Stochastic Systems with Time Delay: Probabilistic and Thermodynamic Descriptions of non-Markovian Processes far From Equilibrium (Springer Theses)

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The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supervisor’s Foreword Abstract Acknowledgements Publications by Sarah A. M. Loos Contents Abbreviations and Symbols Abbreviations Symbols 1 Introduction 1.1 Outline of the Thesis References Part I Theoretical Background and State of the Art 2 The Langevin Equation 2.1 The Stochastic Way of Describing Things 2.1.1 Brownian Motion 2.1.2 Colloidal Suspensions 2.1.3 Side Note: A More General View 2.2 The Markovian Langevin Equation 2.2.1 Gaussian White Noise 2.2.2 Ensemble Averages and Probability Density 2.2.3 Solutions of the Langevin Equation and the Overdamped Limit 2.2.4 Ornstein–Uhlenbeck Process 2.2.5 White Noise—Wiener Process—Stochastic Calculus 2.2.6 Path Integral Representation 2.3 Generalised Langevin Equations—How Stochastic Motion ... 2.3.1 Infinite Harmonic Oscillators Bath—An Example of a Mori–Zwanzig Projection 2.3.2 Coarse-Graining—Forgetting Some Details 2.3.3 Side Node: Taking this Simplified Model Serious 2.3.4 The Markov Assumption 2.3.5 Real-World Complications 2.3.6 Time-Reversal Symmetry and Causality 2.4 Introduction to the Langevin Equation with Time Delay 2.4.1 Optical Traps—An Experimental Tool to Control 2.4.2 Time-Delayed Feedback 2.4.3 The Langevin Equation with Time Delay 2.4.4 Side Note: Delay Differential Equations 2.4.5 Linear Systems with Time Delay 2.5 Nonlinear Example Systems with Time Delay 2.5.1 Bistable System: The Doublewell Potential 2.5.2 Periodic System: The Washboard Potential 2.5.3 Scaling 2.6 Timescales 2.6.1 Kramers Escape Times 2.7 Delay-Induced Oscillations and Coherence Resonance 2.7.1 Delay-Induced Oscillations 2.7.2 Coherence Resonance 2.7.3 Bifurcation Theoretical Perspective on Delay-Induced Oscillations References 3 Fokker-Planck Equations 3.1 Markovian Case 3.1.1 Natural Boundary Conditions 3.1.2 Joint Probability Densities 3.1.3 The Probability Current and Steady States 3.2 Introduction to Fokker-Planck Descriptions of Systems with Time Delay 3.2.1 Earlier Approximation Schemes 3.2.2 Probability Current and Apparent Equilibrium of Time-Delayed Systems 3.2.3 Side Note: Delay in Ensemble-Averaged Quantities References 4 Stochastic Thermodynamics 4.1 Side Note: Some Historical Notes and Where Is Stochastic ... 4.2 Stochastic Energetics 4.2.1 Steady States 4.3 Fluctuating Entropy 4.3.1 Thermal Equilibrium & Nonequilibrium Steady States 4.4 Fluctuation Theorems 4.4.1 Route to First Principles—Axiom of Causality 4.5 Information 4.5.1 Mutual Information and Its Generalization 4.5.2 Information Flow 4.6 Previous Results, Expectations and Apparent Problems for Systems with Time Delay 4.6.1 Energetics in the Presence of Delay 4.6.2 Entropic Description 4.6.3 The Acausality Issue 4.6.4 Short Comment on Effective Thermodynamics 4.7 Side Note: Active Particles & Non-reciprocal Interactions 4.7.1 Active Ornstein-Uhlenbeck Particles 4.7.2 Connection Between Active Matter and Time-Delayed Systems 4.7.3 Non-reciprocal Interactions References Part II Probabilistic Descriptions for Systems with Time Delay 5 Infinite Fokker-Planck Hierarchy 5.1 Derivation of Fokker-Planck Hierarchy from Novikov's Theorem 5.1.1 Alternative Approach with Two Time Arguments 5.2 Exact Probabilistic Solutions for Linear Systems with Time Delay 5.2.1 Derivation of the Second Member of the Fokker-Planck Hierarchy 5.2.2 Steady-State Solutions 5.2.3 The Notion of Effective Temperature 5.2.4 Markovian Versus Non-Markovian Two-Time Probability Density References 6 Markovian Embedding—A New Derivation of the Fokker-Planck Hierarchy 6.1 Markovian Embedding—A Different View on Memory 6.1.1 Projection, Memory Kernel & Colored Noise 6.1.2 Limit ntoinfty 6.1.3 Interpretation of the Xj Variables 6.1.4 Initial Condition 6.1.5 (n+1)-dimensional Markovian Fokker-Planck Equation 6.2 Derivation of First Member of Fokker-Planck Hierarchy 6.3 Derivation of Higher Members via Markovian Embedding 6.3.1 Comparison to Equation from Novikov's Theorem 6.4 Side Note: Discrete Versus Distributed Delay 6.4.1 Distributed Delay 6.4.2 Probability Densities in the Presence of Discrete and Distributed Delay References 7 Force-Linearization Closure 7.1 Details of the Approximation 7.1.1 Linearization of the Deterministic Forces 7.1.2 Analytical Probabilistic Solution for Linearized Forces 7.1.3 Vanishing Steady-State Probability Current 7.1.4 Specification to Linear Delay Force 7.2 Comparison to Earlier Approaches 7.2.1 Small Delay Expansion 7.2.2 Perturbation Theory 7.2.3 Effective Temperatures 7.3 Application to the Periodic Potential 7.3.1 Discussion of Results 7.4 Application to the Bistable Potential 7.4.1 Discussion of Results 7.5 Estimation of Escape Times 7.5.1 Interwell Dynamics 7.5.2 The Kramers-FLC Estimate 7.5.3 Comparison with Numerical Results 7.5.4 Delay-Induced Oscillations 7.5.5 Side Note: Normal Diffusion Despite Non-Markovianity References 8 Approximation for the Two-time Probability density 8.1 Application to the Bistable Delayed System 8.1.1 Comparison with Approaches for One-time Probability Density 8.2 Concluding Remarks References Part III Thermodynamic Notions for Systems with Time Delay 9 The Heat Flow Induced by a Discrete Delay 9.1 Main Idea 9.1.1 Polynomial Energy Landscapes 9.2 Mean Heat Rate & Medium Entropy Production 9.2.1 Linear Systems 9.2.2 Markovian Limits in Nonlinear Systems 9.2.3 Limit of Vanishing Delay Time 9.2.4 Influence of Inertia Term 9.2.5 Discussion of the Behavior for Small Delay Times 9.3 Application to the Bistable Potential 9.3.1 Low Thermal Energy—Intrawell Dynamics 9.3.2 High Thermal Energy—Interwell Dynamics 9.4 Preliminary Numerical Results for Fluctuation of Heat, Work and Internal Energy 9.5 Concluding Remarks References 10 Entropy, Information and Energy Flows 10.1 Emergence of Non-monotonic Memory 10.1.1 Interpretation of Xj>0 in the Case of a Feedback Controller 10.2 The Role of Non-reciprocal Coupling—Connection to Active Matter 10.2.1 A Generic Model with Non-reciprocal Coupling 10.3 Non-reciprocal Coupling and Non-equilibrium 10.3.1 Fluctuation-Dissipation Relation 10.3.2 Total Entropy Production 10.3.3 Analytical Solutions 10.4 Non-reciprocal Coupling and Activity 10.4.1 Mapping Non-reciprocity of the Coupling to a Temperature Gradient Between the Coupled Entities 10.4.2 Reversed Heat Flow 10.5 Non-reciprocal Coupling and Information 10.5.1 Information Flow and Generalized Second Law 10.5.2 Information-Theoretic Perspective on Feedback Control 10.6 Total Entropy Production and Heat Flow in the Presence of Non-monotonic Memory 10.6.1 Limit of Discrete Delay 10.6.2 Impact of Measurement Errors 10.7 Irreversibility and Coarse-Graining References Part IV Concluding Remarks 11 Summary References 12 Outlook—Open Questions and Further Perspectives References Appendix Appendix A.1 Numerical Methods A.2 Derivation of Novikov's Theorem A.3 Green's Function Method A.4 Connection to Fokker–Planck Hierarchy from Novikov's Theorem A.5 Fluctuation-Dissipation Relation for Unidirectional Ring of Arbitrary Length n Appendix About the Author The nonequilibrium behavior of nanoscopic and biological systems, which are typically strongly fluctuating, is a major focus of current research. Lately, much progress has been made in understanding such systems from a thermodynamic perspective. However, new theoretical challenges emerge when the fluctuating system is additionally subject to time delay, e.g. due to the presence of feedback loops. This thesis advances this young and vibrant research field in several directions. The first main contribution concerns the probabilistic description of time-delayed systems; e.g. by introducing a versatile approximation scheme for nonlinear delay systems. Second, it reveals that delay can induce intriguing thermodynamic properties such as anomalous (reversed) heat flow. More generally, the thesis shows how to treat the thermodynamics of non-Markovian systems by introducing auxiliary variables. It turns out that delayed feedback is inextricably linked to nonreciprocal coupling, information flow, and to net energy input on the fluctuating level.