The multiplicity, Ω, is the number of ways a system might be configured given some observable macroscopic state (macrostate). If I flip a coin and catch it in my hand, and I do not look at it, its multiplicity is two(1). A body with no extension (a particle with no length, width, or depth can’t twist, bend, rotate, etc.), contained in some box can occupy some position in that box and have some velocity, and only these two variables contribute to its multiplicity.
If I take a large handful of coins, and toss them up in the air, there is an expectation that “about” half of them will come up heads, correct? What may be less apparent, but can still be intuited, is this: As the number of coins being tossed increases, the percent by which they deviate from a 50/50 split will decrease. This is sometimes called the Weak Law of large numbers, and it is the simplest kind of emergence I can think of; it is a macroscopic, qualitative property that arises from increasing a quantity in the system. It is a statistical impossibility that a large number of fair coin tosses will not reveal the underlying probability of a single toss.
Viewed another way: there exists a set of all possible outcomes for every coin flip. Because they are all equally probable, and there are so many more outcomes with the coins split about 50/50, those outcomes are much more likely. The states that split the coins evenly have a much greater multiplicity.
There are many different kinds of statistical convergence, and I suspect they can all be associated with a type of emergent observable property. If the multiplicity as it appears in formal statistical mechanics, is sufficiently explained, I may give the multiplicity of agency.
Let there exist some macroscopic behavior of a community. This behavior is associated with an effective, or apparent agency which emerges from aggregation of individuals, or the local behavior of components. The multiplicity of this agency is the number of local behaviors which all contribute to reproducing the same macroscopic behavior.
- People shopping for clothes will operate under a number of motivations, and may weigh a number of different things when making purchasing decisions. Different people also go with very different fashion choices, or will prefer certain stores, etc. To the extent that all of these variations in local behavior typically contain a common thread of preferring lower priced goods, the market will generate a downward pressure on costs of production which is robust to all of this variation. This pressure has contributed to the creation of sweatshops. Because people make decisions at a local level and act at that scale, the apparent macroscopic agency, or systemic behavior is indifferent to this.
What I am trying to get at is a way of understanding systemic societal problems in rigorous terms that show the qualitative differences between global scale and local scale behavior.
- This seems closely associated with problems of nonlinearity, and a violation of the basic assumptions people habitually make(2) when trying to understand these systems. It’s just not enough that ‘most people’ would not want some particular system behavior.
- How do wars occur, anyway? It would be too simplistic to attribute this completely to leaders. It would be safe to say that most people do not wish for these events, but we seem to habitually behave in ways that contribute to tensions and conflict; that the greater multiplicity belongs to the emergent agency which creates and maintains hostile divisions between people.
Maybe as a species we are just feckless? After all of our advances in technology and science, why do we still not understand ourselves well enough to solve fundamental problems like poverty and violence? I would hope we are not still holding on to the notion that our individual free will has relevance and power to affect our behavior as a society(3), as the recurrence of civil wars, the tendency of markets to produce dangerous bubbles, and many other phenomena demonstrate that the aggregation of behavior can create systems acting in direct opposition to local intentions.
The companion post is here.
- There is some subtle interpretational issues here with what probability means, but I will gloss over these because they are largely philosophical, and to the extent they relate to underlying issues in classical and quantum probability, etc. I think I can get away with ignoring them for the purposes of this post.
- The prior post about linearity.
- Econophysics, for example, can get pretty far, and reproduce some surprising results assuming individuals’ behavior is entirely random. It hardly seems to matter that people operate under coherent local rules at all, at least for certain properties.