Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond -- Contents -- Preface -- List of Contributors -- Color Plates -- Part I: Fluctuation Relations -- 1 Fluctuation Relations: A Pedagogical Overview -- 1.1 Preliminaries -- 1.2 Entropy and the Second Law -- 1.3 Stochastic Dynamics -- 1.3.1 Master Equations -- 1.3.2 Kramers-Moyal and Fokker-Planck Equations -- 1.3.3 Ornstein-Uhlenbeck Process -- 1.4 Entropy Generation and Stochastic Irreversibility -- 1.4.1 Reversibility of a Stochastic Trajectory -- 1.5 Entropy Production in the Overdamped Limit -- 1.6 Entropy, Stationarity, and Detailed Balance -- 1.7 A General Fluctuation Theorem -- 1.7.1 Work Relations -- 1.7.1.1 The Crooks Work Relation and Jarzynski Equality -- 1.7.2 Fluctuation Relations for Mechanical Work -- 1.7.3 Fluctuation Theorems for Entropy Production -- 1.8 Further Results -- 1.8.1 Asymptotic Fluctuation Theorems -- 1.8.2 Generalizations and Consideration of Alternative Dynamics -- 1.9 Fluctuation Relations for Reversible Deterministic Systems -- 1.10 Examples of the Fluctuation Relations in Action -- 1.10.1 Harmonic Oscillator Subject to a Step Change in Spring Constant -- 1.10.2 Smoothly Squeezed Harmonic Oscillator -- 1.10.3 A Simple Nonequilibrium Steady State -- 1.11 Final Remarks -- References -- 2 Fluctuation Relations and the Foundations of Statistical Thermodynamics: A Deterministic Approach and Numerical Demonstration -- 2.1 Introduction -- 2.2 The Relations -- 2.3 Proof of Boltzmann's Postulate of Equal A Priori Probabilities -- 2.4 Nonequilibrium Free Energy Relations -- 2.5 Simulations and Results -- 2.6 Results Demonstrating the Fluctuation Relations -- 2.7 Conclusion -- References -- 3 Fluctuation Relations in Small Systems: Exact Results from the Deterministic Approach -- 3.1 Motivation -- 3.1.1 Why Fluctuations?.

3.1.2 Nonequilibrium Molecular Dynamics -- 3.1.3 The Dissipation Function -- 3.1.4 Fluctuation Relations: The Need for Clarification -- 3.2 Formal Development -- 3.2.1 Transient Relations -- 3.2.2 Work Relations: Jarzynski -- 3.2.3 Asymptotic Results -- 3.2.4 Extending toward the Steady State -- 3.2.5 The Gallavotti-Cohen Approach -- 3.3 Discussion -- 3.4 Conclusions -- References -- 4 Measuring Out-of-Equilibrium Fluctuations -- 4.1 Introduction -- 4.2 Work and Heat Fluctuations in the Harmonic Oscillator -- 4.2.1 The Experimental Setup -- 4.2.2 The Equation of Motion -- 4.2.2.1 Equilibrium -- 4.2.3 Nonequilibrium Steady State: Sinusoidal Forcing -- 4.2.4 Energy Balance -- 4.2.5 Heat Fluctuations -- 4.3 Fluctuation Theorem -- 4.3.1 FTs for Gaussian Variables -- 4.3.2 FTs for Wt and Qt Measured in the Harmonic Oscillator -- 4.3.3 Comparison with Theory -- 4.3.4 Trajectory-Dependent Entropy -- 4.4 The Nonlinear Case: Stochastic Resonance -- 4.5 Random Driving -- 4.5.1 Colloidal Particle in an Optical Trap -- 4.5.2 AFM Cantilever -- 4.5.3 Fluctuation Relations Far from Equilibrium -- 4.5.4 Conclusions on Randomly Driven Systems -- 4.6 Applications of Fluctuation Theorems -- 4.6.1 Fluctuation-Dissipation Relations for NESS -- 4.6.1.1 Hatano-Sasa Relation and Fluctuation-Dissipation Around NESS -- 4.6.1.2 Brownian Particle in a Toroidal Optical Trap -- 4.6.2 Generalized Fluctuation-Dissipation Relation -- 4.6.2.1 Statistical Error -- 4.6.2.2 Effect of the Initial Sampled Condition -- 4.6.2.3 Experimental Test -- 4.6.3 Discussion on FDT -- 4.7 Summary and Concluding Remarks -- References -- 5 Recent Progress in Fluctuation Theorems and Free Energy Recovery -- 5.1 Introduction -- 5.2 Free Energy Measurement Prior to Fluctuation Theorems -- 5.2.1 Experimental Methods for FE Measurements -- 5.2.2 Computational FE Estimates -- 5.3 Single-Molecule Experiments.

5.3.1 Experimental Techniques -- 5.3.2 Pulling DNA Hairpins with Optical Tweezers -- 5.4 Fluctuation Relations -- 5.4.1 Experimental Validation of the Crooks Equality -- 5.5 Control Parameters, Con.gurational Variables, and the Definition of Work -- 5.5.1 About the Right Definition of Work: Accumulated versus Transferred Work -- 5.6 Extended Fluctuation Relations -- 5.6.1 Experimental Measurement of the Potential of Mean Force -- 5.7 Free Energy Recovery from Unidirectional Work Measurements -- 5.8 Conclusions -- References -- 6 Information Thermodynamics: Maxwell's Demon in Nonequilibrium Dynamics -- 6.1 Introduction -- 6.2 Szilard Engine -- 6.3 Information Content in Thermodynamics -- 6.3.1 Shannon Information -- 6.3.2 Mutual Information -- 6.3.3 Examples -- 6.4 Second Law of Thermodynamics with Feedback Control -- 6.4.1 General Bound -- 6.4.2 Generalized Szilard Engine -- 6.4.3 Overdamped Langevin System -- 6.4.4 Experimental Demonstration: Feedback-Controlled Ratchet -- 6.4.5 Carnot Efficiency with Two Heat Baths -- 6.5 Nonequilibrium Equalities with Feedback Control -- 6.5.1 Preliminaries -- 6.5.2 Measurement and Feedback -- 6.5.3 Nonequilibrium Equalities with Mutual Information -- 6.5.4 Nonequilibrium Equalities with Efficacy Parameter -- 6.6 Thermodynamic Energy Cost for Measurement and Information Erasure -- 6.7 Conclusions -- Appendix 6.A: Proof of Eq. (6.56) -- References -- 7 Time-Reversal Symmetry Relations for Currents in Quantum and Stochastic Nonequilibrium Systems -- 7.1 Introduction -- 7.2 Functional Symmetry Relations and Response Theory -- 7.3 Transitory Current Fluctuation Theorem -- 7.4 From Transitory to the Stationary Current Fluctuation Theorem -- 7.5 Current Fluctuation Theorem and Response Theory -- 7.6 Case of Independent Particles -- 7.7 Time-Reversal Symmetry Relations in the Master Equation Approach.

7.7.1 Current Fluctuation Theorem for Stochastic Processes -- 7.7.2 Thermodynamic Entropy Production -- 7.7.3 Case of Effusion Processes -- 7.7.4 Statistics of Histories and Time Reversal -- 7.8 Transport in Electronic Circuits -- 7.8.1 Quantum Dot with One Resonant Level -- 7.8.2 Capacitively Coupled Circuits -- 7.8.3 Coherent Quantum Conductor -- 7.9 Conclusions -- References -- 8 Anomalous Fluctuation Relations -- 8.1 Introduction -- 8.2 Transient Fluctuation Relations -- 8.2.1 Motivation -- 8.2.2 Scaling -- 8.2.3 Transient Fluctuation Relation for Ordinary Langevin Dynamics -- 8.3 Transient Work Fluctuation Relations for Anomalous Dynamics -- 8.3.1 Gaussian Stochastic Processes -- 8.3.1.1 Correlated Internal Gaussian Noise -- 8.3.1.2 Correlated External Gaussian Noise -- 8.3.2 Levy Flights -- 8.3.3 Time-Fractional Kinetics -- 8.4 Anomalous Dynamics of Biological Cell Migration -- 8.4.1 Cell Migration in Equilibrium -- 8.4.1.1 Experimental Results -- 8.4.1.2 Theoretical Modeling -- 8.4.2 Cell Migration Under Chemical Gradients -- 8.5 Conclusions -- References -- Part II: Beyond Fluctuation Relations -- 9 Out-of-Equilibrium Generalized Fluctuation-Dissipation Relations -- 9.1 Introduction -- 9.1.1 The Relevance of Fluctuations: Few Historical Comments -- 9.2 Generalized Fluctuation-Dissipation Relations -- 9.2.1 Chaos and the FDR: van Kampen's Objection -- 9.2.2 Generalized FDR for Stationary Systems -- 9.2.3 Remarks on the Invariant Measure -- 9.2.4 Generalized FDR for Markovian Systems -- 9.3 Random Walk on a Comb Lattice -- 9.3.1 Anomalous Diffusion and FDR -- 9.3.2 Transition Rates of the Model -- 9.3.3 Anomalous Dynamics -- 9.3.4 Application of the Generalized FDR -- 9.4 Entropy Production -- 9.5 Langevin Processes without Detailed Balance -- 9.5.1 Markovian Linear System -- 9.5.2 Fluctuation-Response Relation -- 9.5.3 Entropy Production.

9.6 Granular Intruder -- 9.6.1 Model -- 9.6.2 Dense Case: Double Langevin with Two Temperatures -- 9.6.3 Generalized FDR and Entropy Production -- 9.7 Conclusions and Perspectives -- References -- 10 Anomalous Thermal Transport in Nanostructures -- 10.1 Introduction -- 10.2 Numerical Study on Thermal Conductivity and Heat Energy Diffusion in One-Dimensional Systems -- 10.3 Breakdown of Fourier's Law: Experimental Evidence -- 10.4 Theoretical Models -- 10.5 Conclusions -- References -- 11 Large Deviation Approach to Nonequilibrium Systems -- 11.1 Introduction -- 11.2 From Equilibrium to Nonequilibrium Systems -- 11.2.1 Equilibrium Systems -- 11.2.2 Nonequilibrium Systems -- 11.2.3 Equilibrium Versus Nonequilibrium Systems -- 11.3 Elements of Large Deviation Theory -- 11.3.1 General Results -- 11.3.2 Equilibrium Large Deviations -- 11.3.3 Nonequilibrium Large Deviations -- 11.4 Applications to Nonequilibrium Systems -- 11.4.1 Random Walkers in Discrete and Continuous Time -- 11.4.2 Large Deviation Principle for Density Profiles -- 11.4.3 Large Deviation Principle for Current Fluctuations -- 11.4.4 Interacting Particle Systems: Features and Subtleties -- 11.4.5 Macroscopic Fluctuation Theory -- 11.5 Final Remarks -- References -- 12 Lyapunov Modes in Extended Systems -- 12.1 Introduction -- 12.2 Numerical Algorithms and LV Correlations -- 12.3 Universality Classes of Hydrodynamic Lyapunov Modes -- 12.4 Hyperbolicity and the Significance of Lyapunov Modes -- 12.5 Lyapunov Spectral Gap and Branch Splitting of Lyapunov Modes in a "Diatomic" System -- 12.6 Comparison of Covariant and Orthogonal HLMs -- 12.7 Hyperbolicity and Effective Degrees of Freedom of Partial Differential Equations -- 12.8 Probing the Local Geometric Structure of Inertial Manifolds via a Projection Method -- 12.9 Summary -- References.

13 Study of Single-Molecule Dynamics in Mesoporous Systems, Glasses, and Living Cells.