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2021 | Buch

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Stochastic Systems with Time Delay

Probabilistic and Thermodynamic Descriptions of non-Markovian Processes far From Equilibrium

verfasst von: Dr. Sarah A.M. Loos

Verlag: Springer International Publishing

Buchreihe : Springer Theses

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Many natural and artificial systems are governed by dynamical equations which involve both, noise and time delay [6‐8]. Noise due to imperfections or random environmental influences is essentially omnipresent in real-world systems and experimental setups. From a mathematical point of view, it often results from the presence of hidden degrees of freedom that have disappeared in the course of a coarse-graining procedure. Stochastic models that account for the noise are well known to be a powerful tool to effectively describe complex systems by using just relatively few, mesoscopic (stochastic) degrees of freedom. Such mesoscale models, for example the Brownian motion studied in this thesis, also play a central role in statistical physics.

Sarah A. M. Loos

Theoretical Background and State of the Art

Frontmatter

Chapter 2. The Langevin Equation

Abstract

While you are reading this thesis, at every instant of time countless particles of the surrounding air hit your skin due to their irregular thermal motion. There are of the order of \({\sim } 10^{23}\) molecules in every liter of air [1, 2].

Sarah A. M. Loos

Chapter 3. Fokker-Planck Equations

Abstract

In the preceding chapter, we have introduced the Langevin equation, which describes the random processes studied in this thesis on a stochastic level. For Markovian systems, it is well known that Fokker-Planck equations (FPE) provide a complementary way of description, on the probabilistic level. These are deterministic equations, whose solutions are probability density functions. In the following, we will briefly introduce this concept, first focusing on the Markovian case.

Sarah A. M. Loos

Chapter 4. Stochastic Thermodynamics

Abstract

Finally, we introduce another theoretical framework employed in this thesis: stochastic thermodynamics. The latter is a rather young theory, which aims at a generalization of thermodynamic notions towards small-scale systems dominated by thermal fluctuations and towards nonequilibrium systems [2]. On this scale, individual fluctuating trajectories can transiently violate the second law of thermodynamics, but in the limit of large systems (or long times), one shall always recover the well-established laws of macroscopic thermodynamics.

Sarah A. M. Loos

Probabilistic Descriptions for Systems with Time Delay

Frontmatter

Chapter 5. Infinite Fokker-Planck Hierarchy

Abstract

In this chapter, we will examine the infinite Fokker-Planck hierarchy. We will explicitly consider the higher members, which play a crucial role to understand the non-Markovian dynamics. Later in Chap. 6, we will present a new derivation of the FP description using a Markovian embedding technique [1‐6]. To better understand the technical and conceptual difference of our approach, we will here also review some earlier work, in particular, Refs. [7‐9].

Sarah A. M. Loos

Chapter 6. Markovian Embedding—A New Derivation of the Fokker-Planck Hierarchy

Abstract

In this Chapter, we derive the FP hierarchy by using a Markovian embedding technique, which is inspired by the linear chain trick [3‐5]. Markovian embeddings are already a well-established tool to treat stochastic systems with memory, for example in the context of generalized Langevin equations [6‐11]. The following considerations are based on Ref. [12], where we have introduced this derivation. Here, we add additional noise terms to the auxiliary variables (different from [12]), which might provide a conceptual advantage for future analytic treatments.

Sarah A. M. Loos

Chapter 7. Force-Linearization Closure

Abstract

In this chapter, we turn to the problem of finding (approximate) probabilistic solutions of systems with delay. In particular, we discuss an approximation scheme for the steady-state one-time PDF, which we have introduced in Ref. [1], called Force-linearization closure (FLC).

Sarah A. M. Loos

Chapter 8. Approximation for the Two-time Probability density

Abstract

So far, we have discussed different approximations of the one-time probability density function, stemming from approaches based on the first member of the Fokker-Planck hierarchy. In particular, we have discussed the perturbation theory and the force-linearization closure (Chap. 7). While these approaches render quite accurate descriptions of the intrawell dynamics, they do not yield any information about the two-time PDF.

Sarah A. M. Loos

Thermodynamic Notions for Systems with Time Delay

Frontmatter

Chapter 9. The Heat Flow Induced by a Discrete Delay

Abstract

Finding thermodynamic notions for non-Markovian systems is a major problem. As a first step in this direction, we consider the heat rate \(\dot{Q}=\langle \delta q /\mathrm {d}t\rangle _\mathrm {ss}\) of nonlinear systems with discrete delay. This quantity can be calculated via well-established concepts from stochastic thermodynamics, in particular, from the framework of stochastic energetics [1]. The heat flow and related medium entropy production are key thermodynamic quantities, which already provide important physical insight. They further constitute a nontrivial part of the total entropy production. Based on this consideration, we can already address several fundamental questions.

Sarah A. M. Loos

Chapter 10. Entropy, Information and Energy Flows

Abstract

In the last chapter, we have studied one component of the steady-state entropy balance, the medium entropy production and associated heat flow. While this investigation has already provided interesting insights, open questions remain. First, we observed that the mean heat rate can vanish at some specific, rare parameter values, despite the presence of time-delayed feedback. On the basis of the energetic consideration alone, we cannot clarify whether these points correspond to equilibrium.

Sarah A. M. Loos

Concluding Remarks

Frontmatter

Chapter 11. Summary

Abstract

This thesis is concerned with the non-Markovian dynamics of stochastic, fluctuation-dominated systems with time delay. The prime example studied here is a colloidal (Brownian) particle subject to a position-dependent, time-delayed feedback force and nonlinear static forces stemming from a potential. We describe its noisy motion by a Langevin equation, which is, due to the delay, non-local in time and infinite-dimensional.

Sarah A. M. Loos

Chapter 12. Outlook—Open Questions and Further Perspectives

Abstract

We close with some comments putting the findings presented in this thesis into a broader perspective and giving an outlook on future research. We begin with formulating a few obvious follow-up questions and take a wider view towards the end.

Sarah A. M. Loos

Backmatter

Titel: Stochastic Systems with Time Delay
verfasst von: Dr. Sarah A.M. Loos
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-80771-9
Print ISBN: 978-3-030-80770-2
DOI: https://doi.org/10.1007/978-3-030-80771-9

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