Atomism, Boltzmann and nonequilibrium statistical mechanics

We often try to infer reality of objects, or concepts, by assuming their existence or consistence and deducing consequences so extreme in appearance, with respect to the starting assumptions, to imply the searched reality, once their truth can be be checked. This process has been behind many of the great advances in Science, in Physics at least. As an example I am thinking of the "aristotelian" view of motion (as composition of uniform circular revolutions, Sc]) which proved to the Ancient Greeks the reality of the quasi periodic planetary motions. Or of Saccheri's attempt, S], to infer the non existence of a non euclidean geometry by assuming the non validity of the fth Euclid's Axiom and proceeding to a breathtaking, almost endless, search for a contradiction, which he thought to have found in the necessary validity of some consequences that he regarded as repellent, failing to see that having found no logical contradictions he had in fact almost established the existence of non{euclidean geometry. A fact recognized later by Gauss, Bolyai, Lobachevsky. I will concentrate, here, on Boltzmann's investigation and connrmation of the atomic hypothesis from the analysis of its prime consequence, statistical mechanics of equilibrium. One reason this is worth examining now is its relation to recent developments that, having been ourselves witnesses to them, can be perhaps better understood via their connection with similar past examples. The aspect of Boltmann's work that I will examine is his heat theorem: he worked all his scientiic life on this problem and one of his last active research works presents a complete solution (a solution usually attributed to Gibbs, who in fact attributed it to Boltzmann, Gi]), B84]. The question is the following: if we suppose matter to consist of many point like atoms interacting pairwise via forces and moving according to Newton's second law, can we deduce any macroscopic consequences that, through their test, could connrm the assumption? Kinetic theory was clearly the key to the answer, as many predecessors and contemporaries of Boltzmann had seen, C]. Boltzmann asks the question: since the quantities that are the object of investigation in equilibrium thermodynamics can be regarded, in a atomistic view of matter, as averages of mechanical quantities, is it then possible that this, by itself, implies that the relation between mechanical averages admitting a thermodynamical interpretation is necessarily such that it implies the second law of thermodynamics? G1], (A,C), G3]. …