Saturday, June 23, 2012

Mixing may not be irreversible

There is an interesting article Entropy: Order or information by Arieh Ben-Naim in the Journal of Chemical Education.
He points out two related and common misconceptions about entropy:
  • mixing is always irreversible (and so must involve an increase in entropy)
  • entropy is related to disorder
This is illustrated with the three processes of mixing shown below. All involve mixing, but only the first is irreversible.
What determines the entropy change is not the mixing (or amount of disorder) but the change in volume of each gas. That can be related to information (or ignorance) about the state of the gas molecules.


  1. Bingo!

    Entropy is all about information. It's a subtle concept that even experts/professors don't always get.

  2. It's a while since I read this article but I remember thinking that a problem was in the definition of "order". Order can be thought of as information - for example, in a crystal, knowing where one atom is gives you information about the conditional distribution for the position of the other atoms. This ordering can be expressed as a conditional entropy (perhaps, as a Kullback-Liebler divergence). What is not always clear in statistical mechanics is what the prior actually is. In the case you point out, the priors are indicated on the left if the temperature and number of particles is the same in both columns. Generally, the prior will depend on the problem. In most cases, the prior is a thermal ensemble with different parameters. In quantum mechanics it is usually the "totally mixed" density matrix, so that the partition function (i.e. the trace of identity on the density matrix) is equal to one (explaining why it usually is not mentioned).