Mobius inversion in physics PDF
Preview Mobius inversion in physics
MÖBIUS INVERSION IN PHYSICS Tsinghua Report and Review in Physics Series Editor: Bangfen Zhu (Tsinghua University, China) Vol. 1 Möbius Inversion in Physics by Nanxian Chen ZhangFang - Mobius Inversion in Physics.pmd2 4/6/2010, 9:00 AM MÖBIUS INVERSION IN PHYSICS Chen Nanxian Tsinghua University, China World Scientifi c NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. MÖBIUS INVERSION IN PHYSICS Tsinghua Report and Review in Physics — Vol. 1 Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4291-62-0 ISBN-10 981-4291-62-5 Printed in Singapore. ZhangFang - Mobius Inversion in Physics.pmd1 4/6/2010, 9:00 AM April2,2010 1:6 WorldScienti(cid:12)cBook-9inx6in mainC To my parents Chen Xuan-shan and Lin De-yin. To my wife He Tong and my daughter Jasmine. v April2,2010 1:6 WorldScienti(cid:12)cBook-9inx6in mainC TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk April2,2010 1:6 WorldScienti(cid:12)cBook-9inx6in mainC Preface August Ferdinand Mo(cid:127)bius gave three profound gifts to the scienti(cid:12)c world: the Mo(cid:127)bius strip in topology, the Mo(cid:127)bius transformation in pro- jective geometry, and the Mo(cid:127)bius inverse formula in number theory. This book focuses on the last, and perhaps the least well known of these gifts, introducing the applications of the Mo(cid:127)bius inverse formula to a variety of (cid:12)eldsin physics,includingmechanics,optics, statisticalphysics,solidstate physics, astrophysics, parallel Fourier transform, and others. In order to give beginners an impression of the range and scope of these applications, several are introduced here at the outset. (1) It is well known that the Fourier transform expands a square wave by a series of sine/cosine waves as 1 t 1 t 1 t 1 t S(t)=sin t sin + sin sin + sin (cid:0) 3 3 5 5 (cid:0) 7 7 9 9 1 t 1 t 1 t sin + sin sin +::: (cid:0) 11 11 13 13 (cid:0) 15 15 The Mo(cid:127)bius inversion further shows how to expand a sine/cosine wave in terms of square waves. This is useful for designing non-orthogonal modu- lations in communication, parallel calculations and signal processing. (2) Also familiar is that the inverse heat capacity problem aims to ex- tract the vibrational frequency spectrum g((cid:23)) inside a solid based on the measured C (T), the temperature dependence of heat capacity. This can v be expressed as an integral equation as 1 (h(cid:23)=kT)2exph(cid:23)=kT C (T)=rk g((cid:23))d(cid:23) v (exph(cid:23)=kT 1)2 Z0 (cid:0) Einstein and Debye had contributed two di(cid:11)erent solutions for this im- portant problem. The Mo(cid:127)bius inversion provides an unexpectedly general vii April2,2010 1:6 WorldScienti(cid:12)cBook-9inx6in mainC viii M(cid:127)obius Inversion inPhysics solution, which not only uni(cid:12)es Einstein’s and Debye’s solutions, but also signi(cid:12)cantly improves upon them. Similarly, and of signi(cid:12)cance to remote sensing and astrophysics, the Mo(cid:127)bius inversion method also solves the in- verse blackbody radiation problem. (3) When the interatomic potentials (cid:8)(x) are known for a crystal, its cohesive energy under varying pressure and temperature can be evaluated directlybysummation. Butamorepracticalproblemishowtopickupthe interatomicpotentialsbasedonab initio dataofthecohesiveenergycurve. Mo(cid:127)biusinversionsolvesthisinverselatticeproblem,andgivesaconciseand general solution for pairwise potentials. In principle, the Mo(cid:127)bius inversion methodisalsousefulforextractingmanybodyinteractionsinclustersand solids. A similar procedure is also adaptable to deducing the interatomic potentials of an interfacial system. In spite of the popular view that the (cid:12)elds are too abstruse to be use- ful, number theory and discrete mathematics have taken on increasingly important roles in computing and cryptography. To provide an example frompersonalexperience,browsingthroughtheshelvesinbookstoresthese days,Inoticeagrowingnumberofvolumesdevotedtothe(cid:12)eldsofdiscrete mathematics, modernalgebra,combinatorics,and elementarynumberthe- ory, manyof which arespeci(cid:12)cally marked asreference study materials for professionaltraininginhighlypractical(cid:12)eldssuchascomputerscienceand communication. This trend linking classical number theory with modern technical practice is inspiring. Yet very few books have related elemen- tary number theory to physics. It is somewhat puzzling that there are so few books on physics addressing number theory, though quantum theory emerged much earlier. This book emphasizes the importance and power of Mo(cid:127)bius inversion, a tiny yet important area in number theory, for the real world. The application of Mo(cid:127)bius inversionto a variety of central and di(cid:14)cult inverse problems in physics quickly yields unexpected results with deep rami(cid:12)cations. Stated more generally,this book aims to build a convenient bridge be- tween elementary number theory and applied physics. As a physicist, I intendforthisbooktobeaccessibletoreaderswithageneralphysicsback- ground,andtheonlyotherprerequisitesarealittle calculus,somealgebra, and the spark of imagination. Thisbookmayalsoprovebene(cid:12)cialtoundergraduateandgraduatestu- dentsin scienceandengineering,andassupplementarymaterialtomathe- maticalphysics,solidstatephysicsandelementarynumbertheorycourses. Illustrations included in the book contain portraits of scientists from April2,2010 1:6 WorldScienti(cid:12)cBook-9inx6in mainC Preface ix stamps and coins of many countries throughout history. I hope they pro- vide pleasant and fruitful breaks during your reading, perhaps triggering further study and re(cid:13)ection on the long, intricate, and multi-faceted his- tory behind the creation and development of many of these theories and applications1. The arrangement of this book is as follows. Chapter 1 presents basic mathematical knowledge, including the Mo(cid:127)bius{Cesa(cid:18)ro inversion formula and a unifying formula for both conventional Mo(cid:127)bius inverse formula and Mo(cid:127)bius series inversion formula. Chapter 2 addresses applications of the Mo(cid:127)bius inversion formula to the inverse boson system problem, such as inverse heat capacity and inverse blackbody radiation problems. Chap- ter 3 concerns applications of the additive Mo(cid:127)bius inversion formula to in- verseproblemsofFermisystems,includingtheFermiintegralequation,the relaxation-time spectrum, and the surface absorption problem. This chap- ter alsoexploresanew method forFourierdeconvolution,andthe additive Mo(cid:127)bius-Cesa(cid:18)ro inversion formula. Chapter 4 demonstrates applications of the Mo(cid:127)bius inversion formula to parallel Fourier transform technique, pre- senting,inparticular,auniformsamplingmethodforthearithmeticFourier transform. Chapter 5 shows the Mo(cid:127)bius inversion formulas on Gaussian’s and Eisenstein’s integers, as well as corresponding applications such as to VLSIprocessing. Chapter6addressesapplicationsoftheMo(cid:127)biusinversion method to 3D lattice systems, interfacial systems and others. Chapter 6 also discusses issues related to the embedded atom method (EAM), the cluster expansion method (CEM), and the Mo(cid:127)bius inversion formula on a partially ordered set (POSET). Then a short epilogue is attached. Theorderforreadingis(cid:13)exible. Formostnon-mathematicians,the ba- sic knowledge in sections 1.2-1.5 is necessary. Beyond that single proviso the chapters may be selected and reviewed in any order the reader, with whatever background, prefers. It should be emphasized that this book is specially designed to be accessible and engaging for non-mathematicians, which includes the author himself. N.X. Chen 1Curious readers can (cid:12)nd supplementary materials on these scientists through http://www-groups.dcs.st-and.ac.uk/history/BiogIndex.html.
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