Why Does Entropy of an Isolated System Always Increase?

Entropy of an isolated system does not necessarily increase in every idealized process: according to the second law of thermodynamics, it increases during an irreversible process and remains constant during an ideal reversible process. It can never spontaneously decrease overall.

This is IB Physics topic B.4 Thermodynamics (HL only). An isolated system exchanges neither matter nor energy with its surroundings, so an entropy decrease within it cannot be offset by a larger entropy increase outside it.

Entropy is related to the number of microscopic arrangements, or microstates, corresponding to the system's observable macrostate:

$$S = k_{\text{B}}\ln \Omega$$