Probabilistic Explanations and the Derivation of Macroscopic Laws

Abstract

We will discuss the link between scientific explanations and probabilities, specially in relationship with statistical mechanics and the derivation of macroscopic laws from microscopic ones. 1 18 See for example figure 4 where i = −1 and i+1 = +1. 19There is a small abuse here, because it seems that we change the laws of motion by changing the orientation (from clockwise to counterclockwise). But one can attach another discrete "velocity" parameter to the particles, having the same value for all of them, and indicating the orientation, clockwise or counterclockwise, of their motion. Then, the motion is truly reversible, and we have simply to assume that the analogue here of the operation I of (9) changes also that extra velocity parameter.20 This is analogous to the Poincaré cycles in mechanics, except that, here, the length of the cycle is the same for all configurations (there is no reason for this feature to hold in general mechanical systems).

The paradox of deterministic probabilities

“…In other words, it seems satisfactory to say that we have explained a phenomenon if we have shown that it holds for typical initial conditions, since we cannot do it for all of them. As Bricmont (2001Bricmont ( , 2020 has pointed out, if something is typical, no further explanation seems to be required because we can expect this phenomenon to happen. For instance, the fact that a gas expands freely is not surprising: this is what we expect to happen, given what we have discussed so far.…”

“…8 See Goldstein (2001). Others instead wish to justify it on Bayesian terms (Bricmont 2001(Bricmont , 2020. always entropy-decreasing microstates.…”

Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

“…And what if the subjects made mistakes, what if they computed the time-evolved distribution ρ • Φ t incorrectly, what if their beliefs were irrational-would that end the validity of subjective entropy? Somebody may be inclined to say that subjective entropy is valid only if it is rational (e.g., Bricmont, 2019), but that means basically to back off from the thought that entropy is subjective. It means that it does not play much of a role whether anybody's actual beliefs follow that particular ρ, but rather that there is a correct ρ that should be used; we will come back to this view at the end of the next subsection.…”

“…This brings us once more to the idea that the ρ in S G (ρ) is the subjective belief of a rational observer as advocated by Bricmont (2019). One could always use the Boltzmann entropy and add a narrative about observers and their beliefs, such as: Whenever X ∈ Γ ν , a rational observer should use the flat distribution over Γ ν , and the Gibbs entropy of that observer's belief is what entropy really means.…”