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Saturday, May 15, 2010

Steepest descent and libertarian philosophy

The evolution of complicated physical systems shares interesting features with social dynamics, especially as viewed through the lens of l ibertarian philosophy. Consider, for example, a system in which the physics of the system is determined by some energy, E, and the evolution is determined by the simple law: minimize the energy. This is called following the path of steepest descent. Water running down a hill is a simple model to keep in mind. In these models the total energy, E, is a sum (integral) of the energies of each constituent particle or point in the system. In most interesting systems, unlike water, you cannot simply tell each particle to minimize its energy, because its energy depends on the states of its neighbors. Consequently, some level of coordination is required among neighbors to minimize energy. In most systems of interest, there is a rule (called the Euler-Lagrange equation) which tells each particle what to do in order to minimize energy, and this rule only requires the particle have knowledge of its neighbors and its neighbors' neighbors. All the particles acting on these simple rules, knowing only what happens around them, pursue a course which approaches the minimal energy (greatest happiness) for the whole system. Thus we frequently find in physics an analog of the notion that individuals acting in their own best interest while respecting basic rules for interacting with others may maximize the happiness of the entire society.

In complicated systems, bad things may happen on the path to minimal energy. With each particle trying to minimize its own energy, following the path for steepest descent, sometimes enormous energy is concentrated in a small region, blowing it up and destroying the system. This blow up can be extremely hard to foresee. Some systems always blow up as they approach minimal energy. This behavior may be a function of the mathematical model of the system. Scientists often suspect that the validity of the idealization involved in creating a mathematical model out of a physical situation loses validity as the system approaches the time of blow up. A new and different model is required to understand what is happening to the system there. A famous example of this arises in the treatment of fluid motion before a sonic boom.

So, the manner in which physical systems optimize their energy is similar to the way libertarian philosophers envision human society. Each individual, acting in his own interest and optimizing his own life without access to a grand plan, optimizes the society. An analog of modern liberal political philosophy, on the other hand, would require physics to be determined by the actions of a bevy of gods on Mount Olympus. Finally, the conservative recognizes that even optimizing behavior, can occasionally lead to the destruction of the society. Hence he reserves the flexibility to deviate from the libertarian optimizing path. In physical systems and probably also in society, such deviations always prevent the evolution to the optimal total energy (happiness) level, but stop the destruction of the system enroute. Unfortunately, one rarely knows in advance whether a great concentration of energy is merely a peak on the way to optimization or is impending blowup. So too, the conservative is rarely sure what is the appropriate juncture to deviate from libertarian minimalism.

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