This is a reference post for my (in-progress) sequences on entropy and optimization. It is intended to be used by readers to catch up on this concept in case it's unfamiliar to them. It is absolutely not intended as a stand-alone introduction to the topic; that would take us far away from its relevance to entropy and optimization. As such, there are countless ideas central to the study of dynamical systems that are omitted.

Both entropy^[1] and optimization take place within an extremely simple formalism called a dynamical system. It is simple in the sense that it has very few parts; often, when you start to work with a specific dynamical system, you'll find that its behavior is extremely interesting.

Definition

A dynamical system is any formally specified system that moves between states over time according to a deterministic^[2] rule which we'll call the evolution rule. This is formalized as a function $f$ that takes every state in a set $X$ to another state in $X$ ; $f : X \to X$ .

If you are treating time as discrete, then each application of the function is moving the system forward in time one unit. If you are treating time as continuous, then you'll need your function to be able to take another parameter t to tell it how far into the future to push the system; $f : T \times X \to X$ . In either case, we'll indicate the amount of time with a little exponent over the function, as in $f^{t} (x)$ . So, in the discrete case, applying it twice would look like $f (f (x)) = f^{2} (x)$ .

The set of states

The set of states represents all states that the system could^[3] be in. There are no further constraints on the set; states do not need to have any properties whatsoever. That said, they almost always do; we'll go into more detail about this in the section about the phase space.

As sets, they can have any cardinality; they could be finite, they could be countably infinite, or they could be uncountably infinite.^[4]

The evolution rule

The rule tells us how the system evolves over time. Any state that the system could be in must be able to evolve to some other state in the specified set; it's not valid to allow your system to simply stop evolving,^[5] or to exit the set of states. In this way, a dynamical system is an abstraction that captures the idea of "laws of physics". (It also just happens to be how validly defined functions work.)

A sequence of states p...