Sunday, 27 October 2013

Optimization in physics

In universal Darwinism, many optimization processes in nature are seen as being Darwinian.

There are many examples of such optimization processes outside biology. Lightning finds the shortest path between storm clouds and the ground. Cracks seek out the path of least resistance. Water on mountains finds the steepest downhill path. Water on flat ground finds the path that maximally erodes the surface it flows over.

If you look at optimization techniques used in computer science, they include "genetic algorithms" and "memetic algorithms" - which are inspired by biology. However, there are also "gradient descent" and "simulated annealing" - which are named after relatively simple physical processes. These non-biological optimization techniques are simple and primitive - compared to biologically-inspired techniques - but they are still capable of performing useful work.

Physicists with knowledge of non-equilibrium thermodynamics typically use "maximum entropy thermodynamics" and the maximum entropy principle to explain these effects. The question physicists will probably have - when contemplating universal Darwinism - will be: what does it offer that maximum entropy ideas do not?

It is an excellent question, but one with some good answers, I think:

The first thing to say on the topic is that there's a massive overlap between maximum entropy thermodynamics and universal Darwinism. These theories cover much the same set of phenomena, and make many of the same predictions.

Universal Darwinism explains some cases where maximum entropy thermodynamics struggles. Increasing entropy isn't the only possible fitness function that can be applied to populations - and adaptations can locally favour other optimization targets in relevant selective environments. These are cases which universal Darwinism can handle but where maximum entropy thermodynamics doesn't typically help very much.

Lastly, maximum entropy thermodynamics and universal Darwinism are quite different-looking theories with different histories. Darwinism is better developed in many respects. It has better models of lock ins, sub-optimality, combining existing solutions and other phenomena. Physics can benefit from Darwinism's maturity.

Other types of optimization in physics

Another example of optimization in physics involves the opposite of hill-climbing: gradient descent. This can be demonstrated with a single ball bearing moving on a landscape - with little sign of a population, of copying, or of selective elimination.

Physics also features the principle of least action - which is probably its best-known optimization process. This works on the same principle as Galton's board (see right).

Putting a ball bearing into Galton's board computes the maximum value of the normal function - using gravity. The more iterative choices the ball faces, the better the board performs its optimizing task. The principle of least action is a broadly similar kind of optimization process. Galton's board illustrates iterated choices - but not terribly much reproduction. The balls and the gravitons involved are both produced at one location - and are clones of each other - but this copying seems rather remote from the optimization process itself.

An example of the principle of least action involves refraction of light. This case represents a challenge for population-level descriptions of optimization processes in physics. However, if considering a photon as a single point particle, refraction makes little sense. Only when a light wave is modeled as a spread-out, distributed phenomenon, can refraction can be understood.

The principle of least action hardly seems Darwinian at all. It can be modeled in simple physical systems devoid of much in the way of populations or copying. However, the search for shared principles that underlie the cases of optimization in physics may yet turn out to be a fruitful one.

There's an interesting link between the principle of least action and the maximizing principles in maximum entropy thermodynamics. After discussing the Hamilton's principle and Maupertuis' Principle, scholarpedia says:

The existence in mechanics of two actions and two corresponding variational principles which determine the true trajectories, with a Legendre transformation between them, is analogous to the situation in thermodynamics (Gray et al. 2004). There, as established by Gibbs, one introduces two free energies related by a Legendre transformation, i.e. the Helmholtz and Gibbs free energies, with each free energy satisfying a variational principle which determines the thermal equilibrium state of the system.

These other cases of optimization in physics raise the issue of what counts as optimization and what doesn't. Practically any physical system could be described as optimizing the function of behaving as it does - though such explanations may violate Occam's razor. The lightning strike in this post's illustration certainly looks as though it is performing search for high ground using a branching tree. However, we really need criteria relating to what qualifies and what doesn't.

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