Not very good.

In 1992 the tau’s mass was measured much more accurately and found to be

$m_\tau$ = 1776.99 ± 0.28 MeV/c2

Much better!

Koide has some more recent thoughts about his formula:

• Yoshio Koide, What physics does the charged lepton mass relation tell us?, 2018.

He points out how difficult it is to explain a formula like this, given how masses depend on an energy scale in quantum field theory.