There are two main theoretical frameworks in statistical mechanics, one associated with Boltzmann and the other with Gibbs. Despite their well-known differences, there is a prevailing view that equilibrium values calculated in both frameworks coincide. We show that this is wrong. There are important cases in which the Boltzmannian and Gibbsian equilibrium concepts yield different outcomes. Furthermore, the conditions under which equilibriums exists are different for Gibbsian and Boltzmannian statistical mechanics. There are, however, special circumstances under which it is true that the equilibrium values coincide. We prove a new theorem providing sufficient conditions for this to be the case.

Many examples of calibration in climate science raise no alarms regarding model reliability. We examine one example and show that, in employing classical hypothesis testing, it involves calibrating a base model against data that are also used to confirm the model. This is counter to the ‘intuitive position’ (in favor of use novelty and against double counting). We argue, however, that aspects of the intuitive position are upheld by some methods, in particular, the general cross-validation method. How cross-validation relates to other prominent classical methods such as the Akaike information criterion and Bayesian information criterion is also discussed.

Boltzmannian statistical mechanics partitions the phase space of a system into macroregions, and the largest of these is identified with equilibrium. What justifies this identification? Common answers focus on Boltzmann’s combinatorial argument, the Maxwell-Boltzmann distribution, and maximum entropy considerations. We argue that they fail and present a new answer. We characterize equilibrium as the macrostate in which a system spends most of its time and prove a new theorem establishing that equilibrium thus defined corresponds to the largest macroregion. Our derivation is completely general and does not rely on assumptions about the dynamics or interparticle interactions.

A gas prepared in a nonequilibrium state will approach equilibrium and stay there. An influential contemporary approach to statistical mechanics explains this behavior in terms of typicality. However, this explanation has been criticized as mysterious as long as no connection with the dynamics of the system is established. We take this criticism as our point of departure. Our central claim is that Hamiltonians of gases that are epsilon-ergodic are typical with respect to the Whitney topology. Because equilibrium states are typical, it follows that typical initial conditions approach equilibrium and stay there.

Why do gases reach equilibrium when left to themselves? The canonical answer, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer is now widely regarded as flawed. We argue that some of the main objections in particular arguments based on the Kolmogorov-Arnold-Moser theorem and the Markus-Meyer theorem are beside the point. We then argue that something close to Boltzmann’s proposal is true: gases behave thermodynamic-like if they are epsilon-ergodic, that is, ergodic on the phase space except for a small region of measure epsilon. This answer is promising because there is evidence that relevant systems are epsilon-ergodic.