Context:I (Daniel C) have been working with Aram Ebtekar on various directions in his work on algorithmic thermodynamics and the causal arrow of time. This post explores some implications of algorithmic thermodynamics on the concept of optimization. All mistakes are my (Daniel's) own.

A typical picture of optimization is when an agent causes the environment to have a convergent attractor: When you have an agent that is trying to steer a system towards particular target configurations, you can predict that the system will likely end up in those target configurations even if you have significant uncertainty over the initial configurations of the subsystem.

Some examples:

When a person tries to build a house, the raw materials might initially be scattered in various possible locations, but they will all end up in a narrow configuration where the house is built according to the architect's plans.
When a chess player is trying to checkmate their opponent, the pieces might start in many possible mid-game configurations, but a skilled player will reliably steer many of these initial positions toward configurations where their opponent's king is trapped.

This view of optimization can be framed as a form of entropy reduction. Initially, we have high uncertainty about which configuration the system occupies: There are many possible initial states the raw materials or chess pieces could be in. The optimizing process reduces this uncertainty, concentrating probability mass from a broad initial distribution into a tight final distribution.

However, note that this entropy reduction cannot occur globally for two related reasons: the second law of thermodynamics and the reversibility of the laws of physics. The second law directly states that the total entropy of an isolated system tends to increase over time, which forbids global entropy reduction. Similarly, reversibility of the microscopic dynamics requires that each initial microstate maps to a distinct final microstate, which is incompatible with our "convergent attractor" picture where many initial states funnel to the same final state. In fact, within the framework of stochastic thermodynamics, the derivation of the second law is closely linked to the reversibility of the underlying physics. Roughly speaking, the reversibility of the underlying laws of physics allows us to define coarse-grained macrostates of the sys...

Entropy

Entropy