Utrecht University Master's Thesis The Polysemic Gibbs Paradox as a Reflection of Foundational Developments in Physics (1876{1911) Supervisors: Author: prof. dr. J.A.E.F. van Dongen, Lara van Zuilen UU, UvA Student ID: prof. dr. J.B.M. Uffink, 3645851 UMN A thesis submitted in fulfilment of the requirements for the Master's degree in History and Philosophy of Science at the Descartes Centre for the History and Philosophy of the Sciences and the Humanities Institute for History and Foundations of Science November 9, 2015 UTRECHT UNIVERSITY Abstract Graduation Colloquium September 18th, 2015: \Gibbs' Paradox" or What various perceptions of a persistent problem in physics can tell us about foundational developments from 1876 to the 1920s. Since 1876, generations of physicists have concerned themselves with the so-called Gibbs paradox on the basis of two excerpts { published in 1876 and 1902 { written by Gibbs himself. A closer look at the offered solutions and other contemplations reveals that there is no such thing as a single, unambiguous Gibbs paradox. Instead, an arrangement of solutions can be found for distinct, yet related interpretations of the paradox. Despite the fact that over 150 works have been dedicated to the paradox, none contain a clear historical analysis of how the perception of the paradox itself has changed. The exception to the rule is Russian history of physics. As a matter of fact, one Russian historian has written an entire book on the solutions to the Gibbs paradox. However, seeing as most scholars don't actually read the Russian language, these sources have remained largely unexplored. Using these Russian sources in addition to the more common sources then not only broadens the scope, but also introduces those sources into the English academic world. To be more precise, my thesis attempts to answer the question of how the perception of the Gibbs paradox has changed from its inception in 1876 until its early inclusion in discussions surrounding quantum theory. Furthermore, partly under the assumption that physicists tend to focus on problems considered relevant in their time, I try to see whether the perspectives on the paradox reflect on developments within foundational concepts in physics. If so, the historical analysis of the paradox offers another means of describing the history of ideas in physics around the turn of the 20th century. Subse- quently, the thesis might then be used to both clarify the notion of the Gibbs paradox and to offer a supporting role in the history of the development from thermodynamics to quantum mechanics. Contents Abstract of the Graduation Colloquiumi Contents ii Symbols iv 1 Introduction1 1.1 Background...................................2 1.2 Research Question and Framework......................4 1.3 Note on Paradoxes...............................5 1.4 Thesis Contents.................................6 2 1876: The Thermodynamic Considerations8 2.1 A Short History of Thermodynamics prior to 1876.............9 2.1.1 The Ideal Gas Laws..........................9 2.1.2 Heat Theory and the Thermodynamic Laws............. 10 2.1.3 Thermodynamic Concepts....................... 12 2.2 Gibbs' First Writings on the Thought Experiment............. 15 2.3 J. Willard Gibbs................................ 27 2.3.1 Scientific Style............................. 28 2.3.2 Writings on Thermodynamics..................... 29 2.3.3 Position in the Scientific Community................. 30 2.4 Conclusion................................... 32 3 1876-1902: The Initial Interpretations 33 3.1 Maxwell's Inclusion of the Concept of Reversibility............. 34 3.2 Neumann's Indication of a Discrepancy................... 36 3.3 Duhem's and Poincar´e'sEntropy Principles Revisions........... 37 3.4 Nernst's Experimental Limit......................... 40 3.5 Wiedeburg's Introduction of a Paradoxical Continuity........... 43 3.6 Planck's Continuous Physical Properties................... 44 3.7 Larmor's Impracticably Large Diffusion Time................ 45 3.8 Van der Waals' Identical Molecules...................... 46 ii Contents iii 3.9 Van Laar's Diffusion Tendency........................ 50 3.10 Ostwald's and Le Ch^atelier'sInfluence on Gibbs' Reach.......... 51 3.11 Analysis of the Paradox Interpretations................... 53 3.11.1 Conclusion............................... 55 4 1902: The Statistical Considerations 56 4.1 A Short History of Statistical Mechanics prior to 1902........... 57 4.1.1 Entropy and Probability........................ 58 4.1.2 Ensembles and Analogies....................... 60 4.2 Gibbs' Second Writings on the Thought Experiment............ 61 4.2.1 Comparison of Gibbs' Writings on the Thought Experiment.... 65 4.3 J. Willard Gibbs................................ 66 4.3.1 Scientific Style............................. 67 4.3.2 Writings on Statistical Mechanics................... 68 4.3.3 Position in the Scientific Community................. 69 4.4 Conclusion................................... 70 5 1902 - 1911: The Subsequent Interpretations 71 5.1 Larmor's Defense of the Molecular Principle................. 73 5.2 Byk's Symmetry in Optically Active Substances............... 75 5.3 Boltzmann's Relatedness of Entropy and Probability............ 77 5.4 Lorentz's Gravity and the Thermodynamic Potentials........... 78 5.5 Van Laar's and Van der Waals' Continued Discussions........... 81 5.6 Postma's `Proof' of the Failure of Analogy.................. 83 5.7 Beyond 1911.................................. 85 5.8 Analysis of the Paradox Interpretations................... 87 5.8.1 Conclusion............................... 89 6 Conclusion 90 6.1 Suggestions for further research........................ 92 A Correspondence and Reprints 93 B Main Responses to the Paradox 100 C Synopsis of Haitun's History of the Gibbs Paradox 103 Bibliography 106 1. Primary Sources.................................. 106 2. Secondary Sources................................. 110 Symbols AA a gas `mixture' of alike or identical gases A and A AA0 a gas mixture of increasingly similar gases A and A0 AB a gas mixture of unlike gases A and B a constant | pressure at t=1 and v=1 or volume at t=1 and p=1 aif interactive forces in the Van Der Waals equation bms molecular size in the Van Der Waals equation c constant | specific heat at constant volume ft time-dependent energy distribution function of particles H entropy of a unit of gas for t = 1 in equation (2.1), otherwise the H-theorem k; kB Boltzmann's constant m; M quantity of a substance n amount in moles or the number of molecules N number of particles p pressure of the system pi probability Q quantity of heat R gas constant r number of molecules S entropy t; T; Θ temperature of the system v; V volume of the system w work W number of microscopic configurations corresponding to a macroscopic state x ratio of the gases in a mixture energy of the system η Gibbs' notation for entropy of the system Θ modulus of a canonical ensemble µi chemical potential ν number of different kinds of particles φ free energy iv 1 Introduction Few topics lend themselves more readily to scientific and philosophical discussion than those related to foundational paradoxes. After all, the effort it takes to validate writing on a subject where the premises clash with the conclusions is less than writing on one that is considered to be internally consistent. This is also the case for physics related paradoxes of the late nineteenth and the early twentieth centuries, of which there are quite a few. One of the popular paradoxes in thermodynamics is the Gibbs paradox. The exact paradoxical element of the Gibbs paradox is not easily indicated and the physicists' perception of it has changed over time. In other words, the term `Gibbs paradox' has become polysemic since its inception. In addition to this, several alter- native terms have been used to describe the paradox. Examples of this are the mixing paradox, the entropic paradox, and the discontinuity paradox. The Gibbs paradox is, however, always related to a thought experiment in which the entropy is calculated after two gases of equal volume are interdiffused into one single volume. The gases in the resulting mixture may either be different (AB), infinitely similar (AA0), or thought to be identical (AA). The value of the entropy of mixing may or may not, depending on the interpretation, rely on the degree of similarity between the gases. Renditions of the paradox almost always deal with comparing the various types of gas mixtures and their entropies. Since its inception in 1876, over 150 articles and book chapters have been written on the subject,1 most of which offer a solution to the paradox. Many well-known physicists considered the paradox at one point in their career. For instance, Boltzmann, Lorentz and Einstein all proposed solutions to the problematic thought experiment. Despite all these contributions, many feel that there still is no suitable solution. The abundance of publications, and the connection the paradox has to various branches in physics, makes it a suitable concept for a historical analysis of foundational issues around 1900. 1Many sources up to and including 2009 on the Gibbs paradox may be found on the webpage http://www.mdpi.org/lin/entropy/gibbs-paradox.htm. 1 Introduction. 2 1.1 Background One of the reasons so many works are dedicated to the Gibbs paradox is that it is not entirely clear what exact aspect of the paradox is supposed to be paradoxical. A quick survey of the articles on this subject reveals that the term `Gibbs paradox' has been used to denote different conceptual difficulties, all related to similar sections in two of J. Willard Gibbs' publications.2 To avoid confusion, the first section discussing the para- dox will be referred to as GP1876 while the second one will be referred to as GP1902. In addition to the paradox having multiple renditions or interpretations, the thought experiment has been extended to account for more and more physical phenomena and chemical substances. For example, Lorentz used gravity in his account of the interdiffu- sion. Having this background information on the paradox is vital when trying to provide a solution to one or more of these renditions.