Self-Consistent Elysian Fields

Material Intelligence as a Field of Fields

Mission first, then myth

You believe, of course," I suggested, "that much remains to be discovered in the realm of electricity?"

"I continue to find my greatest pleasure, and so my reward, in the work that precedes what the world calls success."

"It is the field of fields," he answered. "We can't talk of that, but it holds the secrets which will reorganize the life of the world."

"You have discovered much about it," I said, smiling.

"Yes," he said, "and yet very little in comparison with the possibilities that appear."

The quote above is an excerpt from a Thomas Edison interview in the book How they Succeeded in which he and his "muckers", having invented the underlying infrastructure to practically exploit a technology as foundational as electricity, opened up a new frontier– a terra incognita he envisioned as a field of fields.

And whenever there is unexplored territory, there is war

,as there was in The War of the Currents, (AC vs DC) which:

1. still rages on till this day and,

2. which may eventually reach a hybrid armistice, a minimum energy state predicated on the very means of exploring and exploiting said territory

It would be good to recall that electricity transitioned from a scientific curiosity to an essential tool for modern life– so did the Internet. We must then ask ourselves this question:

In this era of convergence, what scientific curiosities hold secrets that will reorganize the life of the world?

To me– and consistent with my love for scientific history– natural philosophy first started (ab initio) when Democritus (the laughing philosopher) and Leucippus coined “atomos” –meaning uncuttable– as the point at which the philosophical cheese could not be cut into still smaller pieces. Empedocles described matter as consisting not of atoms, but of attractive and repulsive forces (like love and hate). Plato described matter as consisting of elements with polyhedral shapes, but without forces between them. Aristotle rejected all the above, as leading to paradoxes, but offered no mechanism in replacement. It could be very well said that physics was too inchoate, and characterization was impossible at the time– one had to wait for accurate physical and mathematical observations in order to propose accurate laws.

Galileo Galilei then pioneered the scientific method, using the refracting telescope to make astronomical discoveries, culminating in his work on Dialogue Concerning the Two Chief World Systems and anointing him as the father of astronomy. It could very well be said that this was the very first scientific confrontation with the Leviathan of religion because his work alienated the Jesuits and the Pope, who had supported his work prior. The fractal nature of history is reflected in the fight between Aristotle and Galileo on the topic of geocentrism versus heliocentrism. But I digress.

Galileo arguably pioneered modern-day materials science (through his studies on the strength and internal cohesion of materials and their resistance to fracture), and modern day physics ( through his work on kinematics and projectiles upending the Aristotelian school of thought). These can be found in his final book, Discourses and Mathematical Demonstrations Relating to Two New Sciences.

On the idea of interatomic interactions, he– or rather his Socratic jouster–, was against the idea (probably because he was a contrarian of Aristotelianism):

“To begin with let me confess that I do not understand how these large globules of water stand out and hold themselves up, although I know for a certainty, that it is not owing to any internal tenacity acting between the particles of water; whence it must follow that the cause of this effect is external”

It is arguably easier to observe planets than atoms

(case-in-point Tycho Brahe and Johannes Kepler)

Isaac Newton, then used the inverse-square law to not only rationalize all Kepler’s laws, but to explain gravity, a great unification between astronomy and down-to-earth events. He shied away from using them to explain interatomic interactions:

“It seems to me farther, that these Particles have not only a Vis inertiae, accompanied with such passive Laws of Motion as naturally result from that Force, but also that they are moved by certain active Principles, such as that of Gravity, and that which causes Fermentation, and the Cohesion of Bodies. These Principles I consider, not as occult Qualities, supposed to result from the specifick Forms of Things, but as general Laws of Nature, by which the Things themselves are form'd; their Truth appearing to us by Phaenomena, though their Causes be not yet discover'd. For these are manifest Qualities, and their Causes only are occult. And therefore I scruple not to propose the Principles of Motion above-mention'd, they being of very general Extent, and leave their Causes to be found out.”

Roger Boscovich, being a Jesuit and in (stereo)typical Jesuit fashion, described the unification of repulsive and attractive ranges within one force. Inspired by the ideas of Newton’s rival Leibniz, he developed a graphical model with a force alternatively repulsive and attractive between point particles, depending on a distance r between them.

Pierre-Simon Laplace, in line with blue-sky thinking, dreamed of a unified theory of everything, to expose the system of the world by “[bringing] back all phenomena in physics and astronomy to only one general law” as he put forward in Exposition du système du monde:

“I have wanted to establish that the phenomena of nature reduce in the final analysis to action-at-a-distance from molecule to molecule and that the consideration of these actions ought to serve as the basis of the mathematical theory of these phenomena.”

He used capillarity to indirectly study the orders of magnitude of range over which forces between molecules were “sensible” versus “insensible”. Laplace thought that there were attractive forces between molecules, as well as caloric, a mysterious fluid between molecules that tended to keep them apart. Alas, his program arrived too soon. Its lack of success prevented many people from pursuing the path of kinetic theory because it was not consistent and it was no better than what competitive theories could do.

Ludwig Boltzmann (who developed statistical mechanics) believed in the existence of atoms– laying the groundwork for understanding the statistical behavior of particles in systems with a large number of degrees of freedom –and that energy levels of physical systems could be discrete, a forerunner to the development of quantum mechanics.

Johannes Diderik van der Waals, also believed in atoms, and introduced the idea of parameters to characterize molecular size, and attraction, building the very foundations of molecular physics by using many-parameter equations of state.

Funnily enough, Max Planck, father of quantum theory, did not believe in the existence of atoms for a while, and was more in favor of a continuous energetic model, a field of harmonic oscillators which absorbed electromagnetic radiation, changing their energies in discrete, rather than continuous values.

As you may be able to tell, there was a transition between observational means of describing systems to more mathematically rigorous, physically consistent, and directly analytical means.

James Clark Maxwell, inspired by Michael Faraday’s work in describing physical systems as held together by lines of force, described the viscosity of gases using purely repulsive tubes of force. Mie used a purely repulsive potential to study the compressibility of elementary solids, in order to develop a lattice model for liquids– fluids (liquids and gases) were extensively studied but not much attention was given to solids. This was around the same time as the Industrial Revolution.

The first mathematical instance of the combination of repulsive and attractive energies towards some equilibrium was deduced by Eduard Grüneisen who explicitly considered this mathematical form (which resembles Lennard-Jones law) in studying elementary solids:

Sir John Edward Lennard Jones, in studying gas viscosities more accurately, developed analytical formulae for describing the lattice energies of various crystalline structures in 1925, during the Golden Age of Quantum Physics. He would later on go to develop the foundations of molecular orbital theory, and inspired by Douglas Hartree’s idea of the self-consistent field and Vladimir Fock’s corrections based on identical electrons + exchange energy, develop the underpinnings of computational chemistry and computational materials science. These can be found in the classical paper: Wave Functions of Many-Electron Atoms.

A self-consistent field is a representation of equations, constructed for each electron [or particle] and solved numerically so as to make the distribution of charge in the atom reproduce the potential field already assumed.

In his Lectures on Physics, Richard Feynman wondered what statement would contain the most information in the fewest words, if all of scientific knowledge were to be destroyed in some cataclysm, and only one sentence could be passed to the next generations of scientists:

“I believe it is the atomic hypothesis that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.”

Two-body potentials reflect this generic behavior, except those describing two ions carrying a charge of identical sign, since they repel each other even at large separation. From this very statement by Feynman, we can begin to understand the transition from interatomic potentials to fields.

Interatomic potentials approximate the potential energy of atoms as a function of their coordinates. Forces on atoms, as needed in molecular-dynamics simulations, can be obtained by calculating the gradient of these functions with respect to the nuclear coordinates.

Real powerful potentials should mimic nature correctly and account for charge transfer in the appropriate way, so the notion that forces arise as an unambiguous function of nuclear coordinates isn’t all that true, and although the idea of the potential does not come from quantum mechanics, it can be justified by it, as seen by JC Slater in The Self-Consistent Field and the Structure of the Atom, although past a certain point, certain things need to be accounted for.

The root of many-body potentials is that two atoms change their interaction when additional atoms are present, because their electronic structure changes. For systems with more than two atoms, other emergent phenomena occur, and so precisely defining them has to be done by accounting for this emergent behavior using equations of state containing parameters, specific to the kind of system being studied. For instance, two hydrogen atoms prefer to form a strong covalent bond between them rather than to remain lonely. However, as soon as an oxygen enters the scene, the hydrogens happily form a heteronuclear water molecule with the oxygen.

To account for cohesive properties beyond pair potentials, the solution came from solid state quantum mechanics by working in reciprocal space, an ideal space requiring perfect periodicity in perfect crystals which contains a lot of the electron information, but does not account for defects which occurred in real space.

This resulted in the dichotomy between materials science (dirty physics) and solid-state physics (ideal physics), which was reconciled by Jacques Friedel (father of the Friedel-Crafts reaction for cracking oil) by paving the way back from reciprocal space to real space. As you can tell, physics with defects is more complicated than physics of ideal crystals, and natural systems aren’t at all ideal. The need for simple ideas as a foundation is however necessary in order to derive more complex models.

The idea of atoms having to share their electronic energies starting from– and beyond their nearest neighbors proliferated in the 1980s where embedded atoms in an effective medium had embedded energies. Many-body interatomic potentials, as such, can closely approximate real systems, because they are described by many-parameter functions. By describing them in terms of vectors, scalars, and tensors that have values at each point in space and time, we come up with force fields. It is worth emphasizing the possibly obvious point that a force field is nothing but a (possibly very large) collection of functional forms and associated constants. With that collection in hand, the energy of a given molecule (whose atomic connectivity must in general be specified) can be evaluated by computing the energy associated with every defined type of interaction occurring in the molecule. Because there are typically a rather large number of such interactions, the process is facilitated by computation, but the mathematics is really extraordinarily simple and straightforward.

Of course, these parameters need to be optimized, and so the use of optimization algorithms, efficiently based on large databases has arisen to solve these problems. This is the very spirit of machine-learned interatomic potentials, occurring at the intersection between computation, artificial intelligence, and materials science. Now, the idea of optimization– choosing the best solution to a complex problem originally stemmed from war and is the very spirit of operations research (the development of decision-support tools, methods and models to increase awareness and to improve decision-making). The tenets of operations research are:

1. Define a cost function using parameters derived from huge troves of data

2. Have an efficient procedure to search globally for the best solution

What is our modern day terra incognita, and what is this new war?

As for you, Zeus-fostered Menelaus, it's not ordained that you will meet your fate and die in horse-rich Argos. No. The gods will send you off to the Elysian fields, and to the outer limits of the earth— the place where fair-haired Rhadamanthus lives and life for human beings is really easy— there's no snow or heavy storms or even rain, and Oceanus sends a steady breeze, as West Wind blows to keep men cool and fresh. Helen is your wife—that's why they'll do this, because they see you as the man who married Zeus daughter.

Homer, The Odyssey

In Greek mythology, Elysium– or the Elysian Fields– was described as the resting place of the heroic (warriors) and the virtuous (those favored by the gods); it evolved from a designation of a place struck by lightning (enelysios). Homer, in so many words, effectively described it as a land of milk and honey, a field of fields if I may, an idyllic resting place, perfect for an afterlife. But as I mentioned before, whenever there is unexplored territory, there is war. Elysium is also seen as an eternal paradise, where warriors, heroes, and kings relive their living days by engaging in recreational combat.

The beauty of discovery is that it allows us to contrive new forms of local competition, far removed from global physical warfare, but still profoundly affecting it.

So as not to bury the lede, I’ll just state it plainly here: the new war will occur on the field of (force) fields, but it’ll be self-consistent and non-zero sum, in that crunching these numbers, which can then be easily translated into physical space, lifts the very floor of civilization by repositioning every individual, thereby improving their standards of living; the more we are good at it, the better our decision loops, and the more we converge toward Pareto-preferred futures. This is the final war beyond wars of scarcity.

But of course, getting to these Elysian Fields requires a new kind of game between scientists defined in the context of their internal incentives, this new virtual environment, their physical environment, and the broader (geo)political environment. Enter Bordieu’s field theory.

The field is one of the core concepts used by French social scientist Pierre Bourdieu. In his formulation, a field is a setting in which agents and their social positions are located. The position of each particular agent in the field is a result of interaction between the specific rules of the field, their perception, and their capital (social, economic and cultural). Fields interact with each other, and are hierarchical: most are subordinate to the larger field of power and class relations. More complex societies have more fields and more relations between fields.

Fields are constructed according to underlying nomos, fundamental principles of "vision and division" or organizing "laws" of experience that govern practices and experiences within a field. The nomos underlying one field is often irreducible to those underlying another, as in the noted disparity between the nomos of the aesthetic field that values cultural capital and in some sense discourages economic capital, and that of the economic field which values economic capital. Agents subscribe to a particular field not by way of explicit contract, but by their practical acknowledgement of the stakes, implicit in the very "playing of the game.

Fields feature different positions that agents can occupy. The dominant players in the field, called the incumbents, are generally invested in maintaining the field in its current form, as changes to the rules of competition risk destabilizing their dominant position. Fields may also feature insurgents who instead aim to alter the field so they can successfully compete with the incumbents. Dramatic change in previously stable fields can come from either successful insurgents, intrusion from other fields, or from government-imposed rule change.

Unstable fields are defined by rapid change and frequently by destructive forms of competition, such as pure competition over prices that drives profit margins to untenably low levels. Fields thus need to be stabilized with rules which make sure that competition takes non-destructive forms. Stable fields rarely emerge on their own, but must be constructed by skilled entrepreneurs. The government, through policy, plays (or attempts to play) this very role.

In this very final war, its very self-consistent nature confines the war for natural resources and raw materials to virtual space, because its very rules will result in recursive insurgence, agents out-innovating one another till humanity reaches its minimum energy state at which the resultant building blocks become the dominant design, contingent on top-down policy, bottom-up procedures, and mesoscale processes interfering constructively to drive us towards preferred futures. The interaction between the scientific field and the field of force fields will likely result in the Elysian fields… or they may not.

To me, the beauty of discovery is that it allows us to contrive new forms of local competition, far removed from global physical warfare, but still profoundly affecting it. Although, every once in a while, what happens in the dark recesses of a university research lab affects the world ever so profoundly.

Mission first, then myth.