This brief provides a modern pedagogical exposition of the mechanical approach to statistical mechanics initiated by Boltzmann with his early works (1866-1871). Despite the later contribution by Helmholtz, Boltzmann himself (1884-1887), Gibbs, P. Hertz, and Einstein, the mechanical approach remained almost unknown to the modern reader, in favour of the celebrated combinatorial approach, developed by Boltzmann himself during his probabilistic turn (1876-1884). The brief constitutes an ideal continuation of a graduate course of classical mechanics and requires knowledge of basic calculus in many dimension (including differential forms), thermodynamics, probability theory, besides Hamiltonian mechanics. The cornerstone of the whole presentation is the ergodic hypothesis. Special attention is devoted to Massieu potentials (the Legendre transforms of the entropy) which are most natural in statistical mechanics, and also allow for a more direct treatment of the topic of ensemble equivalence.

Preface.- The Constitutive Statements of Thermodynamics.- Minimal Mechanical Model of Thermodynamics.- The Microcanonical Ensemble.- The Canonical Ensemble.- The TP Ensemble.- The Grand Canonical Ensemble.- Ensemble (In)-Equivalence.- Bibliography.

Dr. Michele Campisi is a tenured Researcher at Italy's National Research Council (CNR) working at the Institute for Nanoscience, in Pisa, Italy. He received his PhD in Physics in 2008 from the University of North Texas, USA for which he received the Robert S. Hyer Recognition for "Excellence in Research", along with his supervisor, Prof. Don H. Kobe. He has been research fellow at the University of Augsburg, Germany, where he pursued an intense research programme on non-equilibrium thermodynamics of quantum systems with Prof. P. Hänggi and Prof. P. Talkner. In 2014 he was awarded a Marie Curie Intra-European Fellowship with a project on "Non-equilibrium thermodynamic fluctuations in superconducting devices", which he pursued at Scuola Normale Superiore, Pisa, Italy. From 2017 to 2020 he was Assistant Professor at Università degli studi di Firenze, where he has been teaching general physics for natural science majors, and statistical mechanics for Physics & Astronomy majors.His research interests range from the foundations and the pedagogy of statistical mechanics to quantum technologies.