Graduate Texts in Physics

Transcription

1 Graduate Texts in Physics

2 GRADUATETEXTS IN PHYSICS Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledge of, a research field. Series Editors Professor Richard Needs Cavendish Laboratory JJ Thomson Avenue Cambridge CB3 0HE, UK rn11@cam.ac.uk Professor William T. Rhodes Florida Atlantic University Imaging Technology Center Department of Electrical Engineering 777 Glades Road SE, Room 456 Boca Raton, FL 33431, USA wrhodes@fau.edu Professor H. Eugene Stanley Boston University Center for Polymer Studies Department of Physics 590 Commonwealth Avenue, Room 204B Boston, MA 02215, USA hes@bu.edu For further volumes:

3 Peter Hertel Continuum Physics With 37 Figures 123

4 Peter Hertel F achbereich Physik Universität Osnabrück Osnabrück, Germany ISSN ISSN (electronic) ISBN ISBN (ebook) DOI / Springer Heidelberg New York Dordrecht London Library of Congress Control Number: Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface The subject of this small book is a vast field: continuum physics. Quite naturally, we cannot go into all the details of its many subdomains. Instead, we provide a bird s eye view of the subject. A continuum is made up of material points. Each material point is a tiny region: tiny from a human point of view, where the natural scale are meters, kilograms, and seconds. The same material point, however, contains very many particles, atoms or molecules, such that the statistical laws for infinitely many particles apply. In particular, external parameters change relatively slowly such that each material point is always very close to thermodynamic equilibrium. Thus we deal with fields f D f.t;x/, wheref stands for a property of the material point located at x at time t, such as mass density, temperature, tension, and so forth. Chapter 1 discusses additive and transportable quantities such as mass, electric charge, momentum, kinetic, potential and internal energy as well as entropy. For each such quantity, a balance equation is formulated and related with other balance equations. Each quantity is described by densities for content, flow, and production. The balance equations must be respected in any case; they apply to all kinds of matter in any state. The highlights of this chapter are the precise formulations of the first and second main laws of thermodynamics in terms of partial differential equations. Chapter 2 describes the special continuum under discussion with additional field equations, such as Hooke s law, Ohm s law, optical or acoustical properties, the ideal gas pressure relation, and so forth. Each closed set of field equations characterizes a subdomain of continuum physics: elasticity, hydrodynamics, aerodynamics, optics, rheology, thermo-electricity, diffusion, heat conduction, and so forth. Next we study Linear Response Theory (Chap. 3). The reaction of a system in thermodynamic equilibrium to rapid variations of external parameters is worked out in first order perturbation theory. Linear material equations and explicit expression for their coefficients are arrived at, such as the dielectric susceptibility as a function of frequency. Once a tractable model for the system under consideration has been v

6 vi Preface established, these coefficients can be calculated. Even more importantly, there are general relations, such as the Kramers-Kronig or Onsager relations, that can be shown to be true irrespective of a particular model. A highlight of this chapter is the derivation of the fluctuation-dissipation theorem, which is very close to a proof of the second main law of thermodynamics. In Chap. 4, some standard topics of continuum physics are discussed. These examples serve to familiarize the reader with what she or he may already know, but now on a higher level, such as Archimedes principle, the first article. Some topics, however, may be new to the reader. For instance, the last article presents a simplified model of a white dwarf. There are articles on crystal optics, the electroand magneto-optic effect, and on optical activity as well as dielectric waveguides. Metamaterials and photonic crystals are also introduced. Standard problems of elasticity such as beam-bending and buckling, stress enhancement, and vibrating strings and membranes are discussed. The propagation and attenuation of sound waves in elastic bodies and in air as well as surface acoustic waves are dealt with. There is an article on surface plasmon polaritons, and we discuss Ohm s law and the Hall effect, introducing the Drude model. The reader will find sections on the laws of Bernoulli, Hagen-Poiseuille and Stokes and on Reynold s number. An article on local chemical reactions and diffusion introduces the interesting subject of pattern formation. Fourier s classical treatment of a heat conduction problem is presented and thermoelastic effects explained (Thomson, Seebeck, Peltier). Two articles illustrate the dissipation-fluctuation theorem: one on the thermal noise of an Ohmic resistor, and the other on various aspects of Brownian motion. Diffusion, a standard subject in continuum physics, is nothing else but mass-wise Brownian motion. A model of the earth s atmosphere is also discussed. Chapter 4 could also have been organized into a series of methods of solving partial differential equations. Some can be solved analytically exploiting symmetry, others by rewriting them into systems of ordinary differential equations. The articles on Stokes law and on stress concentration introduce the potential method. Various Computational schemes such as the finite difference method and the finite element method are also discussed. The sample programs presented are in MATLAB because this software package is tailored to the needs of physics. An appendix covers Fields: their transformation properties, how they are differentiated and how they can be integrated over paths, areas, and volumes. After all, continuum physics is formulated in terms of fields and relations between them. There is also a Glossary, which may serve as a summary of this bird s eye view of continuum physics. It provides brief descriptions of the main topics of this book and lists most of the people who have entered the stage. The book systematically presents continuum physics theory (balance equations, material equations, linear response) and substantiates the theory with selected examples. Both parts are of comparable size. On the one hand, the examples cannot be understood without a solid theoretical foundation. On the other hand, physical laws as such are without pertinence if they cannot be shown to be useful for realworld problems from Archimedes principle to white dwarfs.

7 Preface vii This book can be read in practically any order. It is more like a web than an exposition where the next step cannot be taken without going over all the previous steps. However, the first few sections on balance equations should be read first and consecutively because they introduce a precise system of notation that is nevertheless more or less compatible with a century-old tradition. This book on Continuum Physics addresses students of physics, engineering, and related branches of science with a general background in calculus and basic physics. It will also serve graduate students and lecturers who want to embed their special field in a wider context. The book presents the physics of continuously distributed matter in a unified way. We have built and developed a stringent theoretical framework and discussed carefully chosen examples. This will enable the reader to assemble the multitude of principles, rules, laws, effects, and theories of continuum physics the pieces of a jigsaw puzzle into a coherent whole. Osnabrück, Germany February 2012 Peter Hertel

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9 Contents 1 Balance Equations Material Points Summary Densities for Content, Flow, and Production Rate Densities Current Densities Production Generic Balance Equation Summary Particle Numbers, Mass, and Electric Charge Particle Numbers and Currents Chemical Reactions Mass Electric Charge Digression on Electromagnetism Summary Convection and Conduction Conduction Transformation Properties of Currents Material Time Derivative Specific Quantities Summary Linear and Angular Momentum Stress and External Forces Angular Momentum Normal and Shear Forces Summary Energy and the First Law of Thermodynamics Kinetic Energy Potential Energy Internal Energy ix

10 x Contents Digression on Time Reversal First Law of Thermodynamics Summary Entropy and the Second Law of Thermodynamics Thermodynamic Equilibrium Balance Equation for Entropy Second Law of Thermodynamics Summary Material Equations Fluid Media Incompressible Fluids Ideal Gas Newtonian Fluid Medium Hydrodynamics Aerodynamics Summary Solid Media Strain Hooke s Law Structural Mechanics Elastodynamics Summary Heat Conduction Fourier s Law More on Heat Conduction Heat Equation Summary Diffusion Diffusion Currents Chemical Potential Diffusion Equation Digression on Open Systems Summary Charge Transport The Electromagnetic Field in Matter Contributions to the Electric Current Density Ohm s Law Summary Thermoelectric Effects Simultaneous Heat and Charge Transport Forces and Fluxes Kinetic Coefficients for Heat and Charge Transport Summary... 55

11 Contents xi 3 Linear Response Theory Statistical Thermodynamics Quantum Theory Entropy Equilibrium Summary Perturbations Interaction Picture Linear Response Perturbation by Light Susceptibility Tensor Summary Dispersion Relations Causal Functions Kramers-Kronig Relations Summary Fluctuations and Dissipation Wiener-Khinchin Theorem Kubo-Martin-Schwinger Formula Callen-Welton Theorem Interaction with an Electromagnetic Field Summary Onsager Relations Time Reversal Symmetries for Susceptibilities Symmetries for Conductivities Symmetries for Kinetic Coefficients Summary Examples Archimedes Principle Erroneous Reasoning Incompressible Fluid Compressible Fluid Beam Bending and Buckling Beam Theory Equilibrium An Example Buckling Instability Bernoulli s Law Brownian Motion Einstein s Explanation The Diffusion Coefficient Langevin s Approach

12 xii Contents 4.5 Bulk and Surface Acoustic Waves in Solids Wave Equation Bulk Acoustic Waves Attenuation Rayleigh Waves More on SAW Crystal Optics Susceptibility Planar Waves Optical Isotropy Birefringence Polarization Change Due to Birefringence Absorption Dispersion Dielectric Planar Waveguides TE Modes TM Modes Graded Index Waveguides Slab Waveguides Elasticity Moduli Strain and Stress Uniaxial Stress Hydrostatic Pressure Torsion Examples: Steel and Rubber Faraday Effect Susceptibility and Onsager s Relation Rotation of the Polarization Vector Drude Model Optical Isolator Yttrium Iron Garnet Fourier s Solution of a Heat Conduction Problem Representation of a Periodic Function Ordinary Instead of Partial Differential Equations Initial Conditions Hagen-Poiseuille Law Exploiting Symmetry Flux Pump Power Ice Layer Phase Transition Enthalpy The Problem Crude Approximation Dimensionless Quantities The Exact Solution Discussion

13 Contents xiii 4.13 Metamaterials Resonances Wave Propagation Normal Refraction Refraction in a Backward Medium A New Type of Optical Microscope Photonic Crystals Model Atmosphere Adiabatic Process for a Diatomic Ideal Gas Convection Neutral Atmosphere Natural Units SI and Atomic Units Remarks Ohm s Law and Hall Effect Drude Model Onsager Relations Hall Effect Electro-Chemical Potential Optically Active Media Spatial Dispersion Optical Activity Quartz Natural Grape Sugar Pockels and Kerr Effect Dielectric Susceptibility Pockels Effect Crystal Symmetry 3m Symmetry Compatible Tensors Digression on Orders of Magnitude How Light Propagation is Affected Kerr Effect Reactions and Diffusion Gray-Scott Model Similarity Considerations Homogeneous Solutions Pattern Formation Further Remarks Reynold s Number Navier-Stokes Equation Similarity Considerations Laminar Flow and Turbulence Sound in Air Wave Equation Sound Generation

14 xiv Contents 4.22 Stokes Law The Problem Vorticity and Stream Function The Solution Stoke s Formula Stress Concentration Circular Hole in a Thin Plate Airy s Stress Function for Planar Problems The Solution Discussion Surface Plasmon Polaritons The Drude Model Guided TM Mode TE Polarization Thermal Noise of a Resistor Langevin Equation Nyquist Formula Remarks Thomson, Seebeck and Peltier Effect Solid Conductors Charge and Heat Transport Thomson Effect Seebeck and Peltier Effect Vibrating Strings and Membranes Strings Membranes String Eigenmodes Membrane Eigenmodes White Dwarfs The Model Equation of State Orders of Magnitude Numerical Solution Remarks A Fields A.1 Covariance A.1.1 Inertial Systems of Reference A.1.2 Tensor Fields A.2 Differentiating Fields A.2.1 Time Derivative A.2.2 Gradient A.2.3 Divergence A.2.4 Curl

15 Contents xv A.3 Paths, Surfaces and Volumes A.3.1 Paths A.3.2 Surfaces A.3.3 Volumes A.4 Integrating Fields A.4.1 Path Integrals A.4.2 Surface Integrals A.4.3 Volume Integrals A.4.4 Stokes and Gauss Theorems A.5 More on Fields A.5.1 Densities and Current Densities A.5.2 Time Derivatives Glossary References

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17 List of Figures Fig. 1.1 Volumetric production rates for kinetic, potential, and internal energy Fig. 4.1 A thin rod with fixed boundary conditions Fig. 4.2 Random walk of a Brownian particle Fig. 4.3 Cloud of Brownian particles Fig. 4.4 Rayleigh surface acoustic wave Fig. 4.5 Wave propagation in a birefringent medium Fig. 4.6 Modes of a graded index dielectric waveguide Fig. 4.7 Layout of a simple slab waveguide Fig. 4.8 Effective indexes of TE and TM modes of a slab waveguide Fig. 4.9 Depth vs. temperature of an ice layer Fig Split ring resonator Fig Normal refraction Fig Refraction at the interface between an ideal backward medium and vacuum Fig New type of optical microscope Fig Photonic crystal transmission coefficient Fig m symmetry Fig Parameters of the Gray-Scott model Fig Gray-Scott model: early stage Fig Gray-Scott model: intermediate state Fig Gray-Scott model: stationary state Fig Laminar flow around a sphere Fig Circular hole in a thin plate Fig Stress enhancement Fig Permittivity of gold Fig Surface TM mode Fig RC circuit Fig Seebeck and Peltier effect Fig Fourier expansion of the triangle function xvii

18 xviii List of Figures Fig An eigenmode of a rectangular membrane Fig Very crude approximation to the back of a violin Fig Second excited mode of a violin back Fig Pressure and mass of a white dwarf vs. radius Fig. A.1 Composition of Galilei transformations Fig. A.2 Different parameterizations of the same path Fig. A.3 Boundary of a surface Fig. A.4 Boundary of a large spherical disc without a smaller one Fig. A.5 Infinitesimal surface element