A Didactic Journey from Statistical Physics to Thermodynamics

TL;DR

This paper provides a detailed, axiomatic derivation of thermodynamics from statistical physics, clarifying key concepts, correcting misconceptions, and establishing rigorous formal connections between the two fields.

Contribution

It offers a comprehensive formalization of thermodynamic concepts derived from statistical physics, including ensembles, potentials, and the Legendre transform, with rigorous proofs and geometric insights.

Findings

Formal derivation of thermodynamics from statistical physics

Clarification of misconceptions such as Lagrange multipliers

Rigorous proofs of thermodynamic principles from axioms

Abstract

This paper offers a pedestrian guide from the fundamental properties of entropy to the axioms of thermodynamics, which are a consequence of the axiom of statistical physics. It also dismantles flawed concepts, such as assigning physical meaning to Lagrange multipliers and numerous others. This work also provides a comprehensive understanding of the Legendre transform via geometrical, mathematical and physical insights, as well as its connection to the experimental setup. The central result of this paper is the comprehensive formalisation of key concepts, including ensembles, variable dependencies, potentials and natural variables. Furthermore, the framework of thermodynamics, the state function and the Euler inequality are rigorously proven from the axiom of statistical physics.