www.wiley-vch.de

Klages · Just · Jarzynski (Eds.)

Nonequilibrium

Statistical P

hysics of Small System

s

This book offers a comprehensive picture of nonequilibrium phenom-ena in nanoscale systems. Written by internationally recognized experts in the fi eld, it strikes a balance between theory and experiment, and includes in-depth introductions to nonequilibrium fl uctuation relations, nonlinear dynamics and transport, single molecule experiments, and molecular diffusion in nanopores. The authors explore the application of these concepts to nano- and biosystems by cross-linking key methods and ideas from nonequilibrium statistical physics, thermodynamics, stochastic theory, and dynamical systems. By providing an up-to-date survey of small systems physics, the text serves as both a valuable reference for experienced researchers and as an ideal starting point for graduate-level students entering this newly emerging research fi eld.

Part I: Fluctuation relations � Fluctuation relations: A pedagogical overview� Fluctuation relations and the foundations of statistical thermodynamics� Fluctuation relations in small systems� Measuring out of equilibrium fl uctuations� Recent progress in fl uctuation theorems and free energy recovery� Information thermodynamics� Time-reversal symmetry relations� Anomalous fl uctuation relations

Part II: Beyond fl uctuation relations� Out-of-equilibrium generalized fl uctuation-dissipation relations� Anomalous thermal transport in nanostructures� Large deviation approach to nonequilibrium systems� Lyapunov modes in extended systems� Study of single molecule dynamics

Rainer Klages, Reader in Applied Mathematics at Queen Mary University of London, studied phys-ics and philosophy at the Technical University of Berlin. His research stations were Maryland/USA, Budapest, Brussels, and Dresden. His main research interests are nonlinear dynamics, complex systems and nonequilibrium statistical physics with applications to nano- and biosystems.

Wolfram Just, Reader in Applied Mathematics at Queen Mary University of London, studied theo-retical physics in Darmstadt, Fukuoka, Goettingen, and Dresden. His research interests cover topics in statistical physics and dynamical systems theory with special emphasis on synchronisation and control, phase transitions in spatially extended systems, large deviations, and complex networks.

Christopher Jarzynski studied physics in Princeton and Berkeley. He spent ten years at Los Alamos National Laboratory, and since 2006 he has been on the faculty of the University of Maryland in the Department of Chemistry and Biochemistry. His research in-terests include nonequilibrium statistical physics, computational thermodynamics, with the modeling of nanoscale phenomena.

Edited by R. Klages, W. Just, and C. Jarzynski

Nonequilibrium Statistical Physics of Small Systems

Fluctuation Relations and Beyond

Reviews of Nonlinear Dynamics and Complexity

Edited by

Rainer Klages, Wolfram Just,

and Christopher Jarzynski

Nonequilibrium StatisticalPhysics of Small Systems

Reviews of Nonlinear Dynamics and Complexity

Schuster, H. G. (ed.)

Reviews of NonlinearDynamics and ComplexityVolume 12008

ISBN: 978-3-527-40729-3

Schuster, H. G. (ed.)

Reviews of NonlinearDynamics and ComplexityVolume 22009

ISBN: 978-3-527-40850-4

Schuster, H. G. (ed.)

Reviews of NonlinearDynamics and ComplexityVolume 32010

ISBN: 978-3-527-40945-7

Grigoriev, R. and Schuster, H. G. (eds.)

Transport and Mixing inLaminar FlowsFrom Microfluidics to Oceanic Currents2011

ISBN: 978-3-527-41011-8

Lüdge, K. (ed.)

Nonlinear Laser DynamicsFrom Quantum Dots to Cryptography2011

ISBN: 978-3-527-41100-9

Klages, R., Just, W., Jarzynski, C. (eds.)

Nonequilibrium StatisticalPhysics of Small SystemsFluctuation Relations and Beyond2013

ISBN: 978-3-527-41094-1

Pesenson, M. M. (ed.)

Multiscale Analysis andNonlinear DynamicsFrom Genes to the Brain2013

ISBN: 978-3-527-41198-6

Niebur, E., Plenz, D., Schuster,H. G. (eds.)

Criticality in Neural Systems2014

ISBN: 978-3-527-41104-7

ISBN: 978-0-471-66658-5

Edited by Rainer Klages, Wolfram Just,and Christopher Jarzynski

Nonequilibrium Statistical Physicsof Small Systems

Fluctuation Relations and Beyond

The Editors

Dr. Rainer KlagesQueen Mary University of LondonSchool of Mathematical SciencesLondon, UKr.klages@qmul.ac.uk

Dr. Wolfram JustQueen Mary University LondonSchool of Mathematical SciencesLondon, UK

Prof. Christopher JarzynskiUniversity of MarylandDept. of Chemistry & BiochemistryInstitute for Physical Science and TechnologyCollege Park, USA

The Series Editor

Prof. Dr. Heinz Georg SchusterSaarbrücken, Germany

Cover picture

Cover, upper picture by Prof. Dr. Sergio Cilibertoet al., Universite de Lyon, ENSL and CNRSDistributions of the classical work of a Brownianparticle trapped in a nonlinear double-well potentialproduced by two laser beams, and FluctuationTheorem formula.Cover, lower picture by Prof. Dr. Felix Ritort et al.,Universitat de BarcelonaExperimental setup for the mechanical folding/unfolding of a DNA hairpin, and Jarzynski workrelation.

All books published byWiley-VCH are carefully produced.Nevertheless, authors, editors, and publisher do not warrantthe information contained in these books, including thisbook, to be free of errors. Readers are advised to keep inmind that statements, data, illustrations, procedural detailsor other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication DataA catalogue record for this book is available fromthe British Library.

Bibliographic information published by the DeutscheNationalbibliothekThe Deutsche Nationalbibliothek lists this publication in theDeutsche Nationalbibliografie; detailed bibliographic dataare available on the Internet at http://dnb.d-nb.de.

# 2013 Wiley-VCH Verlag & Co. KGaA, Boschstr. 12, 69469Weinheim, Germany

All rights reserved (including those of translation into otherlanguages). No part of this book may be reproduced in anyform – by photoprinting, microfilm, or any other means –nor transmitted or translated into a machine languagewithout written permission from the publishers. Registerednames, trademarks, etc. used in this book, even when notspecifically marked as such, are not to be consideredunprotected by law.

Print ISBN: 978-3-527-41094-1ePDF ISBN: 978-3-527-65873-2ePub ISBN: 978-3-527-65872-5mobi ISBN: 978-3-527-65871-8oBook ISBN: 978-3-527-65870-1

Cover Design Spieszdesign, Neu-Ulm

Typesetting Thomson Digital, Noida, India

Printing and Binding Markono Print Media Pte Ltd,Singapore

Printed in Singapore

Contents

Preface XIIIList of Contributors XVIIColor Plates XXIII

Part I Fluctuation Relations 1

1 Fluctuation Relations: A Pedagogical Overview 3Richard Spinney and Ian Ford

1.1 Preliminaries 31.2 Entropy and the Second Law 51.3 Stochastic Dynamics 81.3.1 Master Equations 81.3.2 Kramers–Moyal and Fokker–Planck Equations 91.3.3 Ornstein–Uhlenbeck Process 111.4 Entropy Generation and Stochastic Irreversibility 131.4.1 Reversibility of a Stochastic Trajectory 131.5 Entropy Production in the Overdamped Limit 211.6 Entropy, Stationarity, and Detailed Balance 251.7 A General Fluctuation Theorem 271.7.1 Work Relations 301.7.1.1 The Crooks Work Relation and Jarzynski Equality 311.7.2 Fluctuation Relations for Mechanical Work 341.7.3 Fluctuation Theorems for Entropy Production 361.8 Further Results 371.8.1 Asymptotic Fluctuation Theorems 371.8.2 Generalizations and Consideration of Alternative Dynamics 391.9 Fluctuation Relations for Reversible Deterministic Systems 411.10 Examples of the Fluctuation Relations in Action 451.10.1 Harmonic Oscillator Subject to a Step Change in Spring Constant 451.10.2 Smoothly Squeezed Harmonic Oscillator 491.10.3 A Simple Nonequilibrium Steady State 521.11 Final Remarks 54

References 55

jV

2 Fluctuation Relations and the Foundations of StatisticalThermodynamics: A Deterministic Approach and NumericalDemonstration 57James C. Reid, Stephen R. Williams, Debra J. Searles,Lamberto Rondoni, and Denis J. Evans

2.1 Introduction 572.2 The Relations 582.3 Proof of Boltzmann’s Postulate of Equal A Priori

Probabilities 622.4 Nonequilibrium Free Energy Relations 672.5 Simulations and Results 692.6 Results Demonstrating the Fluctuation Relations 742.7 Conclusion 80

References 81

3 Fluctuation Relations in Small Systems: Exact Results from theDeterministic Approach 83Lamberto Rondoni and O.G. Jepps

3.1 Motivation 843.1.1 Why Fluctuations? 853.1.2 Nonequilibrium Molecular Dynamics 863.1.3 The Dissipation Function 893.1.4 Fluctuation Relations: The Need for Clarification 923.2 Formal Development 943.2.1 Transient Relations 943.2.2 Work Relations: Jarzynski 963.2.3 Asymptotic Results 983.2.4 Extending toward the Steady State 1013.2.5 The Gallavotti–Cohen Approach 1053.3 Discussion 1083.4 Conclusions 110

References 111

4 Measuring Out-of-Equilibrium Fluctuations 115L. Bellon, J. R. Gomez- Solano, A. Petrosyan, and Sergio Ciliberto

4.1 Introduction 1154.2 Work and Heat Fluctuations in the Harmonic Oscillator 1164.2.1 The Experimental Setup 1164.2.2 The Equation of Motion 1174.2.2.1 Equilibrium 1174.2.3 Nonequilibrium Steady State: Sinusoidal Forcing 1184.2.4 Energy Balance 1194.2.5 Heat Fluctuations 1204.3 Fluctuation Theorem 1214.3.1 FTs for Gaussian Variables 122

VIj Contents

4.3.2 FTs forWt and Qt Measured in the HarmonicOscillator 123

4.3.3 Comparison with Theory 1254.3.4 Trajectory-Dependent Entropy 1254.4 The Nonlinear Case: Stochastic Resonance 1284.5 Random Driving 1324.5.1 Colloidal Particle in an Optical Trap 1324.5.2 AFM Cantilever 1364.5.3 Fluctuation Relations Far from Equilibrium 1394.5.4 Conclusions on Randomly Driven Systems 1424.6 Applications of Fluctuation Theorems 1424.6.1 Fluctuation–Dissipation Relations for NESS 1434.6.1.1 Hatano–Sasa Relation and Fluctuation–Dissipation

Around NESS 1444.6.1.2 Brownian Particle in a Toroidal Optical Trap 1444.6.2 Generalized Fluctuation–Dissipation Relation 1464.6.2.1 Statistical Error 1464.6.2.2 Effect of the Initial Sampled Condition 1474.6.2.3 Experimental Test 1494.6.3 Discussion on FDT 1494.7 Summary and Concluding Remarks 150

References 151

5 Recent Progress in Fluctuation Theorems and Free EnergyRecovery 155Anna Alemany, Marco Ribezzi-Crivellari, and Felix Ritort

5.1 Introduction 1555.2 Free Energy Measurement Prior to Fluctuation Theorems 1565.2.1 Experimental Methods for FE Measurements 1565.2.2 Computational FE Estimates 1585.3 Single-Molecule Experiments 1595.3.1 Experimental Techniques 1605.3.2 Pulling DNA Hairpins with Optical Tweezers 1625.4 Fluctuation Relations 1635.4.1 Experimental Validation of the Crooks Equality 1655.5 Control Parameters, Configurational Variables, and the

Definition of Work 1665.5.1 About the Right Definition of Work: Accumulated versus

Transferred Work 1685.6 Extended Fluctuation Relations 1725.6.1 Experimental Measurement of the Potential of Mean Force 1745.7 Free Energy Recovery from Unidirectional Work

Measurements 1755.8 Conclusions 177

References 177

Contents jVII

6 Information Thermodynamics: Maxwell's Demon in NonequilibriumDynamics 181Takahiro Sagawa and Masahito Ueda

6.1 Introduction 1816.2 Szilard Engine 1826.3 Information Content in Thermodynamics 1846.3.1 Shannon Information 1846.3.2 Mutual Information 1856.3.3 Examples 1876.4 Second Law of Thermodynamics with Feedback Control 1896.4.1 General Bound 1906.4.2 Generalized Szilard Engine 1916.4.3 Overdamped Langevin System 1926.4.4 Experimental Demonstration: Feedback-Controlled Ratchet 1946.4.5 Carnot Efficiency with Two Heat Baths 1966.5 Nonequilibrium Equalities with Feedback Control 1976.5.1 Preliminaries 1976.5.2 Measurement and Feedback 2006.5.3 Nonequilibrium Equalities with Mutual Information 2016.5.4 Nonequilibrium Equalities with Efficacy Parameter 2036.6 Thermodynamic Energy Cost for Measurement and Information

Erasure 2056.7 Conclusions 208

Appendix 6.A: Proof of Eq. (6.56) 208References 209

7 Time-Reversal Symmetry Relations for Currents in Quantum andStochastic Nonequilibrium Systems 213Pierre Gaspard

7.1 Introduction 2137.2 Functional Symmetry Relations and Response Theory 2167.3 Transitory Current Fluctuation Theorem 2207.4 From Transitory to the Stationary Current Fluctuation Theorem 2247.5 Current Fluctuation Theorem and Response Theory 2277.6 Case of Independent Particles 2307.7 Time-Reversal Symmetry Relations in the Master Equation

Approach 2387.7.1 Current Fluctuation Theorem for Stochastic Processes 2387.7.2 Thermodynamic Entropy Production 2417.7.3 Case of Effusion Processes 2417.7.4 Statistics of Histories and Time Reversal 2427.8 Transport in Electronic Circuits 2447.8.1 Quantum Dot with One Resonant Level 2447.8.2 Capacitively Coupled Circuits 2457.8.3 Coherent Quantum Conductor 250

VIIIj Contents

7.9 Conclusions 252References 254

8 Anomalous Fluctuation Relations 259Rainer Klages, Aleksei V. Chechkin, and Peter Dieterich

8.1 Introduction 2598.2 Transient Fluctuation Relations 2608.2.1 Motivation 2608.2.2 Scaling 2628.2.3 Transient Fluctuation Relation for Ordinary Langevin Dynamics 2638.3 Transient Work Fluctuation Relations for Anomalous Dynamics 2658.3.1 Gaussian Stochastic Processes 2658.3.1.1 Correlated Internal Gaussian Noise 2658.3.1.2 Correlated External Gaussian Noise 2668.3.2 L�evy Flights 2678.3.3 Time-Fractional Kinetics 2688.4 Anomalous Dynamics of Biological Cell Migration 2698.4.1 Cell Migration in Equilibrium 2708.4.1.1 Experimental Results 2718.4.1.2 Theoretical Modeling 2728.4.2 Cell Migration Under Chemical Gradients 2758.5 Conclusions 277

References 278

Part II Beyond Fluctuation Relations 283

9 Out-of-Equilibrium Generalized Fluctuation–Dissipation Relations 285G. Gradenigo, A. Puglisi, A. Sarracino, D. Villamaina, and A. Vulpiani

9.1 Introduction 2859.1.1 The Relevance of Fluctuations: Few Historical Comments 2869.2 Generalized Fluctuation–Dissipation Relations 2879.2.1 Chaos and the FDR: van Kampen’s Objection 2879.2.2 Generalized FDR for Stationary Systems 2889.2.3 Remarks on the Invariant Measure 2909.2.4 Generalized FDR for Markovian Systems 2929.3 RandomWalk on a Comb Lattice 2949.3.1 Anomalous Diffusion and FDR 2949.3.2 Transition Rates of the Model 2959.3.3 Anomalous Dynamics 2969.3.4 Application of the Generalized FDR 2979.4 Entropy Production 3009.5 Langevin Processes without Detailed Balance 3019.5.1 Markovian Linear System 3029.5.2 Fluctuation–Response Relation 303

Contents jIX

9.5.3 Entropy Production 3059.6 Granular Intruder 3069.6.1 Model 3079.6.2 Dense Case: Double Langevin with Two Temperatures 3099.6.3 Generalized FDR and Entropy Production 3119.7 Conclusions and Perspectives 313

References 314

10 Anomalous Thermal Transport in Nanostructures 319Gang Zhang, Sha Liu, and Baowen Li

10.1 Introduction 31910.2 Numerical Study on Thermal Conductivity and Heat Energy

Diffusion in One-Dimensional Systems 32010.3 Breakdown of Fourier’s Law: Experimental Evidence 32510.4 Theoretical Models 32710.5 Conclusions 331

References 332

11 Large Deviation Approach to Nonequilibrium Systems 335Hugo Touchette and Rosemary J. Harris

11.1 Introduction 33511.2 From Equilibrium to Nonequilibrium Systems 33611.2.1 Equilibrium Systems 33611.2.2 Nonequilibrium Systems 33911.2.3 Equilibrium Versus Nonequilibrium Systems 34011.3 Elements of Large Deviation Theory 34111.3.1 General Results 34111.3.2 Equilibrium Large Deviations 34311.3.3 Nonequilibrium Large Deviations 34511.4 Applications to Nonequilibrium Systems 34711.4.1 RandomWalkers in Discrete and Continuous Time 34711.4.2 Large Deviation Principle for Density Profiles 34911.4.3 Large Deviation Principle for Current Fluctuations 35011.4.4 Interacting Particle Systems: Features and

Subtleties 35211.4.5 Macroscopic Fluctuation Theory 35411.5 Final Remarks 356

References 357

12 Lyapunov Modes in Extended Systems 361Hong-Liu Yang and G€unter Radons

12.1 Introduction 36112.2 Numerical Algorithms and LV Correlations 36312.3 Universality Classes of Hydrodynamic Lyapunov Modes 36512.4 Hyperbolicity and the Significance of Lyapunov Modes 369

Xj Contents

12.5 Lyapunov Spectral Gap and Branch Splitting of Lyapunov Modes in a“Diatomic” System 372

12.6 Comparison of Covariant and Orthogonal HLMs 37612.7 Hyperbolicity and Effective Degrees of Freedom of Partial Differential

Equations 38012.8 Probing the Local Geometric Structure of Inertial Manifolds via a

Projection Method 38412.9 Summary 388

References 389

13 Study of Single-Molecule Dynamics in Mesoporous Systems, Glasses,and Living Cells 393Stephan Mackowiak and Christoph Br€auchle

13.1 Introduction 39313.1.1 Experimental Method 39313.1.2 Analysis of the Single-Molecule Trajectories 39513.2 Investigation of the Structure of Mesoporous Silica Employing Single-

Molecule Microscopy 39613.2.1 Mesoporous Silica 39613.2.2 Combining TEM and SMM for Structure Determination of Mesoporous

Silica 39813.2.3 Applications of SMM to Improve the Synthesis of Mesoporous

Systems 39913.3 Investigation of the Diffusion of Guest Molecules in Mesoporous

Systems 40213.3.1 A More Detailed Look into the Diffusional Dynamics of Guest

Molecules in Nanopores 40213.3.2 Modification of the Flow Medium in the Nanopores and Its Influence

on Probe Diffusion 40413.3.3 Loading of Cargoes into Mesopores: A Step toward Drug Delivery

Applications 40613.4 A Test of the Ergodic Theorem by Employing Single-Molecule

Microscopy 40713.5 Single-Particle Tracking in Biological Systems 40913.6 Conclusion and Outlook 412

References 413

Index 415

Contents jXI

Preface

The term small systems denotes objects composed of a limited, small number ofparticles, as is typical for matter on meso- and nanoscales. The interest of thescientific community in small systems has been boosted by the recent advent ofmicromanipulation techniques and nanotechnologies. These provide scientificinstruments capable of measuring tiny energies in physical systems under non-equilibrium conditions, that is, when these systems are exposed to externalforces generated by gradients or fields. Prominent examples of small systemsexhibiting nonequilibrium dynamics are biopolymers stretched by optical tweez-ers (as shown in the lower picture on the book cover), colloidal particles draggedthrough a fluid by optical traps, and single molecules diffusing through meso-and nanopores.Understanding the statistical physics of such systems is particularly challenging,

because their small size does not allow one to apply standard methods of statisticalmechanics and thermodynamics, which presuppose large numbers of particles.Small systems often display an intricate interplay between microscopic nonlineardynamical properties and macroscopic statistical behavior leading to highly non-trivial fluctuations of physical observables (cf. the upper picture on the book cover).They can thus serve as a laboratory for understanding the emergence of complexityand irreversibility, in the sense that for a system consisting of many entities thedynamics of the whole is more than the sum of its single parts.Studying the behavior of small systems on different spatiotemporal scales

becomes particularly interesting in view of nonequilibrium transport phe-nomena such as diffusion, heat conduction, and electronic transport. Under-standing these phenomena in small systems requires novel theoretical conceptsthat blend ideas and techniques from nonequilibrium statistical physics, ther-modynamics, stochastic theory, and dynamical systems theory. More recently, ithas become clear that a central role in this field is played by fluctuation relations,which generalize fundamental thermodynamic relations to small systems innonequilibrium situations.The aim of this book is to provide an introduction for both theorists and

experimentalists to small systems physics, fluctuation relations, and the associ-ated research topics listed in the word cloud diagram shown below. The bookshould also be useful for graduate-level students who want to explore this new

jXIII

field of research. The single chapters have been written by internationally recog-nized experts in small systems physics and provide in-depth introductions tothe directions of their research. This approach of a multi-author reference bookappeared to be particularly useful in view of the vast amount of literature availa-ble on different forms of fluctuation relations. While there exist excellentreviews highlighting single facets of fluctuation relations, we feel that the fieldlacks a reference that brings together the most important contributions to thistopic in a comprehensive manner. This book is an attempt to fill the gap. In away, it may act itself as a complex system, in the sense that the book as a wholeideally yields a new picture on small systems physics and fluctuation relationsemerging from a synergy of the individual chapters. Along these lines, ourintention was to embed research on fluctuation relations into a wider context ofsmall systems research by pointing out cross-links to other theories and experi-ments. We thus hope that this book may serve as a catalyst both to fuse existingtheories on fluctuation relations and to open up new directions of inquiry in therapidly growing area of small systems research.

Accordingly, the book is organized into two parts. Part I introduces both thetheoretical and experimental foundations of fluctuation relations. It starts with athreefold opening on basic theoretical ideas. The first chapter features a peda-gogical introduction to fluctuation relations based on an approach that wascoined “stochastic thermodynamics.” The second chapter outlines a fully deter-ministic theory of fluctuation relations by working it out both analytically andnumerically for a particle in an optical trap. The third chapter generalizes thesedeterministic ideas by also establishing cross-links to the Gallavotti–Cohen fluc-tuation theorem, which historically was the first to be established, with mathe-matical rigor, for nonequilibrium steady states. After this theoretical opening,

XIVj Preface

the following two chapters summarize groundbreaking experimental work ontwo fundamental types of fluctuation relations. Along the lines of Gallavotti andCohen, the first subset of them is often referred to as “fluctuation theorems”generalizing the second law of thermodynamics to small systems (see the firstformula on the book cover). This type of fluctuation formulas is tested experi-mentally in systems where particles are confined by optical traps under non-equilibrium conditions. “Work relations,” on the other hand, generalize anequilibrium relation between work and free energy to nonequilibrium (see thesecond formula on the book cover). The result is tested in experiments wheresingle DNA and RNA chains are unzipped by optical tweezers. The remainingthree chapters of Part I elaborate on aspects of fluctuation relations that movedinto the focus of small systems research more recently. The first one introducesthe nonequilibrium thermodynamics of information processing by using feed-back control. The second one reviews quantum mechanical generalizations offluctuation relations applied to electron transport in mesoscopic circuits. Thethird one discusses generalizations of fluctuation relations for stochastic anom-alous dynamics with cross-links to experiments on biological cell migration.Part II goes beyond fluctuation relations by reviewing topics that, while centered

around nonequilibrium fluctuations in small systems, do not elaborate in particu-lar on fluctuation relations. It starts with a discussion of fluctuation–dissipationrelations, which are intimately related to, but may not be confused with, fluctuationrelations. A cross-link to the foregoing chapter is provided in terms of partiallystudying anomalous dynamics, a topic that becomes particularly important for heatconduction in nanostructures, as is demonstrated from both an experimental and atheoretical point of view in the subsequent chapter. Fluctuation relations bear animportant relation to large deviation theory, as is outlined in the next chapter, withapplications to interacting particle systems. The book concludes with a summaryabout Lyapunov modes, which provide important information about the phasespace dynamics in deterministically chaotic interacting many-particle systems,and experiments about diffusion in meso- and nanopores by performing single-molecule spectroscopy.We finally remark that the various points of view expressed in the single chapters

may not always be in full agreement with each other. This became clear in livelydiscussions between different groups of authors when the book was in preparation.As editors, we do not necessarily aim to achieve a complete consensus among allauthors, as differences in opinions are typical for a very active field of researchsuch as the one presented in this book.We are most grateful to Heinz-Georg Schuster, the editor of the series Reviews of

Nonlinear Dynamics and Complexity, in which this book is published as a SpecialIssue, for his invitation to edit this book, and for his help in getting the projectstarted. We also thank Vera Palmer and Ulrike Werner fromWiley-VCH Publishersfor their kind and efficient assistance in editing this book. C.J. gratefullyacknowledges financial support from the National Science Foundation (USA)under grant DMR-0906601. W.J. is grateful for support from the British EPSRC by

Preface jXV

grant EP/H04812X/1. We finally thank all book chapter authors for sharing theirexpertise in this multi-author monograph. Their strong efforts and enthusiasm forthis project were indispensable for bringing it to success.

Summer 2012

London Rainer KlagesLondon Wolfram JustCollege Park, MD Christopher Jarzynski

XVIj Preface

List of Contributors

Anna AlemanyUniversitat de BarcelonaDepartament de Física FonamentalSmall Biosystems LabAvda. Diagonal 64708028 BarcelonaSpain

and

Instituto de Salud Carlos IIICIBER-BBN de BioingenieríaBiomateriales y NanomedicinaC/ Sinesio Delgado 428029 MadridSpain

L. BellonUniversit�e de LyonEcole Normale Sup�erieure de LyonLaboratoire de Physique (CNRSUMR 5672)46 All�ee d’Italie69364 Lyon Cedex 07France

Christoph Br€auchleLudwig-Maximilians-Universit€atM€unchenDepartment ChemieLehrstuhl f€ur PhysikalischeChemie IButenandtstr. 1181377 MunichGermany

Aleksei V. ChechkinNational Science Center “KharkovInstitute of Physics and Technology”(NSC KIPT)Institute for Theoretical PhysicsAkademicheskaya Street 1Kharkov 61108Ukraine

Sergio CilibertoUniversit�e de LyonEcole Normale Sup�erieure de LyonLaboratoire de Physique (CNRSUMR 5672)46 All�ee d’Italie69364 Lyon Cedex 07France

jXVII

Peter DieterichTechnische Universit€at DresdenMedizinische Fakultaet “Carl GustavCarus”Institut f€ur PhysiologieFetscherstrasse 7401307 DresdenGermany

Denis J. EvansAustralian National UniversityResearch School of ChemistryBuilding 35 Science RdCanberra, ACT 0200Australia

Ian FordUniversity College LondonDepartment of Physics andAstronomy and London Centrefor NanotechnologyGower StreetLondon WC1E 6BTUK

Pierre GaspardUniversit�e Libre de BruxellesCenter for Nonlinear Phenomenaand Complex Systems andDepartment of PhysicsCampus PlaineCode Postal 231Boulevard du Triomphe1050 BrusselsBelgium

J.R. Gomez-SolanoUniversit�e de LyonEcole Normale Sup�erieure de LyonLaboratoire de Physique (CNRSUMR 5672)46 All�ee d’Italie69364 Lyon Cedex 07France

G. GradenigoUniversita degli Studi di Roma“La Sapienza”Dipartimento di FisicaPiazzale A. Moro 200185 RomeItaly

Rosemary J. HarrisQueen Mary University of LondonSchool of Mathematical SciencesMile End RoadLondon E1 4NSUK

O.G. JeppsGriffith UniversitySchool of Biomolecular and PhysicalSciencesQueensland Micro- andNanotechnology Centre170 Kessels RoadBrisbane, Qld 4111Australia

Rainer KlagesQueen Mary University of LondonSchool of Mathematical SciencesMile End RoadLondon E1 4NSUK

Baowen LiNational University of SingaporeDepartment of Physics and Centrefor Computational Science andEngineeringScience Drive 2Singapore 117542Singapore

and

NUS Graduate School for IntegrativeSciences and Engineering28 Medical DriveSingapore 117456Singapore

XVIIIj List of Contributors

Sha LiuNational University of SingaporeDepartment of Physics and Centrefor Computational Science andEngineeringScience Drive 2Singapore 117542Singapore

and

NUS Graduate School for IntegrativeSciences and Engineering28 Medical DriveSingapore 117456Singapore

Stephan MackowiakLudwig-Maximilians-Universit€atM€unchenDepartment ChemieLehrstuhl f€ur PhysikalischeChemie IButenandtstr. 1181377 MunichGermany

A. PetrosyanUniversit�e de LyonEcole Normale Sup�erieure de LyonLaboratoire de Physique (CNRSUMR 5672)46 All�ee d’Italie69364 Lyon Cedex 07France

A. PuglisiUniversita degli Studi di Roma“La Sapienza”Dipartimento di FisicaPiazzale A. Moro 200185 RomeItaly

G€unter RadonsChemnitz University of TechnologyInstitute of Mechatronics andInstitute of Physics09107 ChemnitzGermany

James C. ReidThe University of QueenslandAustralian Institute forBioengineering and NanotechnologyAIBN Building (75)Corner Cooper & College RoadsBrisbane, Qld 4072Australia

Marco Ribezzi-CrivellariUniversitat de BarcelonaDepartament de Fisica FonamentalSmall Biosystems LabBarcelonaSpain

and

Instituto de Salud Carlos IIICIBER-BBN de BioingenieriaBiomateriales y NanomedicinaMadridSpain

Felix RitortUniversitat de BarcelonaDepartament de Fisica FonamentalSmall Biosystems LabBarcelonaSpain

and

Instituto de Salud Carlos IIICIBER-BBN de BioingenieriaBiomateriales y NanomedicinaMadridSpain

List of Contributors jXIX

Lamberto RondoniDipartimento di ScienzeMatematichePolitecnico di TorinoCorso Duca degli Abruzzi 2410129 TorinoINFN, Sezione di TorinoVia P. Giuria 110125 Torino, Italy

Takahiro SagawaKyoto UniversityThe Hakubi Center for AdvancedResearchiCeMS Complex 1 West WingYoshida-Ushinomiya-cho, Sakyo-kuKyoto 606-8302Japan

and

Kyoto UniversityYukawa Institute of TheoreticalPhysicsKitashirakawa Oiwake-Cho, Sakyo-kuKyoto 606-8502Japan

A. SarracinoUniversita degli Studi di Roma“La Sapienza”Dipartimento di FisicaPiazzale A. Moro 200185 RomeItaly

Debra J. SearlesThe University of QueenslandAustralian Institute forBioengineering and Nanotechnologyand School of Chemistry andMolecular BiosciencesAIBN Building (75)Corner Cooper & College RoadsBrisbane, Qld 4072Australia

Richard SpinneyUniversity College LondonDepartment of Physics andAstronomy and London Centrefor NanotechnologyGower StreetLondon WC1E 6BTUK

Hugo TouchetteQueen Mary University of LondonSchool of Mathematical SciencesMile End RoadLondon E1 4NSUK

Masahito UedaThe University of TokyoDepartment of Physics7-3-1 Hongo, Bunkyo-kuTokyo 113-0033Japan

D. VillamainaUniversita degli Studi di Roma“La Sapienza”Dipartimento di FisicaPiazzale A. Moro 200185 RomeItaly

A. VulpianiUniversita degli Studi di Roma“La Sapienza”Dipartimento di FisicaPiazzale A. Moro 200185 RomeItaly

Stephen R. WilliamsAustralian National UniversityResearch School of ChemistryBuilding 35 Science RdCanberra, ACT 0200Australia

XXj List of Contributors

Hong-Liu YangChemnitz University of TechnologyInstitute of Mechatronics andInstitute of Physics09107 ChemnitzGermany

Gang ZhangPeking UniversityKey Laboratory for the Physics andChemistry of Nanodevices andDepartment of ElectronicsYiheyuan Road 5Beijing 100871China

and

National University of SingaporeDepartment of Physics and Centrefor Computational Science andEngineeringScience Drive 2Singapore 117542Singapore

List of Contributors jXXI

Color Plates

Viscous fluid(v, T)

(a) (b) (c) (d)

Mirror

Brass wire

Laser beams Brass wire Gold mirror

8 mm

θ

g

– + –+

F F

J

BJ

B

·

Figure 4.1 (a) The torsion pendulum. (b) The magnetostatic forcing. (c) Picture of thependulum. (d) Cell where the pendulum is installed.

0.6(a) (b)

6

5

4

3

2

1

00 1 2 3 4 5 6

100

10–5

–5

–10 –5 0 5 10 15 20 25

0Wcl,n / n

Wcl,n Wcl,n/<Wcl,n>

Sw

cl,n

/<W

cl,n

>

5

0.5

0.4

0.3

PD

F PD

F

0.2

0.1

0

Figure 4.12 (a) Distribution of classical workWcl for different numbers of period n ¼ 1, 2, 4, 8,and 12 (f ¼ 0:25 Hz). Inset: Same data in lin-log. (b) Normalized symmetry function as a functionof the normalized work for n ¼ 1 (þ), 2 (�), 4 (^), 8 (D), and 12 (&).

Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond, First Edition.Edited by Rainer Klages, Wolfram Just, and Christopher Jarzynski.# 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

jXXIII

−5 0 5 10

10−4

10−2

100

wτ

P(w

τ)P

(wτ)

−20 0 20 40 60

10−4

10−2

100

wτ wτ

P(w

τ)

−100 0 100 20010−6

10−4

10−2

100

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

δ f0 (pN)

α

experimental data

quadratic fit

(a)

(b)

(d)(c)

Figure 4.14 (a) Dependence of the parametera on the standard deviation of the Gaussianexponentially correlated external force f 0 actingon the colloidal particle. Probability densityfunctions of the work wt for (b) a ¼ 0:20, (c)

a ¼ 3:89, and (d) a ¼ 10:77. The symbolscorrespond to integration times t ¼ 5 ms (�),55ms (&), 105ms (^), 155ms (3),205ms ("), and 255ms (�). The solid blacklines in (b) and (c) are Gaussian fits.

XXIVj Color Plates

−10 0 10 20

wτ

wτ wτ

0 50 100 150

10−6

10−4

10−2

100

P(w

τ)

10−6

10−4

10−2

100

P(w

τ)

10−8

10−6

10−4

10−2

P(w

τ)

0 200 400 600 800

0 20 40 600

5

10

15

20

25

δ f0 (pN)

α

experimental data

quadratic fit

(a)

(c) (d)

(b)

Figure 4.16 (a) Dependence of the parametera on the standard deviation of the Gaussianwhite external force f 0 acting on the cantilever.Probability density functions of the work wt for(b) a ¼ 0:19, (c) a ¼ 3:03, and (d) a ¼ 18:66.

The symbols correspond to integration timest ¼ 97ms (�), 1:074ms (&), 2.051ms (^),3.027ms (3), 4.004ms ("), and 4.981ms (�).The black dashed lines in (b)–(d) represent theexponential fits of the corresponding tails.

Figure 5.3 The Crooks fluctuation relation. (a)Work distributions for the hairpin shown inFigure 5.1 measured at three different pullingspeeds: 50 nms�1 (blue), 100 nm s�1 (green),and 300 nms�1 (red). Unfolding or forward(continuous lines) and refolding or reverse workdistributions (dashed lines) cross each other ata value of 81.0� 0.2 kBT independent of the

pulling speed. (b) Experimental test of the CFRfor 10 different molecules pulled at differentspeeds. The log of the ratio between theunfolding and refolding work distributions isequal to ðW � DGÞ in kBT units. The insetshows the distribution of slopes for the differentmolecules that are clustered around an averagevalue of 0.96. (Figure taken from Ref. [19].)

Color Plates jXXV

100

10-1

10-2

10-3

10-4

10 0-10

p(x,

t)

x [μm] 100 0-100

x [μm] 200 0-200

x [μm]

OUFKK

100

10-1

10-2

10-3

10-4

10 0-10

p(x,

t)

x [μm] 100 0-100

x [μm] 200 0-200

x [μm]

OUFKK

2 3 4 5 6 7 8 9

500 400 300 200 100 0

kurt

osis

Κ

time [min]

(a)

(b)

(c)

NHE+t = 480 mint = 120 mint = 1 min

NHE-t = 480 mint = 120 mint = 1 min

data NHE+

data NHE-

FKK model NHE+

FKK model NHE-

Figure 8.8 Spatiotemporal probabilitydistributions Pðx; tÞ. (a and b) Experimental datafor both cell types at different times insemilogarithmic representation. The dark lines,labeled FKK, show the long-time asymptoticsolutions of our model (Eq. (8.31)) with the sameparameter set used for the MSD fit. The lightlines, labeled OU, depict fits by the Gaussian

distributions (Eq. (8.11)) representing Brownianmotion. For t ¼ 1 min, both Pðx; tÞ show apeaked structure clearly deviating from a Gaussianform. (c) The kurtosis kðtÞ of Pðx; tÞ (cf.Eq. (8.30)) plotted as a function of time saturatesat a value different from that of Brownian motion(line at k ¼ 3). The other two lines represent kðtÞobtained from the model (Eq. (8.31)) [43].

XXVIj Color Plates

0.0001 0.01 1 100

t/L(a) (b)

2

0.01

1

100<

x(t)

2 > /

LL=16L=32L=64L=128L=256L=512

~t1/2

~t

t*/L2

100 1000 10000

<x(t)2>

0

100

1000

10000

δxd(t

)

1000 10000 1e+05 1e+06 1e+07

t

100

10000

δx(t)

<x(t)2>

0

d=0 ε=0.1

Figure 9.1 (a) hx2ðtÞi0=L versus t=L2 plotted for several values of L in the comb model.(b) hx2ðtÞi0 and the response function dx ðtÞ for L ¼ 512. Inset: The parametric plot of dx ðtÞ versushx2ðtÞi0.

Figure 12.18 Variation of the normalized projection error sðn; rÞ=r with the distance to thereference state r with n ¼ 7, 8, 9, and 11, respectively. To improve the statistics, data from 200reference states are presented together. A line with slope 1 is shown to guide the eyes.

10-3

10-2

10-1

r

10-4

10-2

100

S(n

,r)

^

y=0.38-2.94r

y=0.125+0.78r

y=0.027+1.86r

y=0.010+1.47r

y=0.0038+0.93r

4

5

6

7

8

9

1110

(a)

Figure 12.19 The normalized average projection error Sðn; rÞ=r versus the distance to thereference state r with n from 4 to 11. Linear fittings of data for n < M confirms the saturation ofSðn; rÞ=r to a nonzero constant value.

Color Plates jXXVII

Figure 13.3 (a) Overlay of TEM image (gray)and FFT directors (black bars) with single-molecule trajectories (dark blue). Thepolystyrene beads, used for the overlay, areindicated by the yellow (TEM) and red (SMM)

crosses. The light blue boxes show thepositioning error of the SM trajectories. (b–f)Possible movement patterns of a singlemolecule in various structural features found inthe hexagonal mesoporous films.

XXVIIIj Color Plates