Ok right. It just seemed like the article and jordigh were making contrasting points. Although neither make an explicit broad claim (in which case, what are we doing here other than running an arbitrary MCMC and then stating a Markov property), the article is saying hey look, rich people get richer, while jordigh is saying, nah "you should eventually see each person hoard all of the money in turn.".

So which is it, probably? I think this might be more complex than "everyone will hord all the money eventually", given that the most probable number of sign changes in a 1D random walk is zero.

I don't understand this example. Are you claiming the laws of thermodynamics only work with a very high probability?

Yes, the second law of thermodynamics is a statistical law. It is physically possible that all the particles in an ideal gas container will be moving east all at the same time, for example (it doesn't break any law of mechanics), or that a temperature fluctuation will spontaneously appear in a material in thermal equilibrium. It's just that the probability for such spontaneous entropy decreases happening is astronomically small.

The second law of thermodynamics is intrinsically a statistical law.

I'm not a physicist and I probably could have phrased that a little better. The example of a bowl of soup randomly and unevenly changing temperature implies to me that the physical system behaves randomly which I do not believe it does.

Entropy can decrease, just with vanishingly small probability: https://en.wikipedia.org/wiki/Fluctuation_theorem

You don't have a detailed knowledge about the state of every molecule in the bowl soup and its environment, so you have to rely on statistical mechanics. And the current macroscopic state that you observe could correspond to a particular microscopical state that evolves in surprising ways. But the probability is so low that it just doesn't happen.

To make things worse, quantum mechanics is intrinsically random (at least as far as we know) and having a detailed knowledge of the state of the system allowing for certain prediction of its evolution is impossible in principle (and not just in practice).