How does entropy relate to chaos theory?

Question

Olivia Anton · Accepted Answer

ENTROPY

"Disorder" is the general definition of entropy, which isn't exactly a good definition in and of itself. A more precise definition would be:

#color(blue)(DeltaS = int1/T delq_"rev")#

where

The #del# implies that heat flow is not a state function (path independent), but a path(-dependent) function. Entropy, however, is a path independent function.

THEORY OF CHAOS

The basic idea of chaos theory is that a system can be unpredictable even in the absence of randomness in the process of generating future states in the system. It is beyond the purview of this inquiry to define what constitutes a chaotic system.

An example of a chaotic system is when you work with numbers in computer programming that are near machine precision (just borderline too small, basically); they will be extremely difficult to keep entirely unchanged, even if you are just trying to print out a specific small number (say, near #10^(-16)# on a 64-bit Linux).

So if you try to print #5.2385947493857347xx10^(-16)# multiple times, you might get:

...etc. That makes this chaotic system unpredictable; you expect #5.2385947493857347xx10^(-16)#, but you probably won't get that in a million tries.

ENTROPY VS. CHAPTER THEORY

Chaotic theory's fundamental entropy-related tenet is that the system tends toward "disorder," or something unpredictable. (This is NOT the second law of thermodynamics.)

This suggests that there is chaos in the universe.

It is entropically favorable for a group of non-sticky balls to separate from one another and scatter upon hitting the ground, so you cannot guarantee that they will stay together AND fall onto the same exact spot each time, AND stay in place after falling.

In other words, you can't tell with certainty how they'll fall.

The balls' system lost entropy just by falling and splitting off from the human system, and the human system also lost entropy when the balls left their hands, even if they were forced to stick together.

Reduced entropy for the system is the result of fewer microstates being accessible to it.

It's always accounted for in some way, somehow, and the universe has now increased in entropy because the number of systems considered has doubled (you + balls).

IF ENTROPY FOLLOWS CHAOS THEORY, HOW CAN IT THEN FUNCTION AS A STATE?

The state function status of entropy has been demonstrated previously.

This is reassuring because in a chaotic system, we cannot always predict the final state. That is, we can determine the initial and final state without worrying about the path taken to get there.

Nonetheless, the state function property of entropy enables us to assume that whatever path we take as long as it produces the precise final state we want doesn't matter if we already know the final state we want to get to (that is, we choose it ourselves).

The fundamental principles of chaos theory are overcome when the ultimate state is known in advance.

Mateo Aguilar · Answer

undefined Entropy in chaos theory refers to the measure of disorder or randomness within a system. As chaos theory explores complex, dynamic systems, entropy helps quantify the unpredictability and complexity of these systems. High entropy indicates higher disorder and unpredictability, aligning with the central concepts of chaos theory.