The fact that entropy doesn’t always mean “disorder” or “uniformity”is clear from any bottle of Italian salad dressing.Here are additional demonstrations:

Place a cubic centimeter of air at room temperature and pressure within a 1-L vacuum chamber. The air expands throughout the chamber and becomes uniform. Repeat the experiment with a cubic centimeter of gold instead of air, and the end result is not uniform.

A small sample of gas can be considered uniform. But a large sample cannot: A bottle of air, or even of pure nitrogen, with its base at sea level and its top 10 km above is dense at its bottom and near-vacuum at its top.

Remove several ice cubes from your freezer, smash them into shards, collect the shards in a bowl, and set the bowl on your kitchen counter.The shards melt into liquid water. The shards are disorderly and non-uniform, the water is uniform, but the water has the higher entropy.

Remove an ice cube from your freezer, place it in a saucepan, and heat it. The cube changes from pure ice, to mixed ice plus liquid water, to pure liquid water, to mixed liquid water plus steam, to pure steam. That is, it changes from uniform to non-uniform to uniform to non-uniform to uniform, but at all stages its entropy increases.

1. By “uniform” I mean “uniform on a macroscopic scale.” All phases of all materials are of course granular on the atomic scale.

1. By “uniform” I mean “uniform on a macroscopic scale.” All phases of all materials are of course granular on the atomic scale.

Typical vs. average

2 4 Insight into entropy ,” Am. J. Phys. 68, 1090– 1096 ( Dec. 2000). 4. D. F. Styer, “,” Am. J. Phys., 1090–2000). https://doi.org/10.1119/1.1287353 The two configurations shown in Fig.provide more insight into the distinction between entropy and disorder.

Each configuration has 169 squares tossed at random onto 1225 sites, in accord with certain rules to be revealed in a moment. If I ask for the entropy of either configuration, the answer is of course zero: Each configuration of squares is exactly specified, so there is only one configuration (W = 1) that matches the specification, so the entropy S = k B ln(W) vanishes. Asking for the entropy of a configuration—any configuration—is an uninteresting question.

Instead note that I sampled each configuration at random from a pool of configurations (to use the technical term, an “ensemble” of configurations) that was determined by the as-yet unrevealed rules. The question is, which pool was larger? Which configuration is typical of the larger—i.e., higher entropy—pool? Be sure to make a guess—however ill informed—before reading on.

I have asked this question of hundreds of people—laypeople, physics students, physics teachers, physics researchers —and most of them assign the larger entropy to the pool of the bottom configuration. That configuration is smooth, uniform, and featureless. The top configuration contains irregularities that suggest some sort of pattern: I notice a dog about to catch a ball in the upper left. One of my students pointed out the starship Enterprise in the lower right. Nearly everyone sees the glaring void in the center of the upper half.

It’s time to unveil the rules. To produce the top configuration, I used a computer program to scatter the 169 squares onto the 1225 sites at random, subject only to the restriction that no two squares could fall on the same site. To produce the bottom configuration, I added the restriction that no two squares could fall on adjacent sites. The top configuration was drawn from the larger pool, so it is typical of the pool with larger entropy. (The bottom configuration is of course a member of both pools.)

expect random irregularities from a configuration sampled at random: the first configuration sampled might have a void in the upper left, the second in the lower left, the third in the center. The average configuration will be uniform, but each individual configuration will, typically, be clumpy, and each will be clumpy in a different way. 6 6. This conclusion cannot be avoided by invoking “the law of large numbers” or “the thermodynamic limit.” For if we had a million times more squares scattered on a million times more sites, then the configurations in Fig. 2 would be absolutely typical for a window containing one-millionth of the total sites. The rule prohibiting nearest-neighbor squares gives the bottom configuration a smooth, bland, and (to many) “high-entropy” appearance. The top configuration has voids and clumps and irregularities. But in fact one shouldrandom irregularities from a configuration sampled at random: the first configuration sampled might have a void in the upper left, the second in the lower left, the third in the center. Theconfiguration will be uniform, but each individual configuration will, typically, be clumpy, and each will be clumpy in a different way.

7 no family whatsoever had 3.14 members. See Table II in Daphne Lofquist, Terry Lugaila, Martin O’Connell, and Sarah Feliz, “Households and Families: 2010,” U.S. Census Bureau , issued April 2012, at 7. A more extreme example comes from family size. In the 2010 census, the average American family had 3.14 members—π to three significant digits—despite the fact thathad 3.14 members. See Table II in Daphne Lofquist, Terry Lugaila, Martin O’Connell, and Sarah Feliz, “,”, issued April 2012, at http://www.census.gov/prod/cen2010/briefs/c2010br-14.pdf This is the character of averages. Very few people are of average height: most are either taller or shorter. Any manufacturer of shirts that fit only average people would quickly go bankrupt.In most situations, the average is atypical.

Our minds, however, grasp for patterns even in randomness. This is why we find the starship Enterprise in the top configuration. For the same reason the ancients looked up at the night sky, with stars sprinkled about at random, and saw the gods, heroes, and animals that became today’s constellations.