Tuesday, April 12, 2016

Different types of emergence

I think Daniel Little misunderstands phase transitions here, but it gives me an excellent excuse to talk about the meaning of "emergence". Little says:
What seems to be involved here is a conclusion that is a little bit different from standard ideas about emergent phenomena. The point seems to be that for a certain class of systems, these systems have dynamic characteristics that are formal and abstract and do not require that we understand the micro mechanisms upon they rest at all. It is enough to know that system S is formally similar to a two-dimensional array of magnetized atoms (the "Ising model"); then we can infer that phase-transition behavior of the system will have specific mathematical properties. This might be summarized with the slogan, "system properties do not require derivation from micro dynamics." Or in other words: systems have properties that don't depend upon the specifics of the individual components -- a statement that is strongly parallel to but distinct from the definition of emergence mentioned above. It is distinct, because the approach leaves it entirely open that the system properties are generated by the dynamics of the components.
Well, yes and no. What is being discussed here by SolĂ© are universality classes of phase transitions. Near a phase transition, the correlation length of the order parameter can go to infinity meaning the system becomes "scale free" and depend on some pure numbers (critical exponents). Here's an example with magnetism. Two of the best studied universality classes are the ones containing the Ising model and percolation theory. In plain English what this means is that wildly different systems (with wildly different "micro theories") can have identical behavior (i.e. don't depend on micro theory scales) near a phase transition if they have the same critical exponent.

I should hand the mic to Cosma Shalizi [1]:
... over the last half century or so, the statistical mechanics community has devoted much of its research energy, and its pedagogy, to the theory of phase transitions, such as those between solids and liquids, or liquids and liquid crystals, or between magnetic and non-magnetic materials. ... I don't mean that this is bad for statistical mechanics, because phase transitions are important and the theory developed around them is one of the jewels of the field. But phase transitions have the weird property of "universality". In the vicinity of the critical point, the behavior of the system comes to depend only on a few key parameters, so that any two systems in the same "universality class" are quantitatively similar near the transition, even if they are otherwise as different as chalk and cheese. If what you are interested in is this behavior near the critical point, then, you can get away with analyzing or simulating ridiculously over-simplified models, if only you get their universality class right. The implicit lesson is that details don't matter, and results on toy models should generalize directly to real systems. (Of course, details can matter a lot, even with toy models.) I can't make myself believe it's coincidence that so many of the people active in econophysics come from a background in the theory of critical phenomena.
This represents one way of interpreting "emergence": elements of a macro theory that don't depend on the micro theory are emergent because you are near a phase transition.

Now universality classes aren't just for phase transitions; probability distribution also have universality classes. There's one that many more people are familiar with as well: the Gaussian (normal) distribution universality class. The central limit theorem is directly related to the universality of the normal distribution. There are other distribution universality classes as well [pdf] (related to something I did here).

Instead of universality appearing as you approach a phase transition, the normal distribution appears as the number of samples becomes large regardless of the underlying distribution (well, regardless as long as it has a well-defined mean and variance).

We can say that the normal distribution is "emergent" and the details of the micro system (underlying distribution) don't matter. Coupled with the law of large numbers (that the average of a large number of trials approaches the expected value), this is one of the things I mean when I say something is emergent.

Central limit theorems and the law of large numbers is also one of the ways dimensional reduction can be achieved. Instead of depending on the complex dynamics of the micro theory (at least N-dimensional with N >> 1), the macro theory depends on a smaller number (dimension) of state variables d << N. An example of this is the ideal gas law. The complex properties of the underlying molecules (micro theory) are relevant only as bulk constants to be measured (macro theory).

Let's shift gears to a couple of different types of "emergence" where the details of the micro theory don't matter as much. Near an equilibrium, many complex theories simplify. In physics, near an energy equilibrium, many theories look like a harmonic oscillator (e.g. a pendulum). This applies from plasmons to pogo sticks. I talked about this more here, but the basic idea is a combination of effective field theory and "Taylor expansions" (actually, effective operator expansions). If your system obeys a symmetry, then it will look the same as any system that obeys that symmetry for small enough perturbations around the equilibrium.

There's one more type of emergence that I talk about: macro degrees of freedom or forces that don't exist unless you have more than one of the micro theory degrees of freedom. Examples include nearly all of solid state physics ("electron holes" don't exist without a filled Fermi sphere and phonons don't exist without a lattice), and all of thermodynamics. Entropy does not exist without a large number of degrees of freedom (enough so that the fluctuation theorem isn't significant impact). Therefore entropic (pseudo)forces -- such as the pseudoforce that smooths out a distribution of atoms through diffusion -- do not exist for individual degrees of freedom. I've talked about how sticky prices (nominal rigidity) may be such an entropic force -- it looks like individual prices aren't really sticky at all, but in aggregate they might be.

...

So here is an incomplete list of emergence mechanisms:

  • Unversality classes of phase transitions. Near points of drastic changes from one equilibrium to another, very different complex models can look the same.
  • Unversality classes of probability distributions. Complex agents can obey simple equations that are largely independent of the agents when the system is "statistically large".
  • Small peturbations around an equilibrium (effective field theory). For example, rational agents are the first order effective theory near an economic equilibrium.
  • Entropic forces. Emergent dynamics that derive from a large number of degrees of freedom and entropy maximization that could include things like nominal rigidity.

...

Footnotes:

[1] One complaint Shalizi has about "econophysics" is that this phase transition stuff is just a small part of physics:
Let me also complain that there isn't enough physics: the repertoire of ideas taken from physics is very impoverished. Basically, we see random walks, power laws, and spin systems over and over again.

No comments:

Post a Comment