# Entropy and fluctuation

1. Feb 16, 2014

### japplepie

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?

2. Feb 16, 2014

### DrDu

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

3. Feb 16, 2014

### japplepie

has it ever been resolved?

4. Feb 16, 2014

### Staff: Mentor

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.

5. Feb 16, 2014

### japplepie

What's the resolution?

6. Feb 17, 2014

### DrDu

7. Feb 17, 2014

### japplepie

It would take so long it would almost never happen ?

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

Both of those are just guesses.

8. Feb 17, 2014

### DrDu

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.

9. Feb 17, 2014

### japplepie

No, what i mean't was factorial growth.