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why does entropy increase in an isolated system
Take a gas packed in the left half of a sealed box, wall in the middle. Nothing can leave, nothing can enter. This is an isolated system.
Now pull the wall out. The gas floods across and fills the whole box, and it never gathers back into the left half. Why? No force pushes it.
Here is the odd part. Every single collision is reversible: film it backwards and it obeys the same laws. So the one-way trend must come from somewhere else.
So count. Take just four particles, each equally likely to be left or right. That is sixteen arrangements, and only one has all four on the left.
With a hundred particles the bump gets razor thin: about two to the hundred, a million trillion trillion ways, and nearly all sit close to fifty-fifty.
A mole of gas has six times ten to the twenty-three particles. Then the odds of gathering on one side are one in ten to the power of one point eight times ten to the twenty-three. Never.
Physicists measure that count as W, the number of microscopic arrangements. Entropy is S equals k times the natural log of W. More ways, more entropy.
Nothing pushes entropy upward. The spread-out states are simply so much more numerous that the universe almost never finds its way back.