# Some remarks and refinements in the definition of entropy

As we have given them, the definitions of the phase space volume and entropy are not correct and can give rise to annoying problems and paradoxes.

## The dimension of the phase space volume[edit | edit source]

To begin with we can note that the definition of entropy given in the fundamental postulate of statistical mechanics, makes no sense since is dimensional. In fact is defined as an integral over and:

A legitimate question could now be: what is the value of ? Which constant is it?
Unfortunately, within classical statistical mechanics it is impossible to establish it; only in quantum statistical mechanics the constant acquires a precise meaning, and in particular it turns out that is *Planck's constant*:

^{[1]}, and summing over these cells

^{[2]}: in this sense the entropy gives a measure of the quantity of such possible states.

## Extensivity of the entropy and the Gibbs paradox[edit | edit source]

Even with the introduction of we still have some problems. In particular, from the expression of for an ideal gas we have:

^{[3]}so that in the end:

*not*extensive since it contains terms like and . Furthermore, it gives rise to a catastrophic paradox known as the

*Gibbs paradox*. In order to understand it let us consider a system divided by a wall into two subsystems each of particles of mass , volume and energy (with ). The entropy of the system in this initial state is:

*entropy of mixing*, which is always positive. If this is a correct result, since the mixing of the two gases is an irreversible process. However, since doesn't depend on and this result holds also in the case . But this is a paradox: the mixing of two identical gases is a reversible process (we can recover the initial state reinserting the wall) so the entropy of the whole system shouldn't increase if we remove the wall. Furthermore, the fact that also when is catastrophic because it means that the entropy of a system depends on the

*history*of the system itself, rather than on its

*state*. But this ultimately means that entropy

*doesn't exist at all*: consider a system of energy made of particles contained in a volume ; then we can think that this system has been obtained from the separation of pre-existing subsystems,

*with arbitrarily large*. This means that the entropy of the system has increased an arbitrarily large amount of times from its initial value, and therefore the entropy of the system in its final configuration (energy , volume , particles) is greater than any arbitrary number: in other words, assuming as the entropy of an ideal gas we would conclude that the entropy of

*any*ideal gas is infinite! There's clearly something wrong with the definition of entropy we have given. How can we solve this problem? We know that in classical mechanics identical particles are distinguishable; however in order to solve the Gibbs paradox we must treat them as undistinguishable, just like we would do in quantum mechanics: this way if we exchange two particles the representative point of the system in phase space won't change. Now, since particles can be exchanged in different ways there will be different configurations of the system relative to the same representative point. This means that the we must redefine the phase space volume of the system reducing it by a factor (which is sometimes called

*Boltzmann factor*):

This expression for the entropy is clearly extensive since it is proportional to and the logarithm depends only on and . Furthermore, the computation of the entropy of mixing of two gases now gives:

This solution to the Gibbs paradox, however, is rather an *ad hoc* one. Unfortunately there's no way to understand where does the Boltzmann factor *really* come from within the framework of *classical* statistical mechanics. This can be made clearer within quantum statistical mechanics: in that case the comes from the fact that identical particles are intrinsically indistinguishable and does not "disappear" in the classical limit.

## Conclusions[edit | edit source]

To conclude, the correct definition of the phase space volume that eliminates all the problems that we have mentioned, and the one we will *always* use in the future, is:

- ↑ In other words, we consider all the points inside a cell to represent only a single state. This is just an "approximation" in classical statistical mechanics, but in quantum statistical mechanics it turns out that there
*really*is only one state in every of those cell in phase space. - ↑ To be a bit more precise: the error that we make approximating the integral over the whole phase space with a sum over cells of linear dimension is perfectly negligible. This is due to the fact that is ridiculously small with respect to the common scales of a macroscopic system.
- ↑ Note that this means that we can simply drop the term (which is what we are going to do in the future) in the expression of , since it divides the phase space volume by a negligible amount. In fact, from the fundamental postulate of statistical mechanics we have that multiplying by an intensive factor is equivalent to adding a constant to the entropy; however the entropy of a typical system is so large that adding such a constant doesn't change sensibly its value. In other words the phase space volume is so large that
*multiplying*it by a constant does not change significantly its value.