Is the second law of thermodynamics routinely violated?

[Max_S]

@Half_Man_Half_Wit

But that’s just counting. If there are more ‘more grey’ states than ‘less grey’ states, then a system in a ‘less grey’ state has more ways of evolving towards a ‘more grey’ state.

So assume some ‘less grey’ state G[sub]<[/sub] can be realized by means of white and black marbles in ten different ways (corresponding to microstates G[sub]<[/sub][sup]1[/sup] through G[sub]<[/sub][sup]10[/sup]). Assume there are more ways to realize the ‘more grey’ state G[sub]>[/sub] than there are ways of realizing the ‘even less grey’ state G[sub]<<[/sub]. Then, more of these ten states will evolve towards the more grey state than will evolve towards the less grey one.

Say there are eight states G[sub]>[/sub][sup]1[/sup] through G[sub]>[/sub][sup]8[/sup], and the states G[sub]<<[/sub][sup]1[/sup] and G[sub]<<[/sub][sup]2[/sup]. These are the states available from either of the states realizing G[sub]<[/sub] on the next time-step. Then, eight out of ten times, if the system is in the state G[sub]<[/sub], we will see it evolve into G[sub]>[/sub]. (Perhaps it helps to recall, here, that a reversible evolution always takes different states to different states, as if it didn’t, i. e. by taking two different states to the same following state, you can’t tell which was the original state from looking at the later one, and thus, can’t reverse the evolution.)

Thus, at each time-step, we’re more likely to observe an increase in grey-ness, simply because there are more ways to get more grey.

Let’s say we have a number line from negative ten to positive ten, where negative ten is black, zero is grey, and ten is white. On this number line is a point whose initial position is random, and whose position at every time step is random. The absolute value of the point constitutes the entropy of the system you describe.

We can see now that a point on the leftmost position, -10, has a 19/21 probability of becoming “more grey” or “less entropic” at the next step, and a 2/21 probability of remaining at the same distance from grey or of having no change in entropy. So does a point on the rightmost position. Here are the other values (you may need to switch forum themes at the bottom left of the page):

   Position to Entropy    
Pos | Less  | Same |  More
-10 | 19/21 | 2/21 |  0/21
- 9 | 17/21 | 2/21 |  2/21
- 8 | 15/21 | 2/21 |  4/21
- 7 | 13/21 | 2/21 |  6/21
- 6 | 11/21 | 2/21 |  8/21
- 5 |  9/21 | 2/21 | 10/21
- 4 |  7/21 | 2/21 | 12/21
- 3 |  5/21 | 2/21 | 14/21
- 2 |  3/21 | 2/21 | 16/21
- 1 |  1/21 | 2/21 | 18/21
  0 |  0/21 | 1/21 | 20/21
  1 |  1/21 | 2/21 | 18/21
  2 |  3/21 | 2/21 | 16/21
  3 |  5/21 | 2/21 | 14/21
  4 |  7/21 | 2/21 | 12/21
  5 |  9/21 | 2/21 | 10/21
  6 | 11/21 | 2/21 |  8/21
  7 | 13/21 | 2/21 |  6/21
  8 | 15/21 | 2/21 |  4/21
  9 | 17/21 | 2/21 |  2/21
 10 | 19/21 | 2/21 |  0/21

Total all of those probabilities together and we can see that after one instant, there is a 200/441 probability that the system will move to a less entropic state, a 41/441 probability that the entropy will not change, and a 200/441 probability that entropy will increase. In percentages, that’s about 45% for entropy to decrease, 9% to stay the same, and 45% to increase.

And this is after assuming the equiprobability of microstates at each measure of entropy, which I do not wish to assume.

~Max