| That's not quite right, and the author is reaching for a valid definition, but it also not quite getting it right (entropy is hard). Entropy is defined as: "It quantifies the number of microscopic configurations that are consistent with the macroscopic quantities that characterize the system." Where there are two levels - microscopic - which in this case doesn't necessarily mean invisible to the eye, it means the individual units of the system. In this case shoes. And macroscopic - which is the system as a whole, or in this case, the set of all shoes. A single macroscopic state is the set of shoes in some (maybe random) order. Finally "macroscopic quantities" refers to a property of the macroscopic system. In this case the author is using the macroscopic quantity of "how fast can I find a shoe of a particular order". In this case, entropy measures, for a given macroscopic quantity, how many microscopic configurations exist. For example, lets say the given macroscopic quantity is "I find a given color shoe in O(n) time." Then every configuration of shoes that does not attempt to order the shoes is a possible microscopic state. There are lots of those, so high entropy. Another possible macroscopic quantity is "I find a given color shoe in O(logn) time". In this case the shoes need to be ordered, leading to far fewer states. Hence lower entropy. If this is not clear, it is easier to see if we constrain ourselves to having one pair of shoes for each color. Then the O(logn) configuration has 1 state, while O(n) clearly has many more, so obviously the O(logn) state has lower entropy. However the same logic applies to having >1 pair of shoes for each color. In the O(logn) case, we can only swap shoes of the same color to have the same macro state. In O(n), we can always swap any shoes of any color, as long as this swap isn't the last swap needed to achieve order. |