Some notes outlining progress on a question concerning Gibbs’ Paradox. Not everything will be totally cogent / concise, as they are working notes.

Project supervised by A. Grosberg at NYU, Autumn 2013.

# Statement

There are two paradoxes bearing Gibbs’ name, both arising in the context of entropy in statistical physics.

The first paradox concerns the (apparently spurious) entropy gain from processes which leave the thermodynamic entropy unchanged.

The second paradox is closely related, and concerns the mixing of particles. Since the distinction between identical and non-identical particles can be made arbitrarily small, it is mysterious that there exists a dichotomy between the two cases when dealing with entropy generation.

## Aside: resolution according to Jaynes

In REF?, Jaynes says that entropy increase has to be treated more “subjectively”. Entropy production is not absolute: if we cannot distinguish the properties of two mixing gases, then there is no entropy increase and no work required to un-mix them. If we can distinguish the gases, then this is no longer true. To repeat, if the particles are experimentally indistinguishable for whatever reason, Gibbs’ paradox is resolved.1

At first this seemed absurd: a colour-blind person calculates zero entropy increase when a box of green balls mixes with a box of red balls, but clearly he is wrong. We could use the mixing to do work (see section \ref{sec:workout}): would the colour-blind person also be blind to winches lifting weights?

In Smith et al. (1992), Jaynes elaborates and his story (originally due to Gibbs) makes more sense. The problem is that when the gases go from being non-identical to identical, our definitions of “reversible” and “original state” change; i.e. we have double standards. In the non-identical case, we want to separate all molecules originally in $$V_1$$ and put them back into $$V_1$$, and the same for $$V_2$$; but in the same-gases case we are happy to just reinsert the diaphragm without reference to the particles’ origins. But this is beside the point: remember that “reversible” applies to thermodynamic variables that we can measure, not to microscopic states. The entropy increase associated with the mixing process corresponds to the work required to recover the same family of microstates: in this case, the original separation of molecules into $$V_1$$ and $$V_2$$. The multiplicity $$W$$ is the size of the aforementioned “family”; but this depends on what we’re actually measuring about the macrostate.
Example2: imagine there are two types of argon, but current technology cannot distinguish them, so mixing them gives $$\Delta S=0$$. Then a new solvent, “whifnium”, is synthesised, and it is discovered that one type of argon is soluble in it, but not the other type. By putting this knowledge to use and constructing a setup involving whifnium, we can extract work from the mixing – an observable consequence which produces entropy, i.e. $$\Delta S>0$$. So the work extractable depends on “human” information. Identical (even microscopically) physical processes can be assigned different entropy depending on what we’re interested in. But more knowledge lets us extract more work. Conclusion: entropy not a physical property of microstate (as energy is), but an anthropomorphic quantity.

To do: