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.
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 Î± 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 Î± 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 Î± 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.