Entropy and Microstates

A drop of ink spreads through water because the spread-out arrangement can happen in vastly more ways than the clumped one. Every ink molecule has a position and a velocity; a particular macroscopic state, like all the ink gathered in one corner, corresponds to only a tiny number of ways those molecules could be arranged. 'Ink spread evenly through the glass' corresponds to an astronomically larger number of arrangements. Molecules bounce around randomly, and they are simply far more likely to land in one of the countless spread-out arrangements than in one of the rare clumped ones. Nothing forbids the ink from regathering; it is just absurdly improbable.

The same counting logic explains heat flow. When a hot block touches a cold block, energy exchanges between their atoms in random collisions. The combined system has many more ways to share that total energy evenly across both blocks than to keep it concentrated in one, so the system drifts toward equal temperature because that macrostate has overwhelmingly more microstates, raising total entropy. This is why entropy is not just a word for visual disorder: it is a precise count (via S = k ln(W), where W is the number of microstates) of accessible arrangements, and it rises because probability favors the macrostate with more of them.

This microstate picture is what gives the second law of thermodynamics its teeth: total entropy of an isolated system never decreases, because the system is always wandering toward more probable, higher-count macrostates. It also sets up why local entropy decreases, like water freezing, are allowed, as long as the surroundings pick up the slack.