The change of entropy is defined $$\Delta S = \int \frac{dQ_\mathrm{rev}}{T}.$$ If a system is isolated the heat transfer between the system and the surroundings is zero ($dQ = 0$), thus $\Delta S = 0$.
However, it is commonly stated that the entropy of an isolated system can increase. How is this possible, given the above definition of entropy?
Best Answer
One reason is that different parts of the system can be at different temperatures. If a part of the system at $T_1$ transfers an amount $\Delta Q$ of heat to a part of the system at $T_2<T_1$, the hot part's entropy changes by
$$\Delta S_\text{(hot)} = -\frac{\Delta Q}{T_1}$$
but the entropy of the cold part changes by
$$\Delta S_\text{(cold)} = \frac{\Delta Q}{T_2},$$
so the total entropy change is
$$\Delta S = \Delta S_\text{(cold)} + \Delta S_\text{(hot)} = \Delta Q \left( \frac{1}{T_2} - \frac{1}{T_1} \right) > 0. $$
Another reason, as Ignacio Vergara Kausel pointed out, is that entropy changes can also occur for other reasons than heat flow. For example, chemical reactions can change a system's composition, which affects its entropy.