Statistical Mechanics: Theory and Molecular Simulation This page intentionally left blank Statistical Mechanics: Theory and Molecular Simulation Mark E. Tuckerman Department of Chemistry, New York University and Courant Institute of Mathematical Sciences, New York 1 3 GreatClarendonStreet,Oxfordox26dp OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein Oxford NewYork Auckland CapeTown DaresSalaam HongKong Karachi KualaLumpur Madrid Melbourne MexicoCity Nairobi NewDelhi Shanghai Taipei Toronto Withofficesin Argentina Austria Brazil Chile CzechRepublic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore SouthKorea Switzerland Thailand Turkey Ukraine Vietnam OxfordisaregisteredtrademarkofOxfordUniversityPress intheUKandincertainothercountries PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork (cid:2)c MarkE.Tuckerman2010 Themoralrightsoftheauthorhavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2010 Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethesameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Dataavailable TypesetbySPIPublisherServices,Pondicherry,India PrintedinGreatBritain onacid-freepaperby CPIAntonyRowe,Chippenham,Wiltshire ISBN 978–0–19–852526–4(Hbk.) 1 3 5 7 9 10 8 6 4 2 To my parents, Jocelyn, and Delancey This page intentionally left blank Preface Statistical mechanics is a theoretical framework that aims to predict the observable static and dynamic properties of a many-body system starting from its microscopic constituents and their interactions. Its scope is as broad as the set of “many-body” systems is large: as long as there exists a rule governing the behavior of the fun- damental objects that comprise the system, the machinery of statistical mechanics can be applied. Consequently, statistical mechanics has found applications outside of physics, chemistry, and engineering,including biology,social sciences, economics, and applied mathematics. Because it seeks to establish a bridge between the microscopic and macroscopic realms, statistical mechanics often provides a means of rationalizing observedpropertiesofasystemintermsofthedetailed“modesofmotion”ofitsbasic constituents. An example from physical chemistry is the surprisingly high diffusion constant of an excess proton in bulk water, which is a single measurable number. However, this single number belies a strikingly complex dance of hydrogen bond re- arrangements and chemical reactions that must occur at the level of individual or small clusters of water molecules in order for this property to emerge. In the physical sciences, the technology of molecular simulation, wherein a system’s microscopic in- teractionrules areimplementednumericallyonacomputer,allowsuch“mechanisms” to be extracted and, through the machinery of statistical mechanics, predictions of macroscopic observables to be generated. In short, molecular simulation is the com- putational realization of statistical mechanics. The goal of this book, therefore, is to synthesize these two aspects of statistical mechanics: the underlying theory of the subject, in both its classical and quantum developments, and the practical numerical techniques by which the theory is applied to solve realistic problems. This book is aimed primarily at graduate students in chemistry or computational biology and graduate or advanced undergraduate students in physics or engineering. These students are increasingly finding themselves engaged in researchactivities that cross traditional disciplinary lines. Successful outcomes for such projects often hinge on their ability to translate complex phenomena into simple models and develop ap- proaches for solving these models. Because of its broad scope, statistical mechanics plays a fundamental role in this type of work and is an important part of a student’s toolbox. ThetheoreticalpartofthebookisanextensiveelaborationoflecturenotesIdevel- opedforagraduate-levelcourseinstatisticalmechanicsIgiveatNewYorkUniversity. These courses are principally attended by graduate and advanced undergraduate stu- dents who areplanning to engagein researchintheoreticalandexperimentalphysical chemistry and computational biology. The most difficult question faced by anyone wishing to design a lecture course or a book on statistical mechanics is what to in- clude and what to omit. Because statistical mechanics is an active field of research,it Preface comprises a tremendous body of knowledge, and it is simply impossible to treat the entirety of the subject in a single opus. For this reason, many books with the words “statistical mechanics” in their titles can differ considerably. Here, I have attempted to bring together topics that reflect what I see as the modern landscape of statisti- cal mechanics. The reader will notice from a quick scan of the table of contents that the topics selected are rarely found together in individual textbooks on the subject; these topics include isobaric ensembles, path integrals, classical and quantum time- dependent statistical mechanics, the generalized Langevin equation, the Ising model, andcriticalphenomena.(TheclosestsuchbookIhavefoundisalsooneofmyfavorites, David Chandler’s Introduction to Modern Statistical Mechanics.) The computational part of the book joins synergistically with the theoretical part and is designed to give the reader a solid grounding in the methodology employed to solve problems in statistical mechanics. It is intended neither as a simulation recipe book nor a scientific programmer’s guide. Rather, it aims to show how the develop- ment of computational algorithms derives from the underlying theory with the hope of enabling readers to understand the methodology-oriented literature and develop new techniques of their own. The focus is on the molecular dynamics and Monte Carlotechniquesandthe manynovelextensionsofthese methods thathaveenhanced their applicability to, for example, large biomolecular systems, complex materials, and quantum phenomena. Most of the techniques described are widely available in molecular simulation software packages and are routinely employed in computational investigations.Aswiththetheoreticalcomponent,itwasnecessarytoselectamongthe numerous important methodological developments that have appeared since molecu- lar simulation was first introduced. Unfortunately, severalimportant topics had to be omitted due to space constraints, including configuration-bias Monte Carlo, the ref- erence potential spatial warping algorithm, and semi-classical methods for quantum time correlationfunctions. This omissionwas not made because I view these methods as less important than those I included. Rather, I consider these to be very powerful but highly advanced methods that, individually, might have a narrower target audi- ence. In fact, these topics were slated to appear in a chapter of their own. However, as the book evolved, I found that nearly 700 pages were needed to lay the foundation I sought. In organizing the book, I have made several strategic decisions. First, the book is structured such that concepts are first introduced within the framework of classical mechanicsfollowedbytheirquantummechanicalcounterparts.Thisliescloserperhaps to a physicist’s perspective than, for example, that of a chemist, but I find it to be a particularlynaturalone.Moreover,givenhowwidespreadcomputationalstudiesbased on classical mechanics have become compared to analogous quantum investigations (which have considerably higher computational overhead) this progression seems to be both logical and practical. Second, the technical development within each chapter is graduated, with the level of mathematical detail generally increasing from chapter start to chapter end. Thus, the mathematically most complex topics are reserved for the final sections of each chapter. I assume that readers have an understanding of calculus(throughcalculusofseveralvariables),linearalgebra,andordinarydifferential equations. This structure hopefully allows readers to maximize what they take away Preface fromeachchapterwhilerenderingiteasiertofindastoppingpointwithineachchapter. In short, the book is structured such that even a partial reading of a chapter allows the reader to gain a basic understanding of the subject. It should be noted that I attempted to adhere to this graduated structure only as a general protocol. Where I felt that breaking this progression made logical sense, I have forewarned the reader about the mathematicalarguments to follow,andthe finalresult is generally givenat the outset. Readers wishing to skip the mathematical details can do so without loss of continuity. The third decision I have made is to integrate theory and computational methods within each chapter. Thus, for example, the theory of the classical microcanonical ensembleispresentedtogetherwithadetailedintroductiontothemoleculardynamics method and howthe latter is used to generatea classicalmicrocanonicaldistribution. The other classical ensembles are presented in a similar fashion as is the Feynman path integralformulationof quantum statistical mechanics. The integrationof theory and methodology serves to emphasize the viewpoint that understanding one helps in understanding the other. Throughout the book, many of the computational methods presented are accom- panied by simple numerical examples that demonstrate their performance. These ex- amplesrangefromlow-dimensional“toy”problemsthatcanbeeasilycodedupbythe reader(someoftheexercisesineachchapteraskpreciselythis)toatomicandmolecu- larliquids,aqueoussolutions,modelpolymers,biomolecules,andmaterials.Notevery method presentedis accompaniedbya numericalexample,andingeneralI havetried not to overwhelm the reader with a plethora of applications requiring detailed expla- nations of the underlying physics, as this is not the primary aim of the book. Once the basicsofthe methodologyare understood,readerswishing to exploreapplications particular to their interests in more depth can subsequently refer to the literature. A word or two should be said about the problem sets at the end of each chapter. Math and science are not spectator sports, and the only way to learn the material is to solve problems. Some of the problems in the book require the reader to think con- ceptuallywhileothersaremoremathematical,challengingthereadertoworkthrough various derivations. There are also problems that ask the reader to analyze proposed computational algorithms by investigating their capabilities. For readers with some programming background, there are exercises that involve coding up a method for a simple example in order to explore the method’s performance on that example, and in some cases, reproduce a figure from the text. These coding exercises are included because one can only truly understand a method by programming it up and trying it out on a simple problem for which long runs can be performed and many different parameter choices can be studied. However, I must emphasize that even if a method workswellonasimpleproblem,itisnotguaranteedtoworkwellforrealisticsystems. Readersshouldnot,therefore,na¨ıvelyextrapolatetheperformanceofanymethodthey try on a toy system to high-dimensional complex problems. Finally, in each problem set,someproblemareprecededby anasterisk(∗). Theseareproblemsofa morechal- lengingnaturethatrequiredeeperthinkingoramorein-depthmathematicalanalysis. All of the problems are designed to strengthen understanding of the basic ideas. Let me close this preface by acknowledging my teachers, mentors, colleagues, and