The statistical definition of entropy

Thermodynamics defines entropy through heat and temperature, which tells you how to calculate it but not what it is. Boltzmann's answer is that entropy counts. Fix what you can see about a system — its energy, volume, particle number — and there will be an enormous number of microscopic arrangements consistent with that description. Entropy is the logarithm of that number: S = k ln W.

Once you read it that way, the second law stops being a mysterious arrow and becomes arithmetic. Systems drift towards macroscopic states that can be realised in more ways, simply because there are more ways to land in them. Gas fills its container not because molecules repel — in an ideal gas they barely interact — but because arrangements spread through the whole volume outnumber arrangements confined to one corner by a factor so vast it has no useful name. Nothing forbids the gas from crowding into a corner; it is just overwhelmingly unlikely, and for macroscopic numbers of molecules "overwhelmingly" means never observed in the age of the universe.

Two cautions. "Disorder" is a lossy translation: a tidy room is not a thermodynamic state, and the intuition misleads as often as it helps — think in terms of how many arrangements look the same from outside. And the second law constrains the total, not each part. A refrigerator lowers the entropy of its contents every day without difficulty; it just discharges more entropy into the kitchen than it removes from the food, which is also why it warms the room overall rather than cooling it.