Nib asked
Why does entropy increase in an isolated system?
Nothing pushes entropy up. There are just vastly more ways for energy and particles to be spread out than gathered, so an isolated system drifts to the spread-out state because it is overwhelmingly probable, not forced.
7 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1the setup
Picture a sealed, insulated box split by a wall. All the gas sits on the left and the right half is empty. Because nothing can pass in or out, neither matter nor energy, this is an isolated system.
An isolated system exchanges no matter and no energy with the outside.
2the spread
Remove the wall and the gas spreads until it fills the whole box evenly. Run it forever and it will not return to the left half on its own. Yet no force pulls the molecules apart, since each one is just flying about and bouncing.
The gas spreads out although nothing pushes it to.
3the puzzle
The laws that govern each molecule do not care which way time runs. A film of two molecules colliding looks just as legal when played backwards. So the arrow of time for the whole gas cannot be written into the individual collisions. It comes from counting how many ways things can be arranged.
The one-way trend is not in the microscopic laws. It comes from counting arrangements.
a common mix-upThe second law is not a force and not a new law of motion. It is a statement about probability.
If every collision is reversible, why does the gas not un-spread?
Show the answer
Because spread-out arrangements outnumber gathered ones by an astronomical factor, so the gas essentially never wanders into a gathered one.
4count ways
With four particles there are two to the fourth power, sixteen, left-or-right assignments. One has all four on the left, four have exactly one on the right, and six have a perfect two-and-two split. The even split already has the most arrangements behind it.
Four particles give sixteen arrangements, and the even split has the most.
With four particles, what is the chance all are on the left?
Show the answer
One in sixteen, since only one of the sixteen arrangements has them all there.
5a hundred
With a hundred particles there are two to the hundredth power, about 1.3 times ten to the thirtieth, arrangements. The count peaks sharply at fifty on each side, and about 96 percent of all arrangements have between forty and sixty on the left. All of them on the left is one arrangement in about 10 to the 30.
At 100 particles, 96 percent of arrangements are within ten of an even split.
Roughly how likely is it for all 100 particles to be on the left?
Show the answer
About one in ten to the thirtieth, essentially never.
6a real gas
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.
7S equals k ln W
Boltzmann's entropy is S equals k times the natural logarithm of W, where W counts the microscopic arrangements that look the same from outside and k is Boltzmann's constant. An isolated system wanders among its arrangements at random, and the spread-out ones outnumber the gathered ones so hugely that it ends up spread out and stays there. So entropy increases because it is overwhelmingly probable, not because anything forces it. It can dip, but for a macroscopic system the odds are beyond astronomical.
Entropy is k times the log of the number of arrangements, and a system drifts toward the overwhelmingly larger count.
a common mix-upEntropy is not just disorder and its rise is not forced. It is a count of arrangements, and the rise is overwhelmingly likely.
Why does entropy increase in an isolated system?
Show the answer
There are vastly more spread-out arrangements than gathered ones, so random motion carries the system toward them and almost never back.
Nothing pushes entropy upward. The spread-out states are simply so much more numerous that the universe almost never finds its way back.