Ink in water: microstates

Problem

You mix black ink into a glass of water. The ink molecules spread out uniformly. Explain in terms of microstates and macrostates why you never see the reverse process. How improbable would it be to see it?

Solution

The “mixed” macrostate corresponds to an enormous number of microstates $\Omega$ (the ink molecules can be practically anywhere in the glass), while the “separated” macrostate corresponds to very few microstates (the molecules confined to one corner). The system evolves spontaneously towards the macrostate with more microstates, i.e. the more probable one, in agreement with $S=k_B\ln\Omega$: entropy increases. The reverse process would require a macroscopic decrease in entropy. The relative probability between the two macrostates goes as $e^{-\Delta S/k_B}$; for a number of molecules $N\sim 10^{20}$ this probability is of the order of $\sim 2^{-N}$, a number so small that the event is unobservable even on cosmic timescales. $\ev{\sim 2^{-N}}$

Links

Topics: Entropy and the second law Concepts: Entropy Skills: Micro-macro interpretation