In an otherwise illuminating Reference Frame, Paul Chaikin presents the conundrum that an ordered state has a higher entropy than a disordered state. That in turn raises a question regarding the physics of gravity being omitted from analysis of the problem of whether higher entropy and higher disorder go together.
Chaikin describes the ordered face-centered cubic (FCC) state of packed spheres as being stable because of its “higher entropy than the disordered state, as unintuitive as that may be.” The “ill-defined” disordered state of random packing is experimentally achieved by “tapping” the container in which the spheres reside until the density maximizes, which occurs at lower density and looser packing than the FCC configuration.
However, except in orbit, that procedure is necessarily carried out in a gravitational field, so that the total (potential) energy is minimized, rather than the entropy being maximized, when the system is allowed to come to rest. (The kinetic energy is dissipated and therefore irrelevant.) Yet that gravitational contribution to the physics does not appear to have been included in the analysis. Despite the dissipation, the system will not compact without the presence of the gravitational potential gradient, or some other applied, one-dimensional force, such as that due to entrainment in a fluid flow. How might this problem change if the force were 2D, radiating spherically inward, for example?
Moreover, the FCC packing condition is achieved in one and two dimensions, but not three. That fact invites consideration of the result if a fourth dimension were available during the tapping process, although the final configuration is still restricted to three dimensions.
Of course, such a test can only be carried out by a nontrivial computer simulation. However, just as motion transverse to the plane opens wide the path to the equivalent of the FCC configuration in two dimensions, it seems intuitive that the vast additional phase space available with a fourth dimension can unblock the path for three dimensions. But that would still seem to minimize potential energy rather than maximize entropy.
