Free energy computations : a mathematical perspective

471 Pages·2010·6.687 MB·English

by  Tony Lelièvre, Gabriel Stoltz, Mathias Rousset

Free Energy
Computations
A Mathematical Perspective
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Free Energy
Computations
A Mathematical Perspective
Tony Lelièvre
Ecole des Ponts ParisTech & INRIA Rocquencourt, France
Mathias Rousset
INRIA Lille - Nord Europe, France
Gabriel Stoltz
Ecole des Ponts ParisTech & INRIA Rocquencourt, France
Imperial College Press
ICP
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Published by
Imperial College Press
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Covent Garden
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Distributed by
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Library of Congress Cataloging-in-Publication Data
Lelièvre, Tony.
Free energy computations : a mathematical perspective / by Tony Lelièvre, Gabriel Stoltz &
Mathias Rousset.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-1-84816-247-1 (hardcover : alk. paper)
ISBN-10: 1-84816-247-2 (hardcover : alk. paper)
1. Gibbs' free energy. 2. Statistical physics--Mathematical models. I. Stoltz, Gabriel.
II. Rousset, Mathias. III. Title.
QC318.E57L45 2010
536'.7--dc22
2010005252
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2010 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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For photocopying of material in this volume, please pay a copying fee through the Copyright
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Printed in Singapore.
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Preface
The computation of free-energy differences is a very important and active
research field in computational statistical physics. Being applied mathe-
maticians,wewereunabletofindatextbookthatsuitedourneedswhenwe
first approached numerical problems in computational statistical physics.
The references we found were either theoretical statistical physics books,
focusingonmodelproblemstreatedanalyticallyandperturbationsthereof;
or textbooks for practitioners, with recipes and comments on the physical
models, but almost no mathematical analysis of the numerical techniques.
We hope that the present book contributes to filling this gap and usefully
complements some of the recent references concerning the long-time inte-
gration of Hamiltonian dynamics (such as [Hairer et al. (2006)]).
Theaudiencewehaveinmindwhilewritingtheselinesiscomposedboth
of mathematicians and scientists from the applied communities (physics,
chemistry, biology, etc.), who use free-energy techniquesas one tool among
many to study the complex systems they are interested in. We conceive
thesenotesasaself-containedpresentationofwhatwebelievearecurrently
the most standard computational methods for free-energy computations.
We hope that this presentation will be of interest to researchers in the
field, while still being accessible for graduate students.
Free-energycomputationisanopportunityformathematicianstostudy
many theoretical concepts and numerical strategies, such as techniques to
sample multimodal probability measures, constrained stochastic dynamics
tosamplemeasuresonsubmanifolds,adaptiveimportancesamplingstrate-
gies, etc. In this book, we insist on the numerical analysis of the methods
at hand, giving error estimates or rates of convergence.
Besides, for those interested in the applications, this book is an oppor-
tunitytolearnmoreaboutthemathematicalunderpinningofthenumerical
v
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vi Free Energy Computations: A Mathematical Perspective
techniques used on a daily basis for the computation of free-energy differ-
ences. We want to highlight the similarities between techniques presented
as very different in nature in the current literature. We hope that a more
abstractviewpointontheproblemsathandwillbeinspiringforpractition-
ers. To this audience, this book may also be seen as a companion book,
more biased towards mathematical analysis, of the recent review book on
free energy methods [Chipot and Pohorille (2007b)].
Letusalsoinsistatthispointonthemanypossibleapplicationfieldsof
the techniques presented in this book. Besides the obvious application do-
mainswhereafree-energydifferenceisanimportantquantityperse sinceit
hasanexperimentalmeaning(biology, physics, etc.), therealsoexistscien-
tificfieldswhereratiosofpartitionfunctionsarerequiredforcomputational
purposes. An example is computational statistics (in particular Bayesian
statistics). Moreover, it is often the case that some adequate importance
sampling function is needed to enhance the sampling of multimodal proba-
bilitydistributions,anditisingeneraldifficulttothinkofgoodcandidates
for high-dimensional problems. The free energy can be used as an efficient
and automatic importance sampling function, once some “slowly evolv-
ing”, “frustrated”, or “metastable” degree of freedom has been identified.
In particular, adaptive methods are very interesting since they provide an
adaptive importance sampling strategy.
The book is organized linearly, but sometimes we need to anticipate on
notions presented later on to motivate some techniques or concepts (espe-
cially in the introductory chapter). The notation is homogeneous through-
out the book, and, for the reader’s convenience, it is summarized in the
Appendix. This will hopefully help the reader to keep track of the objects
manipulated, in particular in the sections on constrained processes.
We now briefly describe the structure of the book, which is based on
the mathematical classification explained at the end of Chapter 1. The
introductory chapter presents the notions of statistical physics which will
be constantly used in this book. It also gives the definition of free en-
ergy. This presentation is deliberately very different from that of standard
physics textbooks, since our aim is primarily to describe how to compute
average properties predicted by statistical physics, rather than motivat-
ing the physical relevance of these expressions. Chapter 2 presents in its
first two sections standard techniques to compute averages in computa-
tional statistical physics, before describing a first set of techniques to com-
putefree-energydifferencesusingonlythesestandardmethods(free-energy
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Preface vii
perturbation, histogram methods). Chapter 3 is more technical, and deals
mainlywithconstrained processes,whichareafundamentaltoolfortheso-
called thermodynamic integration method. Nonequilibrium processes are
consideredinChapter4, withvariationsandextensionsaroundthefamous
Jarzynski equality. Adaptive methods are at the heart of Chapter 5. Our
understanding of these recently proposed methods is not complete to date.
We hope that our general mathematical presentation will motivate further
research. The last chapter is a short presentation of selection strategies,
which can be used as a complement to other methods. Chapter 5 is the
part of the book where most of the open problems are listed.
Software deserves some comment. Most research groups in computa-
tional statistical physics are developing an in-house code, or are organized
around the use of a well-known simulation package, dedicated to a fam-
ily of applications. This code is often efficient, flexible, and allows one to
treat challenging systems of current interest. These characteristics mean
that the source files are not easily understood and modified, and that it
is difficult to precisely know which numerical methods are used, and how
they are actually implemented. For some users, the code is a black-box,
whoseresultsaretooquicklytrustedandtoorarelyquestioned. Webelieve
that running a code as a black-box and blindly trusting the outcome is
risky. Questioning the validity of the results is a necessity. Besides, the
reproducibilityofthenumericalsimulationsisasfundamentalaruleasthe
reproducibility of any experiment in experimental sciences. For these rea-
sons, we propose a series of codes for the two running examples considered
throughout the book. They are freely available on Gabriel Stoltz’s web-
page1 so that the numerical computations presented throughout this book
can be double-checked. Our programs are not implemented in a computa-
tionally optimal way. On the other hand, the simplicity of the code and
the documentation should make it simple enough for the reader to fully
understand the details of one possible implementation of the methods and
to test different numerical strategies.
Weconclude thispreface byemphasizingthat computational statistical
physics in general, and free-energy based techniques in particular, are a
relatively recent research domain in applied mathematics. Many questions
thereforeremainopen. Anycommentsaboutthescientificcontent,theped-
agogical or non-pedagogical approaches used, typos, etc., are welcome! We
1Visit the webpage http://cermics.enpc.fr/∼stoltz/ or download directly the file from
http://cermics.enpc.fr/∼stoltz/FreeEnergyCodes.tar.gz.
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viii Free Energy Computations: A Mathematical Perspective
areverymuchindebtedtomanyofourcoworkers,foracarefulre-readingof
parts of this book: Manuel Ath`enes, Chris Chipot, John Chodera, Claude
Le Bris, Fr´ed´eric Legoll, Kimiya Minoukadeh, and Rapha¨el Roux. We fi-
nallyacknowledgethehospitalityoftheHausdorffInstituteofMathematics,
in Bonn (Germany), where we three authors attended a research program
inspring2008andwherethewritingofthisbookwasinitiated. Wearealso
grateful to the Banff International Research Station (Canada), where we
were invited to a timely and stimulating workshop on “Mathematical and
NumericalMethodsforFreeEnergyCalculationsinMolecularSystems”in
June 2008.
Tony Leli`evre, Mathias Rousset and Gabriel Stoltz
Paris, January 2010
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Contents
Preface v
1. Introduction 1
1.1 Computational statistical physics: some landmarks . . . . 1
1.1.1 Some orders of magnitude . . . . . . . . . . . . . 2
1.1.2 Aims of molecular simulation . . . . . . . . . . . . 3
1.2 Microscopic description of physical systems . . . . . . . . 6
1.2.1 Interactions . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Dynamics of isolated systems . . . . . . . . . . . . 13
1.2.3 Thermodynamic ensembles . . . . . . . . . . . . . 20
1.3 Free energy and its numerical computation . . . . . . . . 33
1.3.1 Absolute free energy. . . . . . . . . . . . . . . . . 34
1.3.2 Relative free energies . . . . . . . . . . . . . . . . 37
1.3.3 Free energy and metastability . . . . . . . . . . . 44
1.3.4 Computational techniques . . . . . . . . . . . . . 51
1.4 Summary of the mathematical tools and structure of the
book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2. Sampling methods 61
2.1 Markov chain methods . . . . . . . . . . . . . . . . . . . . 63
2.1.1 Some background material on the theory of
Markov chains . . . . . . . . . . . . . . . . . . . . 64
2.1.2 The Metropolis-Hastings algorithm . . . . . . . . 67
2.1.3 Hybrid Monte-Carlo . . . . . . . . . . . . . . . . . 72
2.1.4 Generalized Metropolis-Hastings variants . . . . . 74
2.2 Continuous stochastic dynamics . . . . . . . . . . . . . . . 77
ix