Reducing Thermodynamics to Boltzmannian Statistical Mechanics: The Case of Macro Values

TL;DR

This paper examines the discontinuity problem in Boltzmannian Statistical Mechanics, showing how it challenges the reduction of thermodynamics to BSM due to the model's inherent macro value discontinuities.

Contribution

It identifies the discontinuity problem in BSM and analyzes its implications for the reduction of thermodynamics to BSM, highlighting a foundational challenge.

Findings

Discontinuity arises from phase space partitioning in BSM.

Thermodynamic macro values do not supervene on BSM macro values.

The reduction of thermodynamics to BSM faces significant conceptual obstacles.

Abstract

Thermodynamic macro variables, such as the temperature or volume macro variable, can take on a continuum of allowable values, called thermodynamic macro values. Although referring to the same macro phenomena, the macro variables of Boltzmannian Statistical Mechanics (BSM) differ from thermodynamic macro variables in an important respect: within the framework of Boltzmannian Statistical Mechanics the evolution of macro values of systems with finite available phase space is invariably modelled as discontinuous, due to the method of partitioning phase space into macro regions with sharp, fixed boundaries. Conceptually, this is at odds with the continuous evolution of macro values as described by thermodynamics, as well as with the continuous evolution of the micro state assumed in BSM. This discrepancy I call the discontinuity problem (DP). I show how it arises from BSM's framework and demonstrate its consequences, in particular for the foundational project of reducing thermodynamics to BSM: thermodynamic macro values are shown to not supervene on the corresponding BSM macro values. With supervenience being a conditio sine qua non for the kind of reduction envisaged by the foundational project, the latter is in jeopardy.