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. (shrink)

In Boltzmannian statistical mechanics macro-states supervene on micro-states. This leads to a partitioning of the state space of a system into regions of macroscopically indistinguishable micro-states. The largest of these regions is singled out as the equilibrium region of the system. What justifies this association? We review currently available answers to this question and find them wanting both for conceptual and for technical reasons. We propose a new conception of equilibrium and prove a mathematical theorem which establishes in full generality (...) -- i.e. without making any assumptions about the system's dynamics or the nature of the interactions between its components -- that the equilibrium macro-region is the largest macro-region. We then turn to the question of the approach to equilibrium, of which there exists no satisfactory general answer so far. In our account, this question is replaced by the question when an equilibrium state exists. We prove another -- again fully general -- theorem providing necessary and sufficient conditions for the existence of an equilibrium state. This theorem changes the way in which the question of the approach to equilibrium should be discussed: rather than launching a search for a crucial factor, the focus should be on finding triplets of macro-variables, dynamical conditions, and effective state spaces that satisfy the conditions of the theorem. (shrink)

We discuss Boltzmann’s probabilistic explanation of the second law of thermodynamics providing a comprehensive presentation of what is called today the typicality account. Countering its misconception as an alternative explanation, we examine the relation between Boltzmann’s H-theorem and the general typicality argument demonstrating the conceptual continuity between the two. We then discuss the philosophical dimensions of the concept of typicality and its relevance for scientific reasoning in general, in particular for understanding the reduction of macroscopic laws to microscopic laws. Finally, (...) we reply to various common criticisms of the typicality account. (shrink)

In the last quarter of the nineteenth century, Ludwig Boltzmann explained how irreversible macroscopic laws, in particular the second law of thermodynamics, originate in the time-reversible laws of microscopic physics. Boltzmann’s analysis, the essence of which I shall review here, is basically correct. The most famous criticisms of Boltzmann’s later work on the subject have little merit. Most twentieth century innovations – such as the identification of the state of a physical system with a probability distribution on its phase space, (...) of its thermodynamic entropy with the Gibbs entropy of , and the invocation of the notions of ergodicity and mixing for the justification of the foundations of statistical mechanics – are thoroughly misguided. (shrink)