What "a disordered arrangement" is.

The answer to this semantic riddle involves nomenclature.

It is true than any specific arrangement of molecules can be considered every bit as "ordered" as any other specific arrangement, but when Feynman speaks of "a disordered arrangement", he is using the words as a code to mean the set of all arrangements that do not correspond to some particular ordered arrangement (or set of ordered arrangements) of interest.

Obviously many more arrangements are lumped together under the name "a disordered arrangement" than are included under the name of any particular ordered arrangement. It is this larger number of disordered arrangements that creates the entropic bias.

Feynman goes on to write (p. I 46-7), "So we now have to talk about what we mean by disorder and what we mean by order... We measure 'disorder' by the number of ways that the insides [of a grid containing black balls and white ones] can be arranged, so that from the outside it looks the same."

[notice the work "looks".  Eye of the beholder.]

Supplement: The Time Problem

Whether this kind of entropic bias gives a direction to the "arrow of time" is another question.

The basis of irreversibility is randomness. If you have some particular arrangement of particles and one of them undergoes a random displacement, there is the possibility that the same particle could undergo the reverse displacement and return the system to the starting arrangement. After a random walk with an even number of steps the original arrangement is always more likely than any other specific arrangement. But the original arrangement becomes increasingly less likely than the set of all other accessible arrangements. Obviously, the more particles there are - the higher the dimensionality of the problem - the less likely is return to the original arrangement.

Consider two dimensions.

A step of length s in an arbitrary direction from position X not only ends up further from X, it is also more likely to be away from any given point P than toward it. (More of the red circle is outside the blue circle than inside it.) The bias becomes stronger in higher dimensions.

So when Nature is left to its own random ways, the tendency is to move away from any specific arrangement, whether one considers it "ordered" or "disordered".

Unless you know the initial arrangement, there is no way to know whether you are going forward in time and drifting away from it, or going backward in time and drifting toward it.

Of course there are very few arrangements whose overall order we can perceive (when the ancients found bears, a scorpion, and a hunter among the stars, they were being quite selective in which stars they considered). So if there is obvious overall order, we can guess that it did not arise at random. There is a clear time sequence between Humpty Dumpty and his fragments.

Still, the tendency to move randomly away from an ordered arrangement can be overwhelmed by a sufficiently strong energetic bias for moving toward it - consider crystal formation or the condensation of steam.

If the energetic bias is not strong enough, the tendency of random steps to move away from any particular arrangement, can make systems move uphill in energy. When water evaporates at room temperature, or ice melts at 0░, the water molecules have to pull energy from the surroundings in order to overcome the mutual attraction they enjoy in the more ordered state. (This is also what makes a rubber band contract and what allows an adiabatic demagnetization refrigerator to cool samples to 0.05 K, or even much lower.)

In many cases of chemical interest, randomization of the distribution of energy within a system is more important for its time development than is the randomization of the spatial distribution of particles, making the increase of any na´vely perceived spatial "disorder" an unreliable measure of increasing time. Supersaturated solutions do spontaneously crystallize.

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copyright 2002 J.M.McBride