# I Resolution to Gibbs' entropy paradox?

1. Jul 11, 2016

### maline

It seems to me that Gibbs' Paradox (that the entropy of a classical ideal gas, calculated by phase-space volume, is not extensive) can be resolved without assuming that particles are indistinguishable.
Suppose instead the opposite: that particles are distinguishable, meaning that each one can in principle be identified- imagine a minuscule serial number stamped on every molecule. Now this should apply not only to the system under consideration, but to the universe as a whole- the serial numbers run from one to A, where A is the total number of molecules (of a particular type) in the universe.
This immediately implies that specifying the position and momentum of each of the N particles in our box does not fully determine the microstate of the system! We must also specify which, out of the A molecules in existence, are in fact the N ones in the box. The total number of microstates should include all such possibilities.
This means multiplying the phase-space volume by "A choose N", that is, A!/(N!(A-N)!). Since A>>N, the factor A!/(A-N)! tends to AN. Thus we are left with the desired factor of 1/N!, giving the standard (extensive) entropy, plus a constant contribution N log(A) that is also extensive and (I think) has no observable effects.

2. Jul 11, 2016

### Andy Resnick

You seem to have stumbled upon Jaynes' solution, which results in a connection between information and entropy:

http://www.santafe.edu/media/workingpapers/07-08-029.pdf

3. Jul 12, 2016

### maline

Where in the book is this?

4. Jul 12, 2016

### Andy Resnick

Chapter 5.

5. Jul 12, 2016

### maline

I don't see my argument there. All I see is the assertion by Gibbs that identical particles must be treated as indistinguishable. There is a completely different conception of entropy there, due to Jaynes, in which the entropy is a function of a probability distribution that we assign, rather than of the physical system per se. I am working with the original phase-space volume concept, and simply pointing out that if particles were distinguishable, then that gives us more possibilities for "different" microstates.

6. Jul 12, 2016

### Andy Resnick

Then keep reading, that includes searching the literature- and there's a lot of it. There's a clear connection with information content and distinguishable particles.

Edit- how about this: a thought experiment. Consider a box partly full of spheres; when you look inside you see a partition separates one half, containing green spheres, from the other half which has red spheres. However, when I look in the box, because I have special glasses that only transmit luminance values, I see a partition separating equal numbers of grey spheres.

Now we remove the partition and shake up the box. When we look inside and compare the final state to the initial state, do we initially report the same or different increase of entropy? (and explain your answer)

Last edited: Jul 12, 2016
7. Jul 12, 2016

### HPt

8. Jul 13, 2016

### maline

Yes, this is a standard statement of the paradox. The answer is well- known: there is an increase in entropy, but the guy with the glasses will not know this. But what exactly is his error? According to the standard account, the problem is that he wrongly thinks the balls are indistinguishable, and therefore undercounts the microstates in the final situation. I am suggesting the opposite: if the balls are in principle distingushable- and macroscopic balls certainly are- then his glasses cause him to overcount the microstates initially, by neglecting the color constraint that limits which balls can be on each side.

9. Jul 13, 2016

### maline

This is great! It restores my faith in the existence of common sense in physics. I especially love this quote:

"Secondly, the resolution above implies that the mere concept of distinguishable identical particles is at odds with thermodynamics [12]. In the opinion of the present author, such a connection between these otherwise unrelated subjects would be rather surprising, to say the least."

Last edited: Jul 13, 2016
10. Jul 13, 2016

### Andy Resnick

I would caution you against using 'error', 'wrongly', etc, because you are making some assumptions. The basic point, the essential starting point, is only that we initially report different values. Our values become equal once you provide me with additional *information*.

There's a deep principle here, because it fundamentally goes to the idea of 'hidden variables' and how we can decide if we have complete information about a system. Because surely you would allow that perhaps you did not notice that there were actually *4* colors instead of 2, so your calculation is 'wrong' as well. This has led some to wonder if entropy is not an objective, external, feature of the universe but instead an anthropomorphic, subjective, one.

Since there are no hidden variables, we instead postulate that the difference in reported values is quantified by information: 1 bit of information has entropy k ln(2).

http://www3.imperial.ac.uk/pls/portallive/docs/1/55905.PDF