Is the second law of thermodynamics routinely violated?

[Max_S]

@Half_Man_Half_Wit

@Max_S

@Half_Man_Half_Wit

Take a coin. Suppose you can only throw it in two different ways—way A and way B. Way A always lands heads up; way B always lands tails up. You don’t have control over whether you’ve thrown it according to way A or way B (random initial conditions, you remember). Then, the probability that it comes up heads is 50% on each throw. No further assumptions necessary.

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?

I stand by both of my statements. In the bolded statement you claimed the probability of a coin throw is 50% each time. That is not in any way implied by the initial state of the coins. Besides, neither throw A nor throw B take into account the previous state of the coin because both throws always give heads or tails respectively. You haven’t told me how A is chosen vs B, but you assume the probability is 50%.

@Half_Man_Half_Wit

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.

I think we’re agreed on the subject of phenomenological laws. You haven’t convinced me that the observer is likely to see only increasing greyness, but if that’s all he saw he could very well formulate a law of increasing greyness. I take no issue with the observer’s logic.

@Half_Man_Half_Wit

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.

If we limit violations to observed violations rather than theoretical violations, I will concede the insurmountable difficulty in falsifying a law if violations are actually so improbable as to make observation unrealistic. But I do not yet concede the improbability of observing decreasing greyness.

@Half_Man_Half_Wit

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.

If you are to limit the observer’s sight in such a way that he can only distinguish between almost black, almost white, and everything-else-is-grey, and if I was to assume significant enough differences in color for the observer to notice are rare (which I do not concede yet), I would concede that the observer can in fact observe grey over time despite microscopic fluctuations. He will assume the left and right boxes are equally grey not because they are, but because his senses are so dull as to fail to recognize the difference and constant fluctuation of lighter and darker shades in one box then the other. But I have not yet conceded the underlying premise.

@Half_Man_Half_Wit

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?

No, having more potential states with one property does not imply higher probability for ‘evolution’ towards a state with said property. Simply having more states says nothing about probability. You have not given me a basis for a probability distribution and, quite to the contrary, you denied that the distribution is random. In fact you say the underlying dynamics are deterministic. How do you come to the conclusion that a system in a state of intermediate grayness will probably evolve into a state of higher greyness? It seems like you are pulling a postulate from thin air.

~Max