# How does entropy relate to chaos theory?

Mar 2, 2016

ENTROPY

Entropy is in general a measure of "disorder". It's not exactly a good definition per se, but that's how it's generally defined. A more concrete definition would be:

$\textcolor{b l u e}{\Delta S = \int \frac{1}{T} \partial {q}_{\text{rev}}}$

where:

• ${q}_{\text{rev}}$ is the reversible (i.e. most efficient) heat flow
• $T$ is temperature
• $S$ is entropy

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

CHAOS THEORY

Chaos theory basically states that a system where no randomness is involved in generating future states in the system can still be unpredictable. We do not need to get into the definition of what makes a chaotic system, because that is way outside the scope of the question.

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 $5.2385947493857347 \times {10}^{- 16}$ multiple times, you might get:

• $2.7634757416249547 \times {10}^{- 16}$
• $9.6239678259758971 \times {10}^{- 16}$
• $7.2345079403769486 \times {10}^{- 16}$

...etc. That makes this chaotic system unpredictable; you expect $5.2385947493857347 \times {10}^{- 16}$, but you probably won't get that in a million tries.

CHAOS THEORY VS. ENTROPY

Essentially, the basic tenents of chaos theory that relate to entropy is the idea that the system leans towards "disorder", i.e. something that is unpredictable. (It is NOT the second law of thermodynamics.)

This implies that the universe is a chaotic system.

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

That is, you cannot predict exactly how they will fall.

Even if you made them stick to each other, the balls system decreased in entropy simply from falling and becoming a system separate from the human system, and the human system has decreased in entropy when the balls left his/her hands.

Less microstates available to the system = smaller entropy for the system.

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

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

It has been proven before that entropy is a state function.

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

But if we already know the final state we want to get to (that is, we choose it ourselves), the state function property of entropy allows us to assume that whatever path we used doesn't matter so long as it generates the exact final state we want.

Knowing the final state ahead of time overcomes the basic tenents of chaos theory.