What is the simplest definition of “entropy”?

The simplest definition of entropy that still retains some precision is: “the degree to which a system is unavailable to produce work.”

The more a system is able to produce work, the lower the entropy. If the system is no longer able to produce work, entropy is maximum.

The problem with this definition is that it configures entropy as the measure of a deficiency, and this makes the concept difficult to understand. For this reason, a second, simpler (but often misleading) definition is sometimes used that qualifies entropy as “the measure of the degree of disorder.”

The problem is that disorder in physics is not the same thing we normally think of.

Take for example a bag of marbles; we can say that it is orderly because the marbles all fit together. If we magically remove the bag, the marbles fall on the floor and scatter. At this point they are all disordered and entropy has increased.

However, let’s take the same marbles in the same “disorder” configuration and put them in a stable orbit around the earth. Give them enough time, and the gravitational attraction between the marbles will put them back together again. By the time they are all back together in a configuration similar to the one they originally had in the bag, the entropy will have increased, even though they are back in the same spatial configuration that we defined as “ordered” in the previous example.

But then which of the two systems is in line with the entropy arrow? Both, because we have to think of order not in the sense we usually mean, but in the sense of potential energy.

The marbles in the bag have a very high potential energy, because of an external force: the earth’s gravity. And therein lies the trick: we cannot analyze two situations, where one is affected by an external force and the other is not, as equal.

When the marbles are put into orbit, another potential energy comes into action (also present in the first example, but too weak to give significant effects), which is the gravitational energy between the marbles themselves. When the marbles in space all come back together again, the system is no longer available to produce work and entropy is increased.