The emergence of randomness

What is the relationship between deterministic and stochastic "laws" of motion?

Subtle. It is possible that random behaviour can emerge from underlying deterministic laws and the converse. Deterministic laws can emerge from a system of many interacting parts that are described by stochastic motion.

This is nicely discussed in a Physics Today, column by Leo Kadanoff from 2002. Here are a few quotes. First, he discusses how random behaviour, such as Brownian motion, can emerge from many particles undergoing deterministic motion, or stochastic motion...

the observation of apparently stochastic features in some behavior does not imply that the underlying laws are themselves probabilistic. Often, deterministic motion is so complex or so sensitively dependent on initial conditions that the motion is indistinguishable from a set of random events. For example, the path taken by an individual molecule in a gas is very well modeled as a random walk, entirely probabilistic in its nature. 

The random walk model can be derived from more fundamental models of molecular scattering. The scattering events could be realized in at least three different ways: using classical mechanics (fully deterministic), using quantum mechanics (partially deterministic), or prespecifying the probabilities of scattering. Thus the probabilistic single-particle model, the random walk, can be equally well obtained from a many-particle model that is entirely deterministic, partially so, or not deterministic at all. Real gases will all show the same behavior independent of the detailed laws governing the scattering. 

We use the word “universality” to describe the rather commonly occurring physical situations in which a set of derived laws remains substantially the same over a wide range of alternative underlying fundamental laws. In these cases, the observable outcome cannot be used to select among the possible underlying laws. 

This universality is a characteristic of emergent phenomena. Laughlin and Pines refer to this as "protectorates": whereby the underlying physics is obscured by the phenomena that is observed at a higher level. For example, observation of acoustic modes in a crystal obscures the underlying atomic structure. 

Kadanoff then discusses how deterministic laws can emerge from underlying stochastic ones.

Conversely,... if you put together many individual stochastic motions, you may well get an essentially deterministic situation. A dilute classical gas obeys the deterministic gas law, PV = NkT. Through the “miracle” of large numbers, many stochastic molecules have produced a deterministic gas. 

Kadanoff's article is actually not primarily concerned with the issues of emergence. It is entitled, Models, Morals, and Metaphors, and is mostly concerned with possible philosophical implications of chance and probability playing a role in evolutionary theory.

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