# Entropy and fluctuation

The 2nd law of thermodynamics state that entropy increases with time and entropy is just a measure of how hard it is to distinguish a state from another state (information theoretical view) or how hard it is to find order within a system (thermodynamic view). There are many ways to view entropy but these are the two that I find most pleasing and they are actually equivalent.

Let's consider a box with 2 kinds of identical but distinguishable (but enough to interfere with the interactions) gas molecules which are initially separated; after a while they mix and become more disorderly due to the random motion of molecules. This seems to agree with the 2nd law of thermodynamics.

But, after a very long time, the randomness would eventually create a fluctuation where the gas would unmix and lead back to the initial state; where they are separated.

Does this mean that entropy could decrease after a long time?

DrDu
Thats called "recurrence paradox". Can you estimate how long it takes for, say, 10 gas molecules to unmix? 100 molecules, 10^23 molecules?

1 person
has it ever been resolved?

Nugatory
Mentor
has it ever been resolved?

Yes, through the methods of statistical mechanics. These give us a crisp mathematical definition of entropy free of the somewhat fuzzy "how hard?" in your original post, and yield the laws of thermodynamics as statistical predictions.

Statistical mechanics might be the most unexpectedly cool thing in physics. Quantum mechanics and relativity are cool too, but even people who don't know them know they're cool; stat mech comes as a surprise.

1 person
What's the resolution?

DrDu
What's the resolution?

It would take so long it would almost never happen ?

Is the time proportional to (number of molecules)! ?

Both of those are just guesses.

DrDu
It would take so long it would almost never happen ?

Is the time proportional to (number of molecules)! ?

Both of those are just guesses.

Yes, the guesses are correct, although the recurrence time increases much faster than linear with particle number.
If you are an aspiring physicist, you should be able to estimate the recurrence time.

Yes, the guesses are correct, although the recurrence time increases much faster than linear with particle number.
If you are an aspiring physicist, you should be able to estimate the recurrence time.

No, what i mean't was factorial growth.