Josiah Willard Gibbs
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Symmetry and Ergodicity Breaking, Mean-Field Study of the Ising Model
Symmetry and ergodicity breaking, mean-field study of the Ising model Rafael Mottl, ETH Zurich Advisor: Dr. Ingo Kirsch Topics Introduction Formalism The model system: Ising model Solutions in one and more dimensions Symmetry and ergodicity breaking Conclusion Introduction phase A phase B Tc temperature T phase transition order parameter Critical exponents Tc T Formalism Statistical mechanical basics sample region Ω i) certain dimension d ii) volume V (Ω) iii) number of particles/sites N(Ω) iv) boundary conditions Hamiltonian defined on the sample region H − Ω = K Θ k T n n B n Depending on the degrees of freedom Coupling constants Partition function 1 −βHΩ({Kn},{Θn}) β = ZΩ[{Kn}] = T r exp kBT Free energy FΩ[{Kn}] = FΩ[K] = −kBT logZΩ[{Kn}] Free energy per site FΩ[K] fb[K] = lim N(Ω)→∞ N(Ω) i) Non-trivial existence of limit ii) Independent of Ω iii) N(Ω) lim = const N(Ω)→∞ V (Ω) Phases and phase boundaries Supp.: -fb[K] exists - there exist D coulping constants: {K1,...,KD} -fb[K] is analytic almost everywhere - non-analyticities of f b [ { K n } ] are points, lines, planes, hyperplanes in the phase diagram Dimension of these singular loci: Ds = 0, 1, 2,... Codimension C for each type of singular loci: C = D − Ds Phase : region of analyticity of fb[K] Phase boundaries : loci of codimension C = 1 Types of phase transitions fb[K] is everywhere continuous. Two types of phase transitions: ∂fb[K] a) Discontinuity across the phase boundary of ∂Ki first-order phase transition b) All derivatives of the free energy per site are continuous -
Vectors and Beyond: Geometric Algebra and Its Philosophical
dialectica Vol. 63, N° 4 (2010), pp. 381–395 DOI: 10.1111/j.1746-8361.2009.01214.x Vectors and Beyond: Geometric Algebra and its Philosophical Significancedltc_1214 381..396 Peter Simons† In mathematics, like in everything else, it is the Darwinian struggle for life of ideas that leads to the survival of the concepts which we actually use, often believed to have come to us fully armed with goodness from some mysterious Platonic repository of truths. Simon Altmann 1. Introduction The purpose of this paper is to draw the attention of philosophers and others interested in the applicability of mathematics to a quiet revolution that is taking place around the theory of vectors and their application. It is not that new math- ematics is being invented – on the contrary, the basic concepts are over a century old – but rather that this old theory, having languished for many decades as a quaint backwater, is being rediscovered and properly applied for the first time. The philosophical importance of this quiet revolution is not that new applications for old mathematics are being found. That presumably happens much of the time. Rather it is that this new range of applications affords us a novel insight into the reasons why vectors and their mathematical kin find application at all in the real world. Indirectly, it tells us something general but interesting about the nature of the spatiotemporal world we all inhabit, and that is of philosophical significance. Quite what this significance amounts to is not yet clear. I should stress that nothing here is original: the history is quite accessible from several sources, and the mathematics is commonplace to those who know it. -
Recent Books on Vector Analysis. 463
1921.] RECENT BOOKS ON VECTOR ANALYSIS. 463 RECENT BOOKS ON VECTOR ANALYSIS. Einführung in die Vehtoranalysis, mit Anwendungen auf die mathematische Physik. By R. Gans. 4th edition. Leip zig and Berlin, Teubner, 1921. 118 pp. Elements of Vector Algebra. By L. Silberstein. New York, Longmans, Green, and Company, 1919. 42 pp. Vehtor analysis. By C. Runge. Vol. I. Die Vektoranalysis des dreidimensionalen Raumes. Leipzig, Hirzel, 1919. viii + 195 pp. Précis de Calcul géométrique. By R. Leveugle. Preface by H. Fehr. Paris, Gauthier-Villars, 1920. lvi + 400 pp. The first of these books is for the most part a reprint of the third edition, which has been reviewed in this BULLETIN (vol. 21 (1915), p. 360). Some small condensation of results has been accomplished by making deductions by purely vector methods. The author still clings to the rather common idea that a vector is a certain triple on a certain set of coordinate axes, so that his development is rather that of a shorthand than of a study of expressions which are not dependent upon axes. The second book is a quite brief exposition of the bare fundamentals of vector algebra, the notation being that of Heaviside, which Silberstein has used before. It is designed primarily to satisfy the needs of those studying geometric optics. However the author goes so far as to mention a linear vector operator and the notion of dyad. The exposition is very simple and could be followed by high school students who had had trigonometry. The third of the books mentioned is an elementary exposition of vectors from the standpoint of Grassmann. -
Selected Bibliography of American History Through Biography
DOCUMENT RESUME ED 088 763 SO 007 145 AUTHOR Fustukjian, Samuel, Comp. TITLE Selected Bibliography of American History through Biography. PUB DATE Aug 71 NOTE 101p.; Represents holdings in the Penfold Library, State University of New York, College at Oswego EDRS PRICE MF-$0.75 HC-$5.40 DESCRIPTORS *American Culture; *American Studies; Architects; Bibliographies; *Biographies; Business; Education; Lawyers; Literature; Medicine; Military Personnel; Politics; Presidents; Religion; Scientists; Social Work; *United States History ABSTRACT The books included in this bibliography were written by or about notable Americans from the 16th century to the present and were selected from the moldings of the Penfield Library, State University of New York, Oswego, on the basis of the individual's contribution in his field. The division irto subject groups is borrowed from the biographical section of the "Encyclopedia of American History" with the addition of "Presidents" and includes fields in science, social science, arts and humanities, and public life. A person versatile in more than one field is categorized under the field which reflects his greatest achievement. Scientists who were more effective in the diffusion of knowledge than in original and creative work, appear in the tables as "Educators." Each bibliographic entry includes author, title, publisher, place and data of publication, and Library of Congress classification. An index of names and list of selected reference tools containing biographies concludes the bibliography. (JH) U S DEPARTMENT Of NIA1.114, EDUCATIONaWELFARE NATIONAL INSTITUTE OP EDUCATION THIS DOCUMENT HAS BEEN REPRO DUCED ExAC ICY AS RECEIVED FROM THE PERSON OR ORGANIZATIONORIGIN ATING IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILYREPRE SENT OFFICIAL NATIONAL INSTITUTEOF EDUCATION POSITION OR POLICY PREFACE American History, through biograRhies is a bibliography of books written about 1, notable Americans, found in Penfield Library at S.U.N.Y. -
Blackbody Radiation: (Vibrational Energies of Atoms in Solid Produce BB Radiation)
Independent study in physics The Thermodynamic Interaction of Light with Matter Mirna Alhanash Project in Physics Uppsala University Contents Abstract ................................................................................................................................................................................ 3 Introduction ......................................................................................................................................................................... 3 Blackbody Radiation: (vibrational energies of atoms in solid produce BB radiation) .................................... 4 Stefan-Boltzmann .............................................................................................................................................................. 6 Wien displacement law..................................................................................................................................................... 7 Photoelectric effect ......................................................................................................................................................... 12 Frequency dependence/Atom model & electron excitation .................................................................................. 12 Why we see colours ....................................................................................................................................................... 14 Optical properties of materials: .................................................................................................................................. -
Ludwig Boltzmann Was Born in Vienna, Austria. He Received His Early Education from a Private Tutor at Home
Ludwig Boltzmann (1844-1906) Ludwig Boltzmann was born in Vienna, Austria. He received his early education from a private tutor at home. In 1863 he entered the University of Vienna, and was awarded his doctorate in 1866. His thesis was on the kinetic theory of gases under the supervision of Josef Stefan. Boltzmann moved to the University of Graz in 1869 where he was appointed chair of the department of theoretical physics. He would move six more times, occupying chairs in mathematics and experimental physics. Boltzmann was one of the most highly regarded scientists, and universities wishing to increase their prestige would lure him to their institutions with high salaries and prestigious posts. Boltzmann himself was subject to mood swings and he joked that this was due to his being born on the night between Shrove Tuesday and Ash Wednesday (or between Carnival and Lent). Traveling and relocating would temporarily provide relief from his depression. He married Henriette von Aigentler in 1876. They had three daughters and two sons. Boltzmann is best known for pioneering the field of statistical mechanics. This work was done independently of J. Willard Gibbs (who never moved from his home in Connecticut). Together their theories connected the seemingly wide gap between the macroscopic properties and behavior of substances with the microscopic properties and behavior of atoms and molecules. Interestingly, the history of statistical mechanics begins with a mathematical prize at Cambridge in 1855 on the subject of evaluating the motions of Saturn’s rings. (Laplace had developed a mechanical theory of the rings concluding that their stability was due to irregularities in mass distribution.) The prize was won by James Clerk Maxwell who then went on to develop the theory that, without knowing the individual motions of particles (or molecules), it was possible to use their statistical behavior to calculate properties of a gas such as viscosity, collision rate, diffusion rate and the ability to conduct heat. -
Geometric-Algebra Adaptive Filters Wilder B
1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions. -
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. -
The Philosophy and Physics of Time Travel: the Possibility of Time Travel
University of Minnesota Morris Digital Well University of Minnesota Morris Digital Well Honors Capstone Projects Student Scholarship 2017 The Philosophy and Physics of Time Travel: The Possibility of Time Travel Ramitha Rupasinghe University of Minnesota, Morris, [email protected] Follow this and additional works at: https://digitalcommons.morris.umn.edu/honors Part of the Philosophy Commons, and the Physics Commons Recommended Citation Rupasinghe, Ramitha, "The Philosophy and Physics of Time Travel: The Possibility of Time Travel" (2017). Honors Capstone Projects. 1. https://digitalcommons.morris.umn.edu/honors/1 This Paper is brought to you for free and open access by the Student Scholarship at University of Minnesota Morris Digital Well. It has been accepted for inclusion in Honors Capstone Projects by an authorized administrator of University of Minnesota Morris Digital Well. For more information, please contact [email protected]. The Philosophy and Physics of Time Travel: The possibility of time travel Ramitha Rupasinghe IS 4994H - Honors Capstone Project Defense Panel – Pieranna Garavaso, Michael Korth, James Togeas University of Minnesota, Morris Spring 2017 1. Introduction Time is mysterious. Philosophers and scientists have pondered the question of what time might be for centuries and yet till this day, we don’t know what it is. Everyone talks about time, in fact, it’s the most common noun per the Oxford Dictionary. It’s in everything from history to music to culture. Despite time’s mysterious nature there are a lot of things that we can discuss in a logical manner. Time travel on the other hand is even more mysterious. -
Episode 1: Phis Wants to Be a Physicist
Episode 1: Phis wants to be a physicist Illustration: Xia Hong Script: Xia Hong 12/2012 Phis is 14 years old. Like most kids of her age, she’s a bit anxious about finding the person inside her… She tried singing. Hmm… “You are very talented, Phis!” Her best friend Lizzy told her. Sports are also not her cup of tea… Episode 1: Phis wants to be a physicist 1 “I wish I could But why I’m short, but does height I’m very be as confident She’s smart. She’s pretty. matter?! confident! as Lizzy!” Phis And she is told her little taller than me! sister Chemi. No wonder she’s “Then I would so confident! know what I can do.” “Who says size doesn’t matter?” Phis is not convinced. “Just read the newspaper…” “It’s always good to be either really small or really big, not in between!” Even her name sounds sharp — it is the same as the Lizzy in Pride and Prejudice ! Chemi, don’t you think our names are a bit strange? Episode 1: Phis wants to be a physicist 2 Phis’s self-search reached a conclusion after a monthly Science Day event in school hosted by Ms. Allen. What I’ll show you today is the It is our fascinating pleasure world of to have quantum Prof. Lee mechanics. here. Imagine, we live in the quantum world. Quantum We have to comply to its rules, and behave very We no longer have a differently from certain answer for things our normal life. -
Viscosity from Newton to Modern Non-Equilibrium Statistical Mechanics
Viscosity from Newton to Modern Non-equilibrium Statistical Mechanics S´ebastien Viscardy Belgian Institute for Space Aeronomy, 3, Avenue Circulaire, B-1180 Brussels, Belgium Abstract In the second half of the 19th century, the kinetic theory of gases has probably raised one of the most impassioned de- bates in the history of science. The so-called reversibility paradox around which intense polemics occurred reveals the apparent incompatibility between the microscopic and macroscopic levels. While classical mechanics describes the motionof bodies such as atoms and moleculesby means of time reversible equations, thermodynamics emphasizes the irreversible character of macroscopic phenomena such as viscosity. Aiming at reconciling both levels of description, Boltzmann proposed a probabilistic explanation. Nevertheless, such an interpretation has not totally convinced gen- erations of physicists, so that this question has constantly animated the scientific community since his seminal work. In this context, an important breakthrough in dynamical systems theory has shown that the hypothesis of microscopic chaos played a key role and provided a dynamical interpretation of the emergence of irreversibility. Using viscosity as a leading concept, we sketch the historical development of the concepts related to this fundamental issue up to recent advances. Following the analysis of the Liouville equation introducing the concept of Pollicott-Ruelle resonances, two successful approaches — the escape-rate formalism and the hydrodynamic-mode method — establish remarkable relationships between transport processes and chaotic properties of the underlying Hamiltonian dynamics. Keywords: statistical mechanics, viscosity, reversibility paradox, chaos, dynamical systems theory Contents 1 Introduction 2 2 Irreversibility 3 2.1 Mechanics. Energyconservationand reversibility . ........................ 3 2.2 Thermodynamics. -
Hamiltonian Mechanics Wednesday, ß November Óþõõ
Hamiltonian Mechanics Wednesday, ß November óþÕÕ Lagrange developed an alternative formulation of Newtonian me- Physics ÕÕÕ chanics, and Hamilton developed yet another. Of the three methods, Hamilton’s proved the most readily extensible to the elds of statistical mechanics and quantum mechanics. We have already been introduced to the Hamiltonian, N ∂L = Q ˙ − = Q( ˙ ) − (Õ) H q j ˙ L q j p j L j=Õ ∂q j j where the generalized momenta are dened by ∂L p j ≡ (ó) ∂q˙j and we have shown that dH = −∂L dt ∂t Õ. Legendre Transformation We will now show that the Hamiltonian is a so-called “natural function” of the general- ized coordinates and generalized momenta, using a trick that is just about as sophisticated as the product rule of dierentiation. You may have seen Legendre transformations in ther- modynamics, where they are used to switch independent variables from the ones you have to the ones you want. For example, the internal energy U satises ∂U ∂U dU = T dS − p dV = dS + dV (ì) ∂S V ∂V S e expression between the equal signs has the physics. It says that you can change the internal energy by changing the entropy S (scaled by the temperature T) or by changing the volume V (scaled by the negative of the pressure p). e nal expression is simply a mathematical statement that the function U(S, V) has a total dierential. By lining up the various quantities, we deduce that = ∂U and = − ∂U . T ∂S V p ∂V S In performing an experiment at constant temperature, however, the T dS term is awk- ward.