An Introduction to Stochastic Thermodynamics: From Basic to Advanced

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