Entropy is Disorder

Well, it’s not exactly, but I thought I’d be argumentative. Here is the problem with entropy and disorder as I see it, possibly somewhat different to Nathaniel.

There are two things that are commonly called entropy, one of which is a specific case of the other. These two types of entropy are physical/thermodynamic entropy and statistical entropy. Thermodynamic entropy is a statistical entropy applied specifically to physical microstates. As physicists generally agree on their definition of the microstates, thermodynamic entropy is well defined physical quantity. Statistical entropy on the other hand can be applied to anything that we can define a probability measure for.

Now lets look at disorder. What’s that about? First of all disorder is subjective:

Introducing Mister A. Mister A has at hand an equation, it is the logistic equation, a chaotic map. He can use this to produce a what could be seen as a sequence of random numbers, but he knows exactly how this sequence of numbers is produced. To him, they are completely ordered (satisfying for some real s in and integer , starting with ). Mister A has a friend Mister B, who is blissfully unaware of the logistic map or Mister As use of it .When Mister B is shown this list of numbers he sees no pattern. And why should he. He sees a string of numbers that seem completely random. Mister B can even try various mathematical transformations, but unless he hits on exactly the conditions that Mister A used to make the sequence, it will appear forever random.

To Mister A, the sequence is completely ordered. To Mister B, the sequence is completely disordered. The conclusion we can draw from this is: when we talk about disorder we have to specify a particular point of view.

Thermodynamic entropy has an inherent point of view: the physical microstates of the system being studied. This point of view is of course completely unintuitive to all but a few (physicists). This microscopic point of view is not the macroscopic point of view where one might see the phase boundaries Nathaniel mentioed in his post on this topic (he discusses the disorder of an emulsion and a layered oil/water mixture). However, this macroscopic level of description is completely reasonable, even though thermodynamic entropy isn’t the correct choice for a correlate of disorder.

We can still come up with some other measure of disorder for the macroscopic system, and it can be an entropy too. We could take the emulsified/seperated system and split it up into macroscopic voxels of sizes smaller than the droplets, and, over a small but not to small period of time measure the probability of each one being mainly full of water or mainly full of oil. We could then find the statistical entropy of this. This entropy  would decrease with time, contrary to what one might be lead to believe from the second law of thermodynamics. But thermodynamic entropy is the only type of entropy that one should expect to obey the second law, and this isn’t that.

So I would like to say something slightly different and far less polemic than title of this post :

Thermodynamic entropy is only disorder if you are a physicist with a physicists conception of disorder. But, whatever your concept of disorder is, there is probably a statistical entropy which corresponds your own particular version of it,  just don’t expect your entropy to increase with time.