Sep 262004

Part 1
Part 2

We’ve established that thermodynamics is based on two fundamental empirical laws: the first law (conservation of energy) and the second law (the entropy law). Any systematic scheme for the description of a physical process (equilibrium or non-equilibrium, discrete or continuous) must also be built upon these two laws. Here we employ a simplified but instructive model of probability that captures our reasoning without overwhelming us with details.

Eustace, our thermodynamic system, hops off the turnip truck with $100 and walks into a casino. The simplest and cheapest games are played downstairs. The complexity and stakes of the games increase with each floor but in order to play, a guest must have the minimum ante.

The proprietor leads Eustace to a small table in the basement and offers him the following proposition. He flips a coin. For every heads, Eustace wins a dollar. For every tails, Eustace loses a dollar. If Eustace accumulates $200, he gets to go upstairs and play a more sophisticated game, like blackjack. If he goes broke he’s back out in the street.

How do we book his chances? Depends on the coin, of course. If it’s fair, then the probability of winning a single trial, p, is .5. And the probability of winning the $100, P, also turns out to be .5, or 50%. Which is just what you’d expect.

But in Las Vegas the coins aren’t usually fair. Suppose the probability of heads is .495, or .49, or .47? What then? James Bernoulli, of the Flying Bernoulli Brothers (and fathers and sons), solved this problem, in the general case, more than three hundred years ago, and his result might save you money.

One round

Chance to win $100

Average # rounds

Unreal. A lousy half a percent bias reduces your chances of winning by a factor of 4. And if the bias is 3%, as it is in many real bets in Vegas, such as black or red on the roulette wheel, you may as well just hand the croupier your money. There is a famous Las Vegas story of a British earl who walked into a casino with half a million dollars, changed it for chips, went to the roulette wheel, bet it all on black, won, and cashed out, never to be seen in town again. This apocryphal earl had an excellent intuitive grasp of probability. He was approximately 80,000 times as likely to win half a million that way as he would have been by betting, say, $5,000 at a time.

Bernoulli trials, the mathematical abstraction of coin tossing, are one of the simplest yet most important random processes in probability. Such a sequence is an indefinitely repeatable series of events each with the same probability (p). More formally, it satisfies the following conditions:

  • Each trial has two possible outcomes, generically called success and failure.
  • The trials are independent. The outcome of one trial does not influence the outcome of the next.
  • On each trial, the probability of success is p and the probability of failure is 1 – p.

Now, what this has to do with α theory is, in a word, everything: Eustace’s thermodynamic state changes can be modeled as a series of Bernoulli trials. Instead of tracking Eustace’s monetary wealth, we’ll track his alpha wealth. And instead of following him through increasingly luxurious levels of the casino, we’ll follow him through increasing levels of complexity, stability and chemical kinetics.

Inside Eustace molecules whiz about. Occasionally there is a transforming collision. We know that for each such collision there are thermodynamic consequences that allow us to calculate alpha. When the alpha of a system increases, its complexity and stability also increase. The probability of success, p, is the chance that a given reaction occurs.

Each successful reaction may create products that have the ability to enable other reactions, in a positive feedback loop, because complex molecules have more ways to interact chemically than simple ones. It takes alpha to make alpha, just as it takes money to make money.

At each floor of the casino, the game begins anew but with greater wealth and a different p. Enzymes, for example, which catalyze reactions, often by many orders of magnitude, can be thought of simply as extreme Bernoulli biases, or increases in p.

If you’ve ever wondered how life sprang from the primordial soup, well, this is how. Remember that Eustace is any arbitrary volume of space through which energy can flow. Untold numbers of Eustaces begin in the basement — an elite few eventually reach the penthouse suite.

Richard Dawkins, in The Blind Watchmaker, describes the process well, if a bit circuitously, since he spares the math. Tiny changes in p produce very large changes in ultimate outcomes, provided you engage in enough trials. And we are talking about a whole lot of trials here. Imagine trillions upon trillions of Eustaces, each hosting trillions of Bernoulli trials, and suddenly the emergence of complexity seems a lot less mysterious. You don’t have to be anywhere near as lucky as you think. Of course simplicity can emerge from complexity too. No matter how high you rise in Casino Alpha, you can always still lose the game.

Alpha theory asserts that the choreographed arrangements we observe today in living systems are a consequence of myriad telescoping layers of alphatropic interactions; that the difference between such systems and elementary Eustaces is merely a matter of degree. Chemists have understood this for a long time. Two hundred years ago they believed that compounds such as sugar, urea, starch, and wax required a mysterious “vital force” to create them. Their very terminology reflected this. Organic, as distinct from inorganic, chemistry was the study of compounds possessing this vital force. In 1828 Friedrich Wohler converted ammonia and cyanate into urea simply by heating the reactants in the absence of oxygen:

NH4 + OCN –> CO(NH2)2

Urea had always come from living organisms, it was presumed to contain the vital force, and yet its precursors proved to be inorganic. Some skeptics claimed that a trace of vital force from Wohler’s hands must have contaminated the reaction, but most scientists recognized the possibility of synthesizing organic compounds from simpler inorganic precursors. More complicated syntheses were carried out and vitalism was eventually discarded.

More recently, scientists grappled with how such reactions that sometimes require extreme conditions can take place in a cell. In 1961, Peter Mitchell proposed that the synthesis of ATP occurred due to a coupling of the chemical reaction with a proton gradient across a membrane. This hypothesis was quickly verified by experiment, and he received a Nobel prize for his work.

I claimed in Part 2 that changes in α can be measured in theory. Thanks to chemical kinetics and probability, they can be pretty well approximated in practice too. Soon, very soon, we will come to assessing the α consequences of actual systems, and these tools will prove their mettle.

One more bit of preamble, in which it will be shown that all randomness is not equally random, and we will begin to infringe philosophy’s sacred turf.

Sep 072004

Tomorrow. Did I say tomorrow? I meant the last syllable of recorded time.

Now where were we? Oh yes: nowhere. Philosophy to date has yielded no explanations, no predictions, no tools unless we classify logic generously, and very little practical advice, much of it bad. And then from “cogito ergo sum,” or “existence exists,” philosophers expect to explain the world, or at least a good chunk of it. Tautologies unpack only so far. No matter how much you cram into a suitcase, you cannot expect to fill a universe with its contents.

I was a little unfair to the Greeks in Part 1. They didn’t have 300 years of dazzling scientific advance to build on. What they had was nothing at all, and as Eddie Thomas pointed out in the comments, you have to start somewhere. But 2500 years later, do we have to start from nothing all over again? In what follows I will take for granted that the external world exists, that we are capable of knowing it, and doubtless many other truths of metaphysics and epistemology that everyone knows but philosophers still hotly dispute. If you want to argue that stuff, the comments are still raging on Part 1.

I propose to begin with the First and Second Laws of thermodynamics. You can follow the links for some helpful refreshers, but in brief, the First Law states that energy is always conserved. It is neither created nor destroyed, merely transferred. And since we know, from relativistic physics, that matter is merely energy in another form, we conclude that everything that ever happens is an energy transfer.

This profound fact about the universe has gone almost entirely unnoticed by philosophers, whether from ignorance or indifference I cannot say. But it leads almost immediately to two other profound facts. First, all events are commensurable at some level. They are all instances of the same thing. Second, all events are measurable, at least in theory. We need only to be able to measure each thermodynamic consequence, and add them all up.

Now let’s set up a little thermodynamic system. Call it Eustace. Eustace need not be biological, or at all fancy; it is best to think of him as just a cube of space. Eustace would be pretty dull without a few things going on, so to liven up matters we will assume that at least something in the way of atomic state change is going on. Particles will dart in and out of our little cube of space.

To describe Eustace, we have recourse to the Second Law. As ordinarily formulated, it states that energy, if unimpeded, always tends to disperse. Frying pans cool when you take them off the stove. Water ripples outward and fades to nothing when you throw a pebble in a still lake. Iron rusts. Perpetual motion machines run down. Rocks don’t roll uphill.

Vast quantities of ink have been spilled in attempts to explain entropy, but really it is nothing more than the measure of this tendency of energy to disperse.

The Second Law is, fortunately, only a tendency. Energy disperses if unimpeded. But it is often impeded, which makes possible life, machines, and anything that does work, in the technical as well as the ordinary sense of the word. The lack of activation energy impedes the Second Law: some external force must push a rock poised atop a cliff, or take the frying pan off the fire. Covalent bond energy impedes the Second Law as well, which is why solid objects hang together. The Second Law has been formulated mathematically in several ways. The most useful for describing Eustace is the Gibbs-Boltzmann equation for free energy, which states:


This is one of the most important equations in the history of science; it has been shown to hold in every context that we know of. The triangles, deltas, represent change. Gibbs-Boltzmann compares two states of a thermodynamic system — Eustace in our case, but it could be anything. As for the terms: G, or free energy, is simply the energy available to do work. The earth, for example, receives new free energy constantly in the form of sunlight. Free energy is the sine qua non; it is why I can write this and you can read it. It does not, unfortunately, necessarily become work, as no one knows better than I. Let alone useful work: this depends on how it is directed. I do work when I paint your car and work when I scratch it.

H is enthalpy, the total heat content of a system. We are interested here in changes (Δ), and since we know from the First Law that energy is neither created nor destroyed, that nothing is for free, any increase in enthalpy has to come from outside the system. T is temperature, and S is entropy, which can be either positive or negative. Negative entropy is, again, good; it leads to more free energy by subtracting a negative from a negative. Positive entropy is what you lose, and one of the consequences of the Second Law is that you always lose something.

To return to Eustace, we know from the First Law, in the terms of the equation, that ΔG >= 0. We will also assign Eustace a constant temperature, which isn’t strictly necessary but simplifies the math a bit. So we have:

ΔH – TΔS >= 0

We are dealing here with sums of discrete quantities here. Not one big thing, but many tiny things. Various particles are darting around inside Eustace, each with its own thermodynamic consequences. Hess’s Law states that we can add these up in any order and the result will always be the same. So we segregate the entropic processes into the positive and the negative:

ΣH – TΣS negative – TΣS positive >= 0

From here it’s just a little algebra. We take the third term, the sum of the positive entropies, add it to both sides, and then divide both sides by that same term, yielding:

α = (ΣH – TΣS negative) / TΣS positive >= 1

And there we have it. Alpha (α) is just an arbitrary term that we assign to the result, like c for the speed of light. The term TΣS negative (the sum of the negative entropy) is always negative, so the higher the negative entropy, the larger the numerator. And alpha is always greater than or equal to 1, as you would expect. One is the alpha number for a system that dissipates every last bit of its enthalpy, retaining no free energy at all.

Alpha turns out to have several interesting properties. First, it is dimensionless. The numerator and denominator are both expressed in units of energy, which divide out. It is a number, nothing more. Second, it is calculable, at least in principle. Third, it is perfectly general. Alpha applies to any two states of any system. Fourth, it is complete. Alpha accounts for everything that has happened inside Eustace between the two states that we’re interested in.

Which leaves the question of what α is, exactly. It can be thought of as the rate at which the free energy in a system is directed toward coherence, rather than dissipation. It is the measure of the stability of a system. And this number, remarkably, will clear up any number of dilemmas that philosophers have been unable to resolve. Not to get too far ahead of ourselves here, but I intend, eventually, to establish that the larger Eustace’s α number is, the better.

Next (I do not say tomorrow): From physics to ethics in one moderately difficult step.

Update: Edited for clarity. So if you still don’t understand it, imagine what it was like before.