Foundation of statistical mechanics: Mechanics by itself

Abstract

Statistical mechanics is a strange theory. Its aims are debated, its methods are contested, its main claims have never been fully proven, and their very truth is challenged, yet at the same time, it enjoys huge empirical success and gives us the feeling that we understand important phenomena. What is this weird theory, exactly? Statistical mechanics is the name of the ongoing attempt to apply mechanics (classical, as discussed in this paper, or quantum), together with some auxiliary hypotheses, to explain and predict certain phenomena, above all those described by thermodynamics. This paper shows what parts of this objective can be achieved with mechanics by itself.It thus clarifies what roles remain for the auxiliary assumptions that are needed to achieve the rest of the desiderata. Those auxiliary hypotheses are described in another paper in this journal, Foundations of statistical mechanics: The auxiliary hypotheses.

Infinite idealizations in physics

“…Such singularities or discontinuities, which correspond to a nonanalytic point where the variable or function in question diverges upon taking a derivative (of some order), mark a crossing of a coexistence line and a corresponding phase transition (see Figure b). The lower‐level, fundamental theory that describes thermodynamic systems is classical or quantum statistical mechanics (see Shenker, 2017a, 2017b, for a philosophically friendly overview of the foundations statistical mechanics). Statistical mechanics follows thermodynamics in representing PT as singularities in the Helmhotlz or Gibbs free energy.…”

Foundation of statistical mechanics: The auxiliary hypotheses

“…Let us begin by describing the basic ontology (for more details, see Shenker, ). According to classical mechanics , at every moment, the universe is in some well‐defined state called a microstate , consisting of the positions and velocities of all the particles .…”

“…Of course, we know from mechanics that a gas that satisfies the Maxwell–Boltzmann distribution evolves with time through a continuous sequence of different mechanical microstates , but (for as long as the gas satisfies the thermodynamic ideal gas law) all of these microstates share an aspect , namely, the all have the same energy distribution among the particles (this is a Boltzmannian picture. See Shenker, for the Gibbsian counterpart of this idea). While an aspect (such as satisfying the Maxwell–Boltzmann energy distribution) pertains to an individual microstate, it also gives rise to a set of microstates, namely the set of the microstates all of which share this aspect and presumably differ in other aspects .…”

“…The main idea of statistical mechanics is that each thermodynamic magnitude is associated with an aspect of the mechanical microstate, and as a system evolves and one of its microstate is replaced by the next according the equation of motion, these aspects exhibit, regularities, and those explain the thermodynamic regularities. (On how the mechanical aspects are associated with thermodynaic magnitudes see Shenker, ).…”

“…Statistical mechanics is the name of the ongoing attempt to explain and predict certain phenomena, above all those described by thermodynamics, on the basis of the fundamental theories of physics, in particular mechanics (classical or quantum) . However, while it is possible to describe the thermodynamic regularities using purely mechanical terms (as shown in Shenker, ), it is impossible to derive these regularities from mechanics. This, for two reasons.…”