Is the second law of thermodynamics routinely violated?

[Half_Man_Half_Wit]

@Max_S

Bolding mine. That sentence is an assumption in and of itself, not implied by the rest of the paragraph. You have assumed that the probability of throwing way A or throwing way B is 50%. You are allowed to make that assumption, but you must acknowledge that you made it to complete your argument.

~Max

But we had already settled that issue, I thought:

@Max_S

I am perfectly fine allowing for equiprobability of the initial microstate of a toy thermodynamic system.

Are you no longer fine with this? And, presuming that you are, do you agree that then, the probability of the coin comes out to 50% for each possibility? And likewise for the die?

@Max_S

I think we understood each other, right up until you said “in any concrete, reasonable large series of trials, we won’t observe a violation.” That’s not true. You would have to say we probably won’t observe a violation, and you can drop the rest of that sentence. But this is a minor issue of semantics. The person formulating the law does not think so, and that is why his law is flawed.

~Max

Well, I could drag the extra verbiage through every statement I make, but I think it’s good enough to say ‘we won’t observe any violation’ instead of ‘for any reasonable time scale, the probability of observing a violation is as small as we care to make it, by increasing the system size’. Because the outcome is the same: if you repeat the experiment some reasonable amount of times, you are astronomically unlikely to ever make an observation contradicting the law of increasing greyness, and thus, you will believe it holds.

You’re right to point out that this is, ultimately, wrong; but the observer has no way to know that, without having the microscopic theory of greyness.

@Max_S

If by ‘the system tends to greyness’ you mean “black and white, if brought into contact, eventually even out to a uniform gray”, I disagree, the probability of a violation is exactly 100%. The colors of the two boxes never “eventually even out to a uniform gray”, and our observer will notice a violation if he is allowed to observe the full period of the system.

Sure. But a full period, for any reasonably macroscopic system, is going to be fantastically huge. So you’re not going to observe anything of the sort.

We’re talking about generalizations made from actually feasible observations. Given this, while there is an astronomically small probability of actually observing a violation of the law of increasing greyness, the far more probable course is going to be that, during the tens or hundreds or thousands of times that the experiment is repeated, no violation is observed, and thus, the law has the status of any other physical law ever formulated.

In the likely case that at least one marble from either box moves into the other box, one or both boxes will always be changing color.

But not on any observable level. The observer has eyes that aren’t significantly better than a human being’s, so, while, again, you’re right in principle, these deviations are not going to be observed. Remember, laws are formulated based on actual observations. And, with overwhelming probability, any actual observation is going to be the system evolving to an equal grey and staying that way, at least, for sufficiently large numbers of marbles.

You did not prove to my satisfaction that, in any random configuration of marbles moving around in a box, entropy is more likely to increase than decrease over time.

That is proven simply by the fact that there are more states of high greyness than there are states of uneven color distribution. For if that’s the case, and the microdynamics is reversible, then, if you take a state of intermediate greyness, there will be more ways (‘more’ meaning here ‘astronomically many more’) to evolve to a state of higher greyness than to evolve to a state of lower greyness. Thus, whenever you find the system in a state of intermediate greyness (i. e. uneven color distribution), then, with astronomic likelihood, the state at the next timestep will be one of higher greyness.

Take my introductory example. Here’s again the distinguishable macrostates, together with the number of their microscopic realizations:

[ul]
li: 6[/li]li: 3[/li]li: 3[/li]li: 3[/li]li: 3[/li]li: 3[/li]li: 3[/li]li: 1[/li]li: 1[/li]li: 1[/li][/ul]

No matter what the microscopic dynamics are, each of the three microstates realizing, say, (A2B1C0) has 6 states corresponding to macrostates of higher entropy it can evolve to, 17 states of equal entropy, but only 3 states of lower entropy. Does that help clear things up?