Do boxes of stars obey thermodynamics? / Boltzmann, Gibbs and the concept of equilibrium (& others)

0:00 Here's an equilibrium state. It depends on whether you choose the forward transition probabilities or the backwards. Let's just take a very crude and simple example. The most simple example you can think of as a microprocess. This is, say, my initial probability over two different states. And this is the transition probability. Now this is a very easy transition probability because in one step it transforms this initial distribution into that distribution which is then the equilibrium distribution because it doesn't change anymore. Now take that first equation and calculate the backward transition probabilities by using Bayes' theorem. You see a funny thing happens, because this thing depends on the P in the initial distribution here. So the transition probabilities are not independent of the initial state. But it has the same property. You can take any arbitrary initial state and you will see that this transition probability will immediately transform it into this, which is then the equilibrium state of the backward transition probability. So, starting here and going to there is moving towards equilibrium or the forward transition probabilities. But going from here to there is an evolution towards an equilibrium distribution too, but the equilibrium of the backwards. So, is there an approach to equilibrium from here to there or from there to here? Both. In both cases, there is. There is no type-deferential, non-inferior result here. Now you might wonder, and this is my last remark, isn't that a very curious thing that the transition probabilities here seem to depend on what was here the initial state? And of course it is. But you can reforce that question too, and in other cases, I mean, I could have taken this to be the forward distribution, and that's what I'm talking about.

2:30 And the funny thing, of course, is that what we have done, in an algebraic approach, is to separate two particular aspects of our probability distributions and give them both completely different scales. This is thought of as a codification of an evolution, and this is thought of as a codification of an initial condition. But if you look at it from a probabilistic point of view, remember that all these probabilities are just aspects of one particular measure that we started with. There's no difference in status between the one-time probability distributions or the two-time probability distributions from which we construct the conditional distributions. They're all just simple probabilities. There's no reason to suppose that we can think of this as a contingent initial condition, whereas these are sort of... These are codified as evolution laws that have a more general validity than the choice of the initial conditions. And so you have the very idea that you can specify the dynamics, the evolutions, without considering the contingent initial distributions. That idea is already something that we have sort of imported from our... Intuitions with other branches of physics, and that's not inherently present in the study of stochastic processes. And with that, I'll take time. All our speakers, and Roman for organizing us. There's a good way to hang around and get questions from people privately. Can I make a brief comment about this paper?

5:00 Einstein basically deduced the stochastic process, position, probability, value, discussed the interaction between photons and radiation. I want to make a motion. That's a stochastic process. He deduced from that stochastic process, he deduced only the equilibrium properties. And the colloquial properties to deduce what Planck's radiation was in complete agreement to date with the experiment. Now, it is very straightforward to generalize Einstein's approach to Markov processes. And it leads to transitions to equilibrium. It leads to the Bose-Einstein distribution. All the physics is absolutely right. Now, the third point is your illusion. Your illusion is based on the fact that there is apparently no time reversal. But that comes about because the transition probabilities... These are the squares of the waveform. The waveform is the amplitude of A, the time-diversal probability, and when you square that and you sum, you get probability conditions which come up. That's of course where the direction of time comes about in quantum mechanics. All of this, by the way, was clearly explained in an article in the American Journal of Physics, but it is interested in a very simple, synagogic way in which an article in the American Journal of Physics can be found in a field. And it is more important of all physics, all of them. I didn't speak about quantum mechanics in this context, but I... I'm not trying to present the most definitive underlying physics in this work. I'm just trying to ask you to study the question whether there is a coherent argument made for deriving irreversible behavior from a particular set of forbade assumptions. And I think that's the only place where I want to claim that there is an illusion, you know.

7:30 I'm not trying to say that quantum chemistry is wrong. End of session on statistical thermodynamics.