DOCSLIB.ORG

The Polysemic Gibbs Paradox As a Reflection of Foundational Developments in Physics (1876–1911)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Polysemic Gibbs Paradox As a Reflection of Foundational Developments in Physics (1876–1911) 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.
Load more

Recommended publications
  • Gibbs' Paradox and the Definition of Entropy
    Entropy 2008, 10, 15-18 entropy ISSN 1099-4300 °c 2008 by MDPI www.mdpi.org/entropy/ Full Paper Gibbs’ Paradox and the Definition of Entropy Robert H. Swendsen Physics Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA E-Mail: [email protected] Received: 10 December 2007 / Accepted: 14 March 2008 / Published: 20 March 2008 Abstract: Gibbs’ Paradox is shown to arise from an incorrect traditional definition of the entropy that has unfortunately become entrenched in physics textbooks. Among its flaws, the traditional definition predicts a violation of the second law of thermodynamics when applied to colloids. By adopting Boltzmann’s definition of the entropy, the violation of the second law is eliminated, the properties of colloids are correctly predicted, and Gibbs’ Paradox vanishes. Keywords: Gibbs’ Paradox, entropy, extensivity, Boltzmann. 1. Introduction Gibbs’ Paradox [1–3] is based on a traditional definition of the entropy in statistical mechanics found in most textbooks [4–6]. According to this definition, the entropy of a classical system is given by the product of Boltzmann’s constant, k, with the logarithm of a volume in phase space. This is shown in many textbooks to lead to the following equation for the entropy of a classical ideal gas of distinguishable particles, 3 E S (E; V; N) = kN[ ln V + ln + X]; (1) trad 2 N where X is a constant. Most versions of Gibbs’ Paradox, including the one I give in this section, rest on the fact that Eq. 1 is not extensive. There is another version of Gibbs’ Paradox that involves the mixing of two gases, which I will discuss in the third section of this paper.
    [Show full text]
  • 5Th World Congress on Physics July 17-18, 2018 Prague, Czech Republic
    Štefan Zajac, J Laser Opt Photonics 2018, Volume:5 DOI: 10.4172/2469-410X-C2-025 5th World Congress on Physics July 17-18, 2018 Prague, Czech Republic Štefan Zajac Czech Technical University, Czech Republic Physics and Prague hysics has been cultivated in Prague since at least 1348, when a University was founded here by Emperor Charles IV. At Pthe beginning the main emphasis was on Astronomy. First astronomers Křišťan from Prachatice (1360 – 1439) and Jan Ondřejov Šindel (1375 – 1456) in co-operation with clockmaker Mikuláš of Kadaň around 1410 had designed and installed the astronomical clock placed on the Old Town Hall tower in Prague. Astronomer and Chief Physician Tadeuš Hájek of Hájek (1525-1600) had inluenced Emperor Rudolph II to invite Tycho Brahe (1546 – 1601) and Johannes Kepler (1571 – 1630) to come to Prague where the first two of his laws of planetary motion have been formulated. Jan Marcus Marci of Kronland (1595 – 1667), a physicist, physician and rector of Prague University made original research in mechanics (impact of bodies) and optics (diffraction of light, explanation of rainbow). In the middle of the 18th century at the Clementinum, the Jesuit College, physicist Josef Stepling (1716 -1778) and his successors at the Prague Faculty of Philosophy promoted Newtonian Physics . Christian Doppler (1803 – 1853) as professor at Prague Polytechnic in 1842 published important phenomenon of the frequency shift due to the velocity of the source of waves relative to the observer. His younger colleague Bohumil Kučera (1874 – 1921) had started pioneering research in radioactivity and had inspired professor Jaroslav Heyrovský (1890 – 1967) to develop polarography for which he in 1959 earned Nobel Prize for chemistry.
    [Show full text]
  • Evolution of Mathematics: a Brief Sketch
    Open Access Journal of Mathematical and Theoretical Physics Mini Review Open Access Evolution of mathematics: a brief sketch Ancient beginnings: numbers and figures Volume 1 Issue 6 - 2018 It is no exaggeration that Mathematics is ubiquitously Sujit K Bose present in our everyday life, be it in our school, play ground, SN Bose National Center for Basic Sciences, Kolkata mobile number and so on. But was that the case in prehistoric times, before the dawn of civilization? Necessity is the mother Correspondence: Sujit K Bose, Professor of Mathematics (retired), SN Bose National Center for Basic Sciences, Kolkata, of invenp tion or discovery and evolution in this case. So India, Email mathematical concepts must have been seen through or created by those who could, and use that to derive benefit from the Received: October 12, 2018 | Published: December 18, 2018 discovery. The creative process of discovery and later putting that to use must have been arduous and slow. I try to look back from my limited Indian perspective, and reflect up on what might have been the course taken up by mathematics during 10, 11, 12, 13, etc., giving place value to each digit. The barrier the long journey, since ancient times. of writing very large numbers was thus broken. For instance, the numbers mentioned in Yajur Veda could easily be represented A very early method necessitated for counting of objects as Dasha 10, Shata (102), Sahsra (103), Ayuta (104), Niuta (enumeration) was the Tally Marks used in the late Stone Age. In (105), Prayuta (106), ArA bud (107), Nyarbud (108), Samudra some parts of Europe, Africa and Australia the system consisted (109), Madhya (1010), Anta (1011) and Parartha (1012).
    [Show full text]
  • Henry Andrews Bumstead 1870-1920
    NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA BIOGRAPHICAL MEMOIRS VOLUME XIII SECOND MEMOIR BIOGRAPHICAL MEMOIR OF HENRY ANDREWS BUMSTEAD 1870-1920 BY LEIGH PAGE PRESENTED TO THE ACADEMY AT THE ANNUAL MEETING, 1929 HENRY ANDREWS BUMSTEAD BY LIUGH PAGE Henry Andrews Bumstead was born in the small town of Pekin, Illinois, on March 12th, 1870, son of Samuel Josiah Bumstead and Sarah Ellen Seiwell. His father, who was a physician of considerable local prominence, had graduated from the medical school in Philadelphia and was one of the first American students of medicine to go to Vienna to complete his studies. While the family was in Vienna, Bumstead, then a child three years of age, learned to speak German as fluently as he spoke English, an accomplishment which was to prove valuable to him in his subsequent career. Bumstead was descended from an old New England family which traces its origin to Thomas Bumstead, a native of Eng- land, who settled in Boston, Massachusetts, about 1640. Many of his ancestors were engaged in the professions, his paternal grandfather, the Reverend Samuel Andrews Bumstead, being a graduate of Princeton Theological Seminary and a minister in active service. From them he inherited a keen mind and an unusually retentive memory. It is related that long before he had learned to read, his Sunday school teacher surprised his mother by complimenting her on the ease with which her son had rendered the Sunday lesson. It turned out that his mother made a habit of reading the lesson to Bumstead before he left for school, and the child's remarkable performance there was due to his ability to hold in his memory every word of the lesson after hearing it read to him a single time.
    [Show full text]
  • Redalyc.On the Sackur-Tetrode Equation in an Expanding Universe
    Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Pereira, S.H. On the Sackur-Tetrode equation in an expanding universe Revista Mexicana de Física, vol. 57, núm. 1, junio, 2011, pp. 11-15 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57024209002 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative ENSENANZA˜ REVISTA MEXICANA DE FISICA´ E 57 (1) 11–15 JUNIO 2011 On the Sackur-Tetrode equation in an expanding universe S.H. Pereira Universidade Federal de Itajuba,´ Campus Itabira, Rua Sao˜ Paulo, 377 – 35900-373, Itabira, MG, Brazil, e-mail: [email protected] Recibido el 5 de abril de 2010; aceptado el 2 de febrero de 2011 In this work we investigate the thermodynamic properties satisfied by an expanding universe filled with a monoatomic ideal gas. We show that the equations for the energy density, entropy density and chemical potential remain the same of an ideal gas confined to a constant volume V . In particular the Sackur-Tetrode equation for the entropy of the ideal gas is also valid in the case of an expanding universe, provided that the constant value that represents the current entropy of the universe is appropriately chosen. Keywords: Expanding universe; ideal gas; Sackur-Tetrode equation. En el presente trabajo investigamos las propriedades termodinamicas´ que son satisfechas por un universo en expansion,´ el cual es lleno por un gas ideal monoatomico.´ Se prueba que las ecuaciones para la densidad de la energ´ıa, la densidad de la entrop´ıa y el potencial qu´ımico son las mismas que las de un gas ideal, el cual se encuentra confinado en un volumen V.
    [Show full text]
  • Gibbs Paradox: Mixing and Non Mixing Potentials T
    Physics Education 1 Jul - Sep 2016 Gibbs paradox: Mixing and non mixing potentials T. P. Suresh 1∗, Lisha Damodaran 1 and K. M. Udayanandan 2 1 School of Pure and Applied Physics Kannur University, Kerala- 673 635, INDIA *[email protected] 2 Nehru Arts and Science College, Kanhangad Kerala- 671 314, INDIA (Submitted 12-03-2016, Revised 13-05-2016) Abstract Entropy of mixing leading to Gibbs paradox is studied for different physical systems with and without potentials. The effect of potentials on the mixing and non mixing character of physical systems is discussed. We hope this article will encourage students to search for new problems which will help them understand statistical mechanics better. 1 Introduction namics of any system (which is the main aim of statistical mechanics) we need to calculate Statistical mechanics is the study of macro- the canonical partition function, and then ob- scopic properties of a system from its mi- tain macro properties like internal energy, en- croscopic description. In the ensemble for- tropy, chemical potential, specific heat, etc. malism introduced by Gibbs[1] there are of the system from the partition function. In three ensembles-micro canonical, canonical this article we make use of the canonical en- and grand canonical ensemble. Since we are semble formalism to study the extensive char- not interested in discussing quantum statis- acter of entropy and then to calculate the tics we will use the canonical ensemble for in- entropy of mixing. It is then used to ex- troducing our ideas. To study the thermody- plain the Gibbs paradox. Most text books Volume 32, Number 3, Article Number: 4.
    [Show full text]
  • Gibb's Paradox for Distinguishable
    Gibb’s paradox for distinguishable K. M. Udayanandan1, R. K. Sathish1, A. Augustine2 1Department of Physics, Nehru Arts and Science College, Kerala-671 314, India. 2Department of Physics, University of Kannur, Kerala 673 635, India. E-mail: [email protected] (Received 22 March 2014, accepted 17 November 2014) Abstract Boltzmann Correction Factor (BCF) N! is used in micro canonical ensemble and canonical ensemble as a dividing term to avoid Gibbs paradox while finding the number of states and partition function for ideal gas. For harmonic oscillators this factor does not come since they are considered to be distinguishable. We here show that BCF comes twice for harmonic oscillators in grand canonical ensemble for entropy to be extensive in classical statistics. Then we extent this observation for all distinguishable systems. Keywords: Gibbs paradox, harmonic oscillator, distinguishable particles. Resumen El factor de corrección de Boltzmann (BCF) N! se utiliza en ensamble microcanónico y ensamble canónico como un término divisor para evitar la paradoja de Gibbs, mientras se busca el número de estados y la función de partición para un gas ideal. Para osciladores armónicos este factor no viene sin que éstos sean considerados para ser distinguibles. Mostramos aquí que BCF llega dos veces para osciladores armónicos en ensamble gran canónico para que la entropía sea ampliamente usada en la estadística clásica. Luego extendemos esta observación para todos los sistemas distinguibles. Palabras clave: paradoja de Gibbs, oscilador armónico, partículas distinguibles. PACS: 03.65.Ge, 05.20-y ISSN 1870-9095 I. INTRODUCTION Statistical mechanics is a mathematical formalism by which II. CANONICAL ENSEMBLE one can determine the macroscopic properties of a system from the information about its microscopic states.
    [Show full text]
  • August/September 2009
    August-September 2009 Volume 18, No. 8 TM www.aps.org/publications/apsnews PhysicsQuest APS NEWS Goes to Kenya A PublicAtion of the AmericAn PhysicAl society • www.aps.org/PublicAtions/apsnews See page 5 APS Awards Three Hildred Blewett Scholarships Presidents Two This year APS has announced electrical properties. three women as recipients of the “It was amazing how many ses- M. Hildred Blewett scholarship. sions there were on graphene at the Chosen by the Committee on March Meeting,” Guikema said. the Status of Women in Physics, She plans to use funds from the three are Janice Guikema at the Blewett Scholarship to further Johns Hopkins University, Marija research the feasibility of using Nikolic-Jaric at the University of graphene as a sensitive magnetic Manitoba, and Klejda Bega at Co- detector. She said that graphene has lumbia University. a lot of potential for use as a Hall Each year the committee se- effect detector to detect nanoscale lects women who are returning to particles and map out magnetic their research careers that had been Janice guikema structures. Currently she is continu- interrupted for family or other rea- ing to look for ways to make the sons. The scholarship is a one-year as a second-time recipient of the material as sensitive as possible. grant of up to $45,000 that can be Blewett Scholarship. She currently In addition she will use scanning used towards a wide range of ne- has a part-time research position at probe microscopy to further ex- cessities, including equipment pro- Johns Hopkins University, where plore the nature of graphene.
    [Show full text]
  • Classical Particle Indistinguishability, Precisely
    Classical Particle Indistinguishability, Precisely James Wills∗1 1Philosophy, Logic and Scientific Method, London School of Economics Forthcoming in The British Journal for the Philosophy of Science Abstract I present a new perspective on the meaning of indistinguishability of classical particles. This leads to a solution to the problem in statisti- cal mechanics of justifying the inclusion of a factor N! in a probability distribution over the phase space of N indistinguishable classical par- ticles. 1 Introduction Considerations of the identity of objects have long been part of philosophical discussion in the natural sciences and logic. These considerations became particularly pertinent throughout the twentieth century with the develop- ment of quantum physics, widely recognized as having interesting and far- reaching implications concerning the identity, individuality, indiscernibility, indistinguishability1 of the elementary components of our ontology. This discussion continues in both the physics and philosophy literature2. ∗[email protected] 1. This menagerie of terms in the literature is apt to cause confusion, especially as there is no clear consensus on what exactly each of these terms mean. Throughout the rest of the paper, I will be concerned with giving a precise definition of a concept which I will label `indistinguishability', and which, I think, matches with how most other commentators use the word. But I believe it to be a distinct debate whether particles are `identical', `individual', or `indiscernible'. The literature I address in this paper therefore does not, for example, overlap with the debate over the status of Leibniz's PII. 2. See French (2000) for an introduction to this discussion and a comprehensive list of references.
    [Show full text]
  • Lecture 6: Entropy
    Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 6: Entropy 1 Introduction In this lecture, we discuss many ways to think about entropy. The most important and most famous property of entropy is that it never decreases Stot > 0 (1) Here, Stot means the change in entropy of a system plus the change in entropy of the surroundings. This is the second law of thermodynamics that we met in the previous lecture. There's a great quote from Sir Arthur Eddington from 1927 summarizing the importance of the second law: If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equationsthen so much the worse for Maxwell's equations. If it is found to be contradicted by observationwell these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of ther- modynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. Another possibly relevant quote, from the introduction to the statistical mechanics book by David Goodstein: Ludwig Boltzmann who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. There are many ways to dene entropy. All of them are equivalent, although it can be hard to see. In this lecture we will compare and contrast dierent denitions, building up intuition for how to think about entropy in dierent contexts. The original denition of entropy, due to Clausius, was thermodynamic.
    [Show full text]
  • December 4, 1954 NATURE 1037
    No. 4440 December 4, 1954 NATURE 1037 COPLEY MEDALLISTS, 1915-54 is that he never ventured far into interpretation or 1915 I. P. Pavlov 1934 Prof. J. S. Haldane prediction after his early studies in fungi. Here his 1916 Sir James Dewar 1935 Prof. C. T. R. Wilson interpretation was unfortunate in that he tied' the 1917 Emile Roux 1936 Sir Arthur Evans word sex to the property of incompatibility and 1918 H. A. Lorentz 1937 Sir Henry Dale thereby led his successors astray right down to the 1919 M. Bayliss W. 1938 Prof. Niels Bohr present day. In a sense the style of his work is best 1920 H. T. Brown 1939 Prof. T. H. Morgan 1921 Sir Joseph Larmor 1940 Prof. P. Langevin represented by his diagrams of Datura chromosomes 1922 Lord Rutherford 1941 Sir Thomas Lewis as packets. These diagrams were useful in a popular 1923 Sir Horace Lamb 1942 Sir Robert Robinson sense so long as one did not take them too seriously. 1924 Sir Edward Sharpey- 1943 Sir Joseph Bancroft Unfortunately, it seems that Blakeslee did take them Schafer 1944 Sir Geoffrey Taylor seriously. To him they were the real and final thing. 1925 A. Einstein 1945 Dr. 0. T. Avery By his alertness and ingenuity and his practical 1926 Sir Frederick Gow­ 1946 Dr. E. D. Adrian sense in organizing the Station for Experimental land Hopkins 1947 Prof. G. H. Hardy Evolution at Cold Spring Harbor (where he worked 1927 Sir Charles Sherring- 1948 . A. V. Hill Prof in 1942), ton 1949 Prof. G.
    [Show full text]
  • Cesta Ke Hvězdám I Do Nitra Molekul
    Matematický ústav Akademie věd Èeské republiky Cesta ke hvězdám i do nitra molekul Osudy Vladim´ıra Vanda, konstruktéra počítačů Alena Solcovˇ ´a, Michal Kˇr´ıˇzek Praha 2011 Recenzenti RNDr. Pavla Pavlíková, Ph. D. RNDr. Václav Vopravil Vydal Matematický ústav AV ÈR, Praha Sazbu programem TEX pøipravil Michal Křížek c Alena ©olcová, Michal Křížek Obálka Pavel Křížek Vydání 1., Praha 2011 ISBN 978-80-85823-56-1 Věnováno prof. Vladimíru Vandovi ke 100. výročí jeho narození. 4 A. ©olcová, M. Křížek: Cesta ke hvězdám i do nitra molekul Obsah Pøedmluva : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1. Úvod : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :9 2. Období dospívání : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.1. ®ivot na Ukrajině : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.2. Rodina a dětství : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.3. Studium na Univerzitě Karlově : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.4. Vzpomínka Zdeňka Kopala : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 3. Vladimír Vand v Říši hvězd : : : : : : : : : : : : : : : : : : : : : : : : : :
    [Show full text]