An Introduction to Stochastic Thermodynamics: From Basic to Advanced

Description

This book presents the fundamentals of stochastic thermodynamics, one of the most central subjects in non-equilibrium statistical mechanics. It also explores many recent advances, e.g., in information thermodynamics, the thermodynamic uncertainty relation, and the trade-off relation between efficiency and power. The content is divided into three main parts, the first of which introduces readers to fundamental topics in stochastic thermodynamics, e.g., the basics of stochastic processes, the fluctuation theorem and its variants, information thermodynamics, and large deviation theory. In turn, parts two and three explore advanced topics such as autonomous engines (engines not controlled externally) and finite speed engines, while also explaining the key concepts from recent stochastic thermodynamics theory that are involved. To fully benefit from the book, readers only need an undergraduate-level background in statistical mechanics and quantum mechanics; no background in information theory or stochastic processes is needed. Accordingly, the book offers a valuable resource for early graduate or higher-level readers who are unfamiliar with this subject but want to keep up with the cutting-edge research in this field. In addition, the author’s vivid descriptions interspersed throughout the book will help readers grasp ‘living’ research developments and begin their own research in this field. Preface Contents 1 Background 1.1 Aims of Stochastic Thermodynamics 1.2 Overview of This Textbook 1.2.1 Overview of Part I 1.2.2 Overview of Part II 1.2.3 Overview of Part III 1.2.4 Overview of Part IV 1.2.5 How to Read This Textbook? 1.3 Notation, Terminologies and Remarks Part I Basic Framework 2 Stochastic Processes 2.1 Markov Process and Discrete-Time Markov Chain 2.2 Continuous Time Markov Jump Process on Discrete System 2.3 Convergence Theorem 2.4 Formal Introduction of Markov Process 3 Stochastic Thermodynamics 3.1 Shannon Entropy 3.1.1 Stochastic Entropy 3.1.2 Shannon Entropy 3.2 Definition of Heat 3.2.1 Time-Reversal Symmetry of Equilibrium State 3.2.2 Heat in Discrete-State Systems and Detailed-Balance Condition 3.3 Entropy Production 3.4 Differences Between Conventional Thermodynamics and Stochastic Thermodynamics 3.4.1 Summary of Conventional Thermodynamics 3.4.2 Summary of Stochastic Thermodynamics 3.4.3 Entropy 3.4.4 Reversible Adiabatic Processes 3.4.5 How to Derive Results for Macroscopic Systems from Stochastic Thermodynamics 4 Stochastic Processes in Continuous Space 4.1 Mathematical Foundations 4.1.1 Wiener Process 4.1.2 Stochastic Differential Equations and Integrals 4.1.3 Differential Chapman-Kolmogorov Equation 4.2 Description of Langevin Dynamics 4.2.1 Langevin Equation 4.2.2 Experimental Verification of Langevin Description 4.3 Heat in Langevin System 4.4 Entropy Production and Mean Local Velocity 4.5 Multi-dimensional Cases 4.6 Discretization and Continuum Limit 4.6.1 Decomposition of Operator 4.6.2 Discretization of the Stochastic Part 4.6.3 Discretization of the Deterministic Part 4.6.4 Space Discretization and Time Discretization Part II Equalities 5 Fluctuation Theorem 5.1 Detailed Fluctuation Theorem 5.1.1 Stochastic Case 5.1.2 Deterministic Case 5.2 Integral Fluctuation Theorem 5.2.1 Integral Fluctuation Theorem 5.2.2 Jarzynski Equality 5.3 Entropy Production as Phase Volume Change and Expression with KL Divergence 5.3.1 Kullback-Leibler Divergence 5.3.2 Phase Volume 5.3.3 Deterministic Case 5.3.4 Stochastic Case 5.3.5 Absolute Irreversibility 5.4 Thermodynamic Quantities with Strong-Coupling 6 Reduction from Fluctuation Theorem to Other Thermodynamic Relations 6.1 Second Law of Thermodynamics 6.1.1 Standard Derivation of the Second Law 6.1.2 Large Deviation Analysis 6.2 Fluctuation-Dissipation Theorem 6.2.1 Fluctuation-Dissipation Theorem at Zero Frequency 6.2.2 Fluctuation-Dissipation Theorem with Finite Frequency 6.2.3 Higher-Order Relations 6.2.4 Difference from Conventional Linear Response Theory 6.3 Onsager Reciprocity Theorem 7 Fluctuation-Theorem-Type Equalities 7.1 Hatano-Sasa Relation 7.1.1 Dual Transition 7.1.2 Hatano-Sasa Relation and Generalized Second Law 7.1.3 Framework of Steady State Thermodynamics 7.1.4 Hatano-Sasa Inequality and Monotonicity of Kullback-Leibler Divergence 7.2 Entropy Production Under Coarse-Graining 7.2.1 Case Without Nonequilibrium Driving 7.2.2 Case with Nonequilibrium Driving and Hidden Entropy Production 7.2.3 Invariance of Extended Entropy Through Coarse-Graining 8 Various Aspects of Symmetry in Entropy Production 8.1 Introduction to Large Deviation Property and Generating Function 8.1.1 Moments and Cumulants 8.1.2 Counting Field 8.1.3 Large Deviation Theory and Rate Function 8.1.4 Gärtner-Ellis Theorem 8.2 Lebowitz-Spohn Fluctuation Theorem 8.2.1 Symmetry in Cumulant Generating Function of Entropy Production 8.2.2 Fluctuation-Dissipation Theorem Derived from the Symmetry of Cumulant Generating Function 8.3 Waiting Time Statistics 8.3.1 Martingale Property 8.3.2 First Passage Time Statistics 8.4 Work-Heat Rate Function and Stochastic Efficiency 8.4.1 Stochastic Current and Stochastic Efficiency 8.4.2 Carnot Efficiency as Least Probable Efficiency 9 Information Thermodynamics 9.1 Maxwell's Demon Problem 9.1.1 Maxwell's Original Problem Setting 9.1.2 Breakthrough by Szilard 9.1.3 Arguments by Brillouin and Gabor 9.1.4 Arguments by Landauer and Bennett 9.1.5 Is Maxwell's Demon Problem Solved? 9.2 Second Law of Information Thermodynamics 9.2.1 Mutual Information 9.2.2 Second Law of Information Thermodynamics 9.2.3 Clarification of Maxwell's Demon 9.3 Sagawa-Ueda Relation 9.3.1 Sagawa-Ueda Relation 9.3.2 Additivity 9.4 Problem of Autonomous Maxwell's Demon 9.4.1 Autonomous Maxwell's Demon: 4-State Model 9.4.2 Second Law of Information Thermodynamic in General Information Processes 9.4.3 Limitation of Sagawa-Ueda Relation 9.5 Partial Entropy Production and IFT for General Information Processes 9.5.1 Partial Entropy Production 9.5.2 Fluctuation Theorem for Partial Entropy Production 9.5.3 Fluctuation Theorem for General Information Processes 9.6 Another Extension: Ito-Sagawa Relation 9.6.1 Bayesian Network 9.6.2 Transfer Entropy 9.6.3 Ito-Sagawa Relation and Its Derivation 9.7 Remarks 9.7.1 Partial Entropy Production with Broken Time-Reversal Symmetry 9.7.2 Inequality for Partial Entropy Production 9.7.3 Definition of Heat in Discrete-Time Markov Chains 9.7.4 Information Reservoir 10 Response Relation Around Nonequilibrium Steady State 10.1 Fluctuation-Response Relation at Stalling State 10.1.1 Fluctuation-Response Relation on Current at Stalling State 10.1.2 Fluctuation-Response Relation on Time-Symmetric Current at Stalling State 10.2 Response Theory of Stationary Distribution 10.2.1 Expression of Stationary Distribution by Matrix-Tree Theorem 10.2.2 Response Equality and Inequality for Stationary Distribution 10.3 Remarks 10.3.1 Alternative Proof of Eq. (10.5) Based on the Generating Function 10.3.2 Proof of Eq. (10.46) 11 Some Results on One-Dimensional Overdamped Langevin Systems 11.1 Path Probability of Dynamics 11.1.1 Onsager-Machlup Functional 11.1.2 Fluctuation Theorem and Hatano-Sasa Relation 11.1.3 Harada-Sasa Relation 11.2 Stationary State of One-Dimensional Overdamped Langevin Systems 11.2.1 Expression of Stationary Distribution 11.2.2 Generating Function of Velocity 11.2.3 Diffusion Constant and Mobility Part III Intermission: Interesting Models 12 Externally-Controlled Systems: Flashing Ratchet and Pump 12.1 Ratchet and Asymmetric Pumping 12.1.1 Flashing Ratchet and Curie Principle 12.1.2 Reversible Transport 12.2 Hidden Pumping 13 Direction of Transport 13.1 Brownian Motor and Adiabatic Piston 13.1.1 Brownian Motor 13.1.2 Adiabatic Piston Problem 13.1.3 Heuristic Argument 13.2 Parrondo's Paradox 13.2.1 Problem and Examples 13.2.2 Similarity to Simpson's Paradox 14 Stationary Systems: From Brownian Motor to Autonomous Macroscopic Engines 14.1 Autonomous Ratchet Model 14.1.1 Feynman's Ratchet 14.1.2 Büttiker-Landauer Model 14.1.3 Unattainability of Carnot Efficiency 14.2 Small Autonomous Models Attaining the Carnot Efficiency 14.3 Macroscopic Autonomous Engines 14.3.1 Setup and Its Coarse-Grained Description 14.3.2 Maximum Efficiency 14.3.3 Attainability of Carnot Efficiency 14.4 Necessary Condition to Attain Carnot Efficiency 14.4.1 Questions 14.4.2 General Principle 14.4.3 Nonlinear Tight-Coupling Window Part IV Inequalities 15 Efficiency at Maximum Power 15.1 Endoreversible Processes and Curzon-Ahlborn Efficiency 15.2 Onsager Matrix Approach 15.3 Linear Expansion with Velocity 15.4 Remarks 16 Trade-Off Relation Between Efficiency and Power 16.1 Carnot Efficiency and Finite Power: Prelude 16.1.1 No Restriction from General Frameworks 16.1.2 Model Analyses 16.2 Trade-Off Relation Between Heat Current and Entropy 16.2.1 Main Inequalities 16.2.2 Proofs 16.3 Trade-Off Relation Between Efficiency and Power 16.4 Notion of Finite Speed and Finite Power 16.4.1 Inherent Time Scale 16.4.2 Time-Scale Separation 16.5 Remarks 16.5.1 Inequality for General Conserved Quantities 16.5.2 Evaluation of Θ 17 Thermodynamic Uncertainty Relation 17.1 Thermodynamic Uncertainty Relation 17.1.1 Main Claim 17.1.2 Proof Based on Generalized Cramér-Rao Inequality 17.2 TUR-Type Inequalities 17.2.1 Generalization of Thermodynamic Uncertainty Relation 17.2.2 Kinetic Uncertainty Relation 17.2.3 The Optimal TUR-Type Inequality 17.2.4 Attainability of Equality in TUR-Type Inequalities 17.3 Thermodynamic Uncertainty Relation for Ballistic Transport with Broken Time-Reversal Symmetry 17.4 Remarks 17.4.1 TUR in Langevin Systems 17.4.2 Alternative Derivation of Thermodynamic Uncertainty Relation with Large Deviation Techniques 17.4.3 Weaker Relation Derived from Time-Reversal Symmetry 17.4.4 Statistical Meaning of the Cramér-Rao Inequality and the Fisher Information 18 Speed Limit for State Transformation 18.1 Geometric Viewpoint for Speed Limit Inequalities 18.2 Speed Limit for Overdamped Langevin System 18.2.1 Linear Expansion by Speed 18.2.2 Optimal Bound with Wasserstein Distance 18.3 Speed Limit for General Markov Processes on Discrete States 18.3.1 Speed Limit with Entropy Production 18.3.2 Numerical Demonstration 18.3.3 Optimal Speed Limit with Pseudo Entropy Production 18.4 Remarks 18.4.1 Quantum Speed Limit for Isolated Systems 19 Variational Aspects of Entropy Production 19.1 Variational Expression of Entropy Production Rate and Bounds in Relaxation Processes 19.1.1 Variational Expression of Entropy Production Rate 19.1.2 Bound for Relaxation Processes 19.1.3 Generalization 19.2 Variational Expression of Excess and Housekeeping Entropy Productions 19.2.1 Excess/Housekeeping Decomposition 19.2.2 Bound of Excess Entropy Production in Relaxation Processes 19.3 Entropy Production with Inappropriate Initial Distribution Part V Notes and History 20 Notes and History 20.1 Notes and History of Part I 20.2 Notes and History of Part II 20.3 Notes and History of Part III 20.4 Notes and History of Part IV Appendix A Derivation of Eqs. (14.15) and (14.18) Appendix B Proof of Eq. (16.49) Appendix C Evaluation of Θ in Several Systems C.1 Finiteness of Θ C.2 Θ in Langevin System C.3 Θ in Linear Response Regime References Index