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E-raamat: Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond

Edited by (Queen Mary University of London, UK, and TU Darmstadt, Germany), Edited by (University of Maryland), Edited by (Queen Mary University of London), Series edited by (University of Kiel, Germany)
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Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and BeyondSuurem pilt
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This book offers a comprehensive picture of nonequilibrium phenomena in nanoscale systems. Written by internationally recognized experts in the field, this book strikes a balance between theory and experiment, and includes in-depth introductions to nonequilibrium fluctuation 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 field.

Preface xiii
List of Contributors
xvii
Color Plates xxiii
Part I Fluctuation Relations
1(282)
1 Fluctuation Relations: A Pedagogical Overview
3(54)
Richard Spinney
Ian Ford
1.1 Preliminaries
3(2)
1.2 Entropy and the Second Law
5(3)
1.3 Stochastic Dynamics
8(5)
1.3.1 Master Equations
8(1)
1.3.2 Kramers-Moyal and Fokker-Planck Equations
9(2)
1.3.3 Ornstein-Uhlenbeck Process
11(2)
1.4 Entropy Generation and Stochastic Irreversibility
13(8)
1.4.1 Reversibility of a Stochastic Trajectory
13(8)
1.5 Entropy Production in the Overdamped Limit
21(4)
1.6 Entropy, Stationarity, and Detailed Balance
25(2)
1.7 A General Fluctuation Theorem
27(10)
1.7.1 Work Relations
30(1)
1.7.1.1 The Crooks Work Relation and Jarzynski Equality
31(3)
1.7.2 Fluctuation Relations for Mechanical Work
34(2)
1.7.3 Fluctuation Theorems for Entropy Production
36(1)
1.8 Further Results
37(4)
1.8.1 Asymptotic Fluctuation Theorems
37(2)
1.8.2 Generalizations and Consideration of Alternative Dynamics
39(2)
1.9 Fluctuation Relations for Reversible Deterministic Systems
41(4)
1.10 Examples of the Fluctuation Relations in Action
45(9)
1.10.1 Harmonic Oscillator Subject to a Step Change in Spring Constant
45(4)
1.10.2 Smoothly Squeezed Harmonic Oscillator
49(3)
1.10.3 A Simple Nonequilibrium Steady State
52(2)
1.11 Final Remarks
54(3)
References
55(2)
2 Fluctuation Relations and the Foundations of Statistical Thermodynamics: A Deterministic Approach and Numerical Demonstration
57(26)
James C. Reid
Stephen R. Williams
Debra J. Searles
Lamberto Rondoni
Denis J. Evans
2.1 Introduction
57(1)
2.2 The Relations
58(4)
2.3 Proof of Boltzmann's Postulate of Equal A Priori Probabilities
62(5)
2.4 Nonequilibrium Free Energy Relations
67(2)
2.5 Simulations and Results
69(5)
2.6 Results Demonstrating the Fluctuation Relations
74(6)
2.7 Conclusion
80(3)
References
81(2)
3 Fluctuation Relations in Small Systems: Exact Results from the Deterministic Approach
83(32)
Lamberto Rondoni
O.G. Jepps
3.1 Motivation
84(10)
3.1.1 Why Fluctuations?
85(1)
3.1.2 Nonequilibrium Molecular Dynamics
86(3)
3.1.3 The Dissipation Function
89(3)
3.1.4 Fluctuation Relations: The Need for Clarification
92(2)
3.2 Formal Development
94(14)
3.2.1 Transient Relations
94(2)
3.2.2 Work Relations: Jarzynski
96(2)
3.2.3 Asymptotic Results
98(3)
3.2.4 Extending toward the Steady State
101(4)
3.2.5 The Gallavotti-Cohen Approach
105(3)
3.3 Discussion
108(2)
3.4 Conclusions
110(5)
References
111(4)
4 Measuring Out-of-Equilibrium Fluctuations
115(40)
L. Bellon
J. R. Gomez-Solano
A. Petrosyan
Sergio Ciliberto
4.1 Introduction
115(1)
4.2 Work and Heat Fluctuations in the Harmonic Oscillator
116(5)
4.2.1 The Experimental Setup
116(1)
4.2.2 The Equation of Motion
117(1)
4.2.2.1 Equilibrium
117(1)
4.2.3 Nonequilibrium Steady State: Sinusoidal Forcing
118(1)
4.2.4 Energy Balance
119(1)
4.2.5 Heat Fluctuations
120(1)
4.3 Fluctuation Theorem
121(7)
4.3.1 FTs for Gaussian Variables
122(1)
4.3.2 FTs for Wπ and Qπ Measured in the Harmonic Oscillator
123(2)
4.3.3 Comparison with Theory
125(1)
4.3.4 Trajectory-Dependent Entropy
125(3)
4.4 The Nonlinear Case: Stochastic Resonance
128(4)
4.5 Random Driving
132(10)
4.5.1 Colloidal Particle in an Optical Trap
132(4)
4.5.2 AFM Cantilever
136(3)
4.5.3 Fluctuation Relations Far from Equilibrium
139(3)
4.5.4 Conclusions on Randomly Driven Systems
142(1)
4.6 Applications of Fluctuation Theorems
142(8)
4.6.1 Fluctuation-Dissipation Relations for NESS
143(1)
4.6.1.1 Hatano-Sasa Relation and Fluctuation-Dissipation Around NESS
144(1)
4.6.1.2 Brownian Particle in a Toroidal Optical Trap
144(2)
4.6.2 Generalized Fluctuation-Dissipation Relation
146(1)
4.6.2.1 Statistical Error
146(1)
4.6.2.2 Effect of the Initial Sampled Condition
147(2)
4.6.2.3 Experimental Test
149(1)
4.6.3 Discussion on FDT
149(1)
4.7 Summary and Concluding Remarks
150(5)
References
151(4)
5 Recent Progress in Fluctuation Theorems and Free Energy Recovery
155(26)
Anna Alemany
Marco Ribezzi-Crivellari
Felix Ritort
5.1 Introduction
155(1)
5.2 Free Energy Measurement Prior to Fluctuation Theorems
156(3)
5.2.1 Experimental Methods for FE Measurements
156(2)
5.2.2 Computational FE Estimates
158(1)
5.3 Single-Molecule Experiments
159(4)
5.3.1 Experimental Techniques
160(2)
5.3.2 Pulling DNA Hairpins with Optical Tweezers
162(1)
5.4 Fluctuation Relations
163(3)
5.4.1 Experimental Validation of the Crooks Equality
165(1)
5.5 Control Parameters, Configurational Variables, and the Definition of Work
166(6)
5.5.1 About the Right Definition of Work: Accumulated versus Transferred Work
168(4)
5.6 Extended Fluctuation Relations
172(3)
5.6.1 Experimental Measurement of the Potential of Mean Force
174(1)
5.7 Free Energy Recovery from Unidirectional Work Measurements
175(2)
5.8 Conclusions
177(4)
References
177(4)
6 Information Thermodynamics: Maxwell's Demon in Nonequilibrium Dynamics
181(32)
Takahiro Sagawa
Masahito Ueda
6.1 Introduction
181(1)
6.2 Szilard Engine
182(2)
6.3 Information Content in Thermodynamics
184(5)
6.3.1 Shannon Information
184(1)
6.3.2 Mutual Information
185(2)
6.3.3 Examples
187(2)
6.4 Second Law of Thermodynamics with Feedback Control
189(8)
6.4.1 General Bound
190(2)
6.4.2 Generalized Szilard Engine
192(1)
6.4.3 Overdamped Langevin System
192(2)
6.4.4 Experimental Demonstration: Feedback-Controlled Ratchet
194(2)
6.4.5 Carnot Efficiency with Two Heat Baths
196(1)
6.5 Nonequilibrium Equalities with Feedback Control
197(8)
6.5.1 Preliminaries
197(3)
6.5.2 Measurement and Feedback
200(2)
6.5.3 Nonequilibrium Equalities with Mutual Information
202(1)
6.5.4 Nonequilibrium Equalities with Efficacy Parameter
203(2)
6.6 Thermodynamic Energy Cost for Measurement and Information Erasure
205(3)
6.7 Conclusions
208(5)
Appendix 6.A Proof of Eq. (6.56)
208(1)
References
209(4)
7 Time-Reversal Symmetry Relations for Currents in Quantum and Stochastic Nonequilibrium Systems
213(46)
Pierre Gaspard
7.1 Introduction
213(3)
7.2 Functional Symmetry Relations and Response Theory
216(4)
7.3 Transitory Current Fluctuation Theorem
220(4)
7.4 From Transitory to the Stationary Current Fluctuation Theorem
224(3)
7.5 Current Fluctuation Theorem and Response Theory
227(3)
7.6 Case of Independent Particles
230(8)
7.7 Time-Reversal Symmetry Relations in the Master Equation Approach
238(6)
7.7.1 Current Fluctuation Theorem for Stochastic Processes
238(3)
7.7.2 Thermodynamic Entropy Production
241(1)
7.7.3 Case of Effusion Processes
241(1)
7.7.4 Statistics of Histories and Time Reversal
242(2)
7.8 Transport in Electronic Circuits
244(8)
7.8.1 Quantum Dot with One Resonant Level
244(1)
7.8.2 Capacitively Coupled Circuits
245(5)
7.8.3 Coherent Quantum Conductor
250(2)
7.9 Conclusions
252(7)
References
254(5)
8 Anomalous Fluctuation Relations
259(24)
Rainer Klages
Aleksei V. Chechkin
Peter Dieterich
8.1 Introduction
259(1)
8.2 Transient Fluctuation Relations
260(5)
8.2.1 Motivation
260(2)
8.2.2 Scaling
262(1)
8.2.3 Transient Fluctuation Relation for Ordinary Langevin Dynamics
263(2)
8.3 Transient Work Fluctuation Relations for Anomalous Dynamics
265(4)
8.3.1 Gaussian Stochastic Processes
265(1)
8.3.1.1 Correlated Internal Gaussian Noise
265(1)
8.3.1.2 Correlated External Gaussian Noise
266(1)
8.3.2 Levy Flights
267(1)
8.3.3 Time-Fractional Kinetics
268(1)
8.4 Anomalous Dynamics of Biological Cell Migration
269(8)
8.4.1 Cell Migration in Equilibrium
270(1)
8.4.1.1 Experimental Results
271(1)
8.4.1.2 Theoretical Modeling
272(3)
8.4.2 Cell Migration Under Chemical Gradients
275(2)
8.5 Conclusions
277(6)
References
278(5)
Part II Beyond Fluctuation Relations
283(132)
9 Out-of-Equilibrium Generalized Fluctuation-Dissipation Relations
285(34)
G. Gradenigo
A. Puglisi
A. Sarracino
D. Villamaina
A. Vulpiani
9.1 Introduction
285(2)
9.1.1 The Relevance of Fluctuations: Few Historical Comments
286(1)
9.2 Generalized Fluctuation-Dissipation Relations
287(7)
9.2.1 Chaos and the FDR: van Kampen's Objection
287(1)
9.2.2 Generalized FDR for Stationary Systems
288(2)
9.2.3 Remarks on the Invariant Measure
290(2)
9.2.4 Generalized FDR for Markovian Systems
292(2)
9.3 Random Walk on a Comb Lattice
294(6)
9.3.1 Anomalous Diffusion and FDR
294(1)
9.3.2 Transition Rates of the Model
295(1)
9.3.3 Anomalous Dynamics
296(1)
9.3.4 Application of the Generalized FDR
297(3)
9.4 Entropy Production
300(1)
9.5 Langevin Processes without Detailed Balance
301(5)
9.5.1 Markovian Linear System
302(1)
9.5.2 Fluctuation-Response Relation
303(2)
9.5.3 Entropy Production
305(1)
9.6 Granular Intruder
306(7)
9.6.1 Model
307(2)
9.6.2 Dense Case: Double Langevin with Two Temperatures
309(2)
9.6.3 Generalized FDR and Entropy Production
311(2)
9.7 Conclusions and Perspectives
313(6)
References
314(5)
10 Anomalous Thermal Transport in Nanostructures
319(16)
Gang Zhang
Sha Liu
Baowen Li
10.1 Introduction
319(1)
10.2 Numerical Study on Thermal Conductivity and Heat Energy Diffusion in One-Dimensional Systems
320(5)
10.3 Breakdown of Fourier's Law Experimental Evidence
325(2)
10.4 Theoretical Models
327(4)
10.5 Conclusions
331(4)
References
332(3)
11 Large Deviation Approach to Nonequilibrium Systems
335(26)
Hugo Touchette
Rosemary J. Harris
11.1 Introduction
335(1)
11.2 From Equilibrium to Nonequilibrium Systems
336(5)
11.2.1 Equilibrium Systems
336(3)
11.2.2 Nonequilibrium Systems
339(1)
11.2.3 Equilibrium Versus Nonequilibrium Systems
340(1)
11.3 Elements of Large Deviation Theory
341(6)
11.3.1 General Results
341(2)
11.3.2 Equilibrium Large Deviations
343(2)
11.3.3 Nonequilibrium Large Deviations
345(2)
11.4 Applications to Nonequilibrium Systems
347(9)
11.4.1 Random Walkers in Discrete and Continuous Time
347(2)
11.4.2 Large Deviation Principle for Density Profiles
349(1)
11.4.3 Large Deviation Principle for Current Fluctuations
350(2)
11.4.4 Interacting Particle Systems: Features and Subtleties
352(2)
11.4.5 Macroscopic Fluctuation Theory
354(2)
11.5 Final Remarks
356(5)
References
357(4)
12 Lyapunov Modes in Extended Systems
361(32)
Hong-Liu Yang
Gunter Radons
12.1 Introduction
361(2)
12.2 Numerical Algorithms and LV Correlations
363(2)
12.3 Universality Classes of Hydrodynamic Lyapunov Modes
365(4)
12.4 Hyperbolicity and the Significance of Lyapunov Modes
369(3)
12.5 Lyapunov Spectral Gap and Branch Splitting of Lyapunov Modes in a "Diatomic" System
372(4)
12.6 Comparison of Covariant and Orthogonal HLMs
376(4)
12.7 Hyperbolicity and Effective Degrees of Freedom of Partial Differential Equations
380(4)
12.8 Probing the Local Geometric Structure of Inertial Manifolds via a Projection Method
384(4)
12.9 Summary
388(5)
References
389(4)
13 Study of Single-Molecule Dynamics in Mesoporous Systems, Glasses, and Living Cells
393(22)
Stephan Mackowiak
Christoph Brauchle
13.1 Introduction
393(3)
13.1.1 Experimental Method
393(2)
13.1.2 Analysis of the Single-Molecule Trajectories
395(1)
13.2 Investigation of the Structure of Mesoporous Silica Employing Single-Molecule Microscopy
396(6)
13.2.1 Mesoporous Silica
396(2)
13.2.2 Combining TEM and SMM for Structure Determination of Mesoporous Silica
398(1)
13.2.3 Applications of SMM to Improve the Synthesis of Mesoporous Systems
399(3)
13.3 Investigation of the Diffusion of Guest Molecules in Mesoporous Systems
402(5)
13.3.1 A More Detailed Look into the Diffusional Dynamics of Guest Molecules in Nanopores
402(2)
13.3.2 Modification of the Flow Medium in the Nanopores and Its Influence on Probe Diffusion
404(2)
13.3.3 Loading of Cargoes into Mesopores: A Step toward Drug Delivery Applications
406(1)
13.4 A Test of the Ergodic Theorem by Employing Single-Molecule Microscopy
407(2)
13.5 Single-Particle Tracking in Biological Systems
409(3)
13.6 Conclusion and Outlook
412(3)
References
413(2)
Index 415
Rainer Klages, Reader in Applied Mathematics at Queen Mary University of London, studied physics 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 theoretical 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, dynamics with time delay, phase transitions in spatially extended systems, derivation of transport equations, large deviations and extreme events, and complex networks.

Christopher Jarzynski studied physics at Princeton University and the University of California, Berkeley. After a postdoctoral appointment at the Institute for Nuclear Theory in Seattle, he spent ten years at Los Alamos National Laboratory, and since 2006 he has been on the faculty of the University of Maryland, College Park, in the Department of Chemistry and Biochemistry. His research interests include nonequilibrium statistical physics, computational thermodynamics, with the modeling of nanoscale phenomena.
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