As far as I can tell, the concept of entropy is a purely statistical one. In my engineering thermodynamics course we were told that the second law of Thermodynamics states that "the entropy of an isolated system never decreases". However, this doesn't make much sense to me.
By counter-example: Consider a gas-filled isolated system where the gas has maximum entropy (it is at equilibrium). Since the molecular motion is considered to be random, at some point in the future there will be a pressure gradient formed by pure chance. At this point in time, entropy has decreased.
According to Wikipedia, the second law purely states that systems tend toward thermodynamic equilibrium which makes sense. I then ask a) is the second law as we were taught it wrong (in general), and b) what is the use of entropy (as a mathematical value) if it's effectively an arbitrary definition (i.e. what implications can we draw from knowing the change in entropy of a system)?
Thanks in advance for your help.