It may seem illogical that randomness could be the driver behind the development of ordered, complex structures like plants, animals, and humans. The reason why structure can arise from randomness is that randomness is not necessarily the same thing as chaos. While chaos is a complete lack of any type of predictability or order, randomness is governed by rules of probability.
A familiar example of a random process is the rolling of a pair of dice. The sum of the two dice is a number between 2 and 12 and the occurrence of these sums follows the distribution of probabilities shown in Figure 1.
An individual roll of the dice is random, and if the dice are fair there is no way to predict the outcome. However, the non-equal distribution of probabilities means that accurate predictions can be made about the relative numbers of different outcomes one would obtain for many rolls of the dice. The larger the number of rolls, the more accurate the predictions become. This is an example of structured randomness. First of all, the results are all of a common predictable type: the sum of numbers on two dice. Second, this sum will always be between 2 and 12, and it will statistically follow a known probability distribution. This situation is to be contrasted with chaos, in which a roll of the dice might yield 426, a poker hand, or a nuclear explosion (i.e. total unpredictability).
The structured probability distributions in casino games allow the house to always come out ahead, because it has staked out higher probability positions for itself, provided a large number of games are played. Random events with unequal probability distributions lead to the emergence of persistent structure in the systems they control. In the case of gambling, this structure is a huge worldwide industry in cities such as Las Vegas and Macau. The fact that any individual random event can produce any possible outcome, even the most unlikely, is an agent for transformation and change. A gambler may strike it rich by betting big on a low probability result. Although this is rare compared to the number of losers, the huge number of games played guarantees that there are always some whose lives are transformed by a big win.
A familiar example of a random process is the rolling of a pair of dice. The sum of the two dice is a number between 2 and 12 and the occurrence of these sums follows the distribution of probabilities shown in Figure 1.
An individual roll of the dice is random, and if the dice are fair there is no way to predict the outcome. However, the non-equal distribution of probabilities means that accurate predictions can be made about the relative numbers of different outcomes one would obtain for many rolls of the dice. The larger the number of rolls, the more accurate the predictions become. This is an example of structured randomness. First of all, the results are all of a common predictable type: the sum of numbers on two dice. Second, this sum will always be between 2 and 12, and it will statistically follow a known probability distribution. This situation is to be contrasted with chaos, in which a roll of the dice might yield 426, a poker hand, or a nuclear explosion (i.e. total unpredictability).
The structured probability distributions in casino games allow the house to always come out ahead, because it has staked out higher probability positions for itself, provided a large number of games are played. Random events with unequal probability distributions lead to the emergence of persistent structure in the systems they control. In the case of gambling, this structure is a huge worldwide industry in cities such as Las Vegas and Macau. The fact that any individual random event can produce any possible outcome, even the most unlikely, is an agent for transformation and change. A gambler may strike it rich by betting big on a low probability result. Although this is rare compared to the number of losers, the huge number of games played guarantees that there are always some whose lives are transformed by a big win.
To illustrate how randomness based on a distribution of probabilities can give rise to structure, consider the following simple game:
(1) Repeatedly roll a pair of dice and plot the frequency of occurrence of their sums on a graph.
(2) Whenever snake eyes—2— is rolled, shift all the numbers on the horizontal axis to the right by a number determined by another roll of the two dice.
(3) Go back to Step (1).
The result of applying these rules for about 150 rolls is shown in Figure 2. A clear structure has emerged, consisting of roughly, equally spaced clumps, which are higher in the middle and lower on their sides. Of course, this is a trivial example and thus is very limited in the degree of complexity it can generate. However, it is clear that structure can be generated by a random process, as long as that randomness follows a structured distribution of probabilities.
(1) Repeatedly roll a pair of dice and plot the frequency of occurrence of their sums on a graph.
(2) Whenever snake eyes—2— is rolled, shift all the numbers on the horizontal axis to the right by a number determined by another roll of the two dice.
(3) Go back to Step (1).
The result of applying these rules for about 150 rolls is shown in Figure 2. A clear structure has emerged, consisting of roughly, equally spaced clumps, which are higher in the middle and lower on their sides. Of course, this is a trivial example and thus is very limited in the degree of complexity it can generate. However, it is clear that structure can be generated by a random process, as long as that randomness follows a structured distribution of probabilities.
Now consider a bottle full of air. Increasing the temperature of the air will cause the gas molecules to zoom around at higher speed, and the collisions of these faster molecules with the sides of the bottle will result in an increase in air pressure. Figure 3 illustrates how the distribution of speeds of the gas molecules in the air changes as the temperature is changed. These are known as Boltzmann distributions. For a specific gas molecule in the bottle at a given temperature, it gives the distribution of probabilities that the molecule will be traveling at a particular speed at a given instant. This is analogous to the distribution of probabilities for dice, as shown in Figure 1.
Since the probability distribution in Figure3 applies to all the molecules in the bottle, at any instant it is also the distribution of speeds of all the molecules. Pick a particular molecular speed on the horizontal axis of the graph and go up to the curve corresponding to the temperature and the graph tells you the relative number of molecules traveling that fast. The Boltzmann distribution curves are bell shaped, with peaks in their centers; molecular speeds in those peak regions are the most common for a given temperature. However, the distributions contain higher molecular speeds in the flared bases of the bells; these represent less likely fluctuations of the molecular speed. Faster moving molecules are not as likely as those at the peak of the distribution, but the vast number of molecules present means that some much more energetic molecules always exist. As the temperature increases, energy is added to the bottle of air, and the distribution of speeds become broader and increasingly includes a substantial proportion of higher molecular speeds.
Boltzmann distributions are specifically for gases such as air, but similar probability distributions apply to molecules comprising all the liquids and solids, inorganic and biological, that makeup the world. On a very small scale, every substance, including the substances that comprise us humans, is comprised of jiggling molecules moving wildly and randomly, but randomly according to unequal probability distributions similar to the Boltzmann distributions in Figure 3. The speeds at which the molecules jiggle and move, in the bottle of air or any other object, are intimately connected to heat and temperature. Heat is a form of energy, and the main thing this energy does is make molecules wiggle and move. It is actually the speed of molecular motion that we feel as temperature. Hot air is hotter because the gas molecules comprising it are moving faster on average than in cooler air. Likewise, when we get sick and have a fever of 104˚F, this is an indication that the molecules that make up our bodies are wiggling about a little bit faster than when our temperature is normal.
Uneven probability distributions, such as the Boltzmann distribution (Figure 3), lead to organization and structure in the systems they influence. In the case of a bottle of air, this structure is a consistent and predictable response to adding heat to the air, producing a change in temperature and air pressure. Temperature and pressure may seem pretty minimal as far as organization or structure, but remember, this is just a gas. Having a consistent temperature and pressure related to the input of heat is as organized as a gas gets.
Since the probability distribution in Figure3 applies to all the molecules in the bottle, at any instant it is also the distribution of speeds of all the molecules. Pick a particular molecular speed on the horizontal axis of the graph and go up to the curve corresponding to the temperature and the graph tells you the relative number of molecules traveling that fast. The Boltzmann distribution curves are bell shaped, with peaks in their centers; molecular speeds in those peak regions are the most common for a given temperature. However, the distributions contain higher molecular speeds in the flared bases of the bells; these represent less likely fluctuations of the molecular speed. Faster moving molecules are not as likely as those at the peak of the distribution, but the vast number of molecules present means that some much more energetic molecules always exist. As the temperature increases, energy is added to the bottle of air, and the distribution of speeds become broader and increasingly includes a substantial proportion of higher molecular speeds.
Boltzmann distributions are specifically for gases such as air, but similar probability distributions apply to molecules comprising all the liquids and solids, inorganic and biological, that makeup the world. On a very small scale, every substance, including the substances that comprise us humans, is comprised of jiggling molecules moving wildly and randomly, but randomly according to unequal probability distributions similar to the Boltzmann distributions in Figure 3. The speeds at which the molecules jiggle and move, in the bottle of air or any other object, are intimately connected to heat and temperature. Heat is a form of energy, and the main thing this energy does is make molecules wiggle and move. It is actually the speed of molecular motion that we feel as temperature. Hot air is hotter because the gas molecules comprising it are moving faster on average than in cooler air. Likewise, when we get sick and have a fever of 104˚F, this is an indication that the molecules that make up our bodies are wiggling about a little bit faster than when our temperature is normal.
Uneven probability distributions, such as the Boltzmann distribution (Figure 3), lead to organization and structure in the systems they influence. In the case of a bottle of air, this structure is a consistent and predictable response to adding heat to the air, producing a change in temperature and air pressure. Temperature and pressure may seem pretty minimal as far as organization or structure, but remember, this is just a gas. Having a consistent temperature and pressure related to the input of heat is as organized as a gas gets.