https://orcid.org/0009-0005-1770-3479
Foundations of Statistical Mechanics, which was conscientiously written by Roman Frigg and Charlotte Werndl is a book that focuses on one of the three pillars of modern physics besides quantum theory and relativity theory, that is Statistical Mechanics (SM). This book is explained in four chapters, which are Introduction, Mechanics and Probability, Boltzmannian Statistical Mechanics (BSM), and Gibbsian Statistical Mechanics (GSM), and each chapter is also divided into several sections.
Chapter 1 explains about the introduction of the book. The desideratum of SM is to expound the connection between microscopic-physics and macroscopic-physics, as well as the macroscopic system’s behavior in terms of the dynamical laws that regulate the microscopic constituents of a system and probabilistic assumptions. The motivation for using probabilities is based on the fact that systems in SM have a large number of microscopic constituents that are of the order of Avogadro’s number. It also explains the focus of SM which is equilibrium, that is behavior of macrosystems’ special aspect. Chapter 1 also explains that SM is different from other theories such as quantum mechanics and relativity theory because SM has not yet found a generally accepted theoretical framework, specifically unique canonical formalism. BSM and GSM are used as two broad theoretical approaches in SM.
Chapter 2 explains mechanics and probability and is divided into several sections, which coherently discuss the theoretical background of SM to the relationship between mechanics and probability. As SM can be formulated from classical mechanics which will produce classical SM then this chapter will discuss classical mechanics. First, this chapter discussed an introduction to dynamical systems. Several basic terms and notations from mechanics are also introduced. Hamiltonian systems are a special class of dynamical systems. Then, explain the basic concepts of Hamiltonian mechanics, the equations, and the notations used in the equations. As an important aspect of Hamiltonian dynamics, the concept of time reversal invariance is also introduced. It is also shown how Hamiltonian mechanics behaves in time reversal transformations. The second discussed aspect of the Hamiltonian system is the Poincare Recurrence. It is shown how the Hamiltonian system behaves in the long run. This behavior is explained by the recurrence theorem contained in this chapter. As one of the properties that plays an important role in dynamic systems, ergodicity is also explained. It also explains how ergodicity plays an important role using illustrations, equations, and theorems that explain this. It is also explained how the ergodicity of the Hamiltonian system is. Last, probability is discussed. It explains the concepts of formal probability theory, probability densities, and answers to the question of what probability describes. This chapter also explained how mechanics and probability are related.
Chapter 3 describes BSM and is divided into several parts, such as the core BSM approach that identifies the basic structure of BSM and its variations. A more detailed explanation of the BSM is explained in a structured manner in nine subchapters. It gives the core of BSM itself, which distinguishes between microstates and macrostates. BSM states that macrostates depend on microstates, meaning that two systems cannot be in different macrostates without also being in different microstates. Each microstate corresponds to one macrostate. To construct macrostates and identify equilibria in the BSM, they explain that the Boltzmann combinatorial argument is used, which provides the basis for characterizing the macrostate and describes the system’s approach to equilibrium. Then, this book described another approach, ergodicity, which explains how a system approaches an equilibrium state. However, this approach faces two challenges known as the Loschmidt rejection and the Zermelo rejection, which question the validity of the ergodicity approach. Next, these two objections are discussed. The Loschmidt rejection is based on time reversal invariance and argues that the microstate that causes the equilibrium state can evolve backward in time. Meanwhile, Zermelo’s rejection is based on Poincaré’s recurrence. It states that a system can return to its previous state after a considerable time, which implies that the ergodicity approach does not fully explain the behavior of the system toward equilibrium. Furthermore, they introduce the Residence Time Account as an alternative approach to explain equilibrium by replacing the combinatorial argument and ergodicity with the concept of residence time. This provides a new perspective in understanding the phenomenon of thermodynamic equilibrium. Later, they use another approach to explain equilibrium, which uses the concept of typicality to argue that systems tend to reach equilibrium because micro-states that correspond to more probable macro-states have higher probabilities. Next, they explain ways to incorporate probabilities into the BSM. This provides a foundation for understanding how probabilities are incorporated into the BSM framework. Then, this book discusses a typical approach to solving some problems in BSM, using Metaculus and the Past Hypothesis. By combining conditional probability with the Past Hypothesis, this approach provides a way to combine probability with system dynamics in BSM. Lastly, this book acknowledges the problems and limitations faced by BSM. It highlights that while there has been significant progress, challenges remain in consistently combining probabilities with system dynamics. Overall, this chapter provides an important foundation for further understanding of statistical physics.
Chapter 4 explains about GSM and is divided into 10 sections. This chapter begins by introducing the formalism of GSM and its interpretation. One of the important principles of GSM is the Averaging Principle (AP). However, this principle fails to be well articulated based on ergodicity. This chapter explains how this failure occurred and what approaches have been taken to correct it. To add more perspectives to the interpretation of the formalism, probabilism is discussed. It is explained what if we view GSM as a theory of probability. By using probability, AP is then re-articulated by using fluctuations. It also discusses how to approach equilibrium in GSM and how to characterize it. However, there is a problem in the characterization of GSM so to carry out the characterization, restrictions are made on GSM which are discussed in the Coarse-Graining subchapter. To expand the assumptions of the case discussed, it is also discussed what if we look at our approach from an interventionist perspective. It is explained what problems arise when looking from this point of view. Next, there is also an explanation of epistemic accounts. It explains how epistemic accounts require SM to reconceptualize. Finally, in this chapter, the relationship between GSM and BSM is explained and several open questions about GSM are included which were obtained from previous discussions.
There are some minor weaknesses of this book that could be improved, as they do not significantly affect the content. First, the table of contents is incomplete, as it does not list all the sections in each chapter, which could impede readers’ ability to find the content that they want. Then, the quality of the figures is not HD, as a result, attempts to zoom the figure result in a blurred picture. Next, the structure of the book might be improved with chapter separation for easier reading. Further, with the extensive use of abbreviations in this book, adding a list of abbreviations could enhance the reading experience. Also, not having a list of figures may be a disturbance for the readers to find the right figure. Finally, enhancing the visual appeal by improving how figures are placed, such as arranging the figures without using a lot of space would be good. This book is recommended for anyone who wants to learn about the foundation of SM because this book gives details about theories that are used as the basis of SM.
Department of Mathematics, Institut Teknologi Bandung, Indonesia
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Andriko
Department of Mathematics, Institut Teknologi Bandung, Indonesia
Nathan
Department of Aerospace Engineering, Institut Teknologi Bandung, Indonesia
Zulfaidil
Department of Mathematics, Institut Teknologi Bandung, Indonesia
Maya Nabila
Combinatorial Mathematics Research Group, Institut Teknologi Bandung, Indonesia