Lectures on the Mechanical Foundations of Thermodynamics

The mechanical approach to statistical mechanics, developed by Boltzmann, was unknown to the modern reader until recently. It requires knowledge of basic calculus, thermodynamics, probability theory, and Hamiltonian mechanics and is based on the ergodic hypothesis. Massieu potentials are used to treat ensemble equivalence.

Format: Paperback / softback
Length: 91 pages
Publication date: 31 October 2021
Publisher: Springer Nature Switzerland AG

This concise overview offers a contemporary pedagogical exploration of the mechanical approach to statistical mechanics, which was pioneered by Boltzmann in his early works (1866-1871). Despite the subsequent contributions of Helmholtz, Boltzmann himself (1884-1887), Gibbs, P. Hertz, and Einstein, the mechanical approach remained largely unknown to the modern reader, in favor of the renowned combinatorial approach developed by Boltzmann during his probabilistic turn (1876-1884). The brief serves as an ideal continuation of a graduate-level course in classical mechanics and necessitates a foundational understanding of basic calculus in numerous dimensions (including differential forms), thermodynamics, probability theory, in addition to Hamiltonian mechanics. The ergodic hypothesis forms the cornerstone of the entire presentation. Special emphasis is given to Massieu potentials (the Legendre transforms of entropy), which are particularly natural in statistical mechanics and facilitate a more direct treatment of the topic of ensemble equivalence.

Introduction:
The mechanical approach to statistical mechanics, initiated by Boltzmann with his early works (1866-1871), provides a fundamental framework for understanding the behavior of complex systems. Despite the later contributions of Helmholtz, Boltzmann himself (1884-1887), Gibbs, P. Hertz, and Einstein, this approach has remained relatively unknown to the modern reader, overshadowed by the celebrated combinatorial approach developed by Boltzmann during his probabilistic turn (1876-1884). This brief aims to rectify this oversight by presenting a comprehensive and accessible exposition of the mechanical approach.

Ergodic Hypothesis:
The ergodic hypothesis is a central concept in statistical mechanics, and it forms the basis of the mechanical approach. According to the ergodic hypothesis, a system in thermodynamic equilibrium will eventually reach a state of statistical equilibrium, where its properties are consistent with the probabilities of its states. This hypothesis is supported by extensive experimental evidence and has played a crucial role in the development of statistical mechanics.

Massieu Potentials:
Massieu potentials are a fundamental concept in statistical mechanics, and they play a crucial role in the mechanical approach. Massieu potentials are the Legendre transforms of the entropy, and they provide a natural way to describe the interactions between particles in a system. By studying the behavior of particles near a given massieu potential, one can gain insights into the thermodynamic properties of the system and the behavior of ensembles of particles.

Ensemble Equivalence:
Ensemble equivalence is a key concept in statistical mechanics, and it is closely related to the mechanical approach. Ensemble equivalence states that two systems with the same thermodynamic properties can be described in terms of different ensembles of particles. This allows for a more efficient and accurate description of complex systems, as it avoids the need to consider all possible states of the system.

Conclusion:
In conclusion, this brief provides a modern pedagogical exposition of the mechanical approach to statistical mechanics. It highlights the ergodic hypothesis as the cornerstone of the approach, discusses the significance of Massieu potentials, and emphasizes the importance of ensemble equivalence. The mechanical approach offers a valuable alternative to the combinatorial approach and provides a deeper understanding of the behavior of complex systems. By exploring the mechanical approach, researchers can gain a more comprehensive understanding of the principles underlying statistical mechanics and apply these principles to a wide range of scientific and engineering problems.

Weight: 180g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030871628
Edition number: 1st ed. 2021