# Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Model

- (i)
- By changing n, we can model various common types of delay with distinct characteristics, see Figure 2. For $n=1$, the kernel is a simple exponential decay. For $n>1$, $K(\Delta t)$ is peaked around $\Delta t=\tau $, with peaks becoming sharper upon increasing n. In the limit $n\to \infty $, it reaches a delta-distribution, ${lim}_{n\to \infty}K(\Delta t)=k\delta (\Delta t-\tau )$ [41]. Thus the ansatz Equation (3) includes the case of a discrete (single) delay.
- (ii)
- The weight of K, which defines the feedback gain k, is identical for all n: ${\int}_{0}^{\infty}K(\Delta t)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\Delta t=k$.
- (iii)
- The mean delay time τ is identical for all n:$$\begin{array}{c}\hfill \frac{{\int}_{0}^{\infty}K(\Delta t)\Delta t\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\Delta t}{{\int}_{0}^{\infty}K(\Delta t)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}\Delta t}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\tau .\end{array}$$
- (iv)

#### 2.1. Markovian Representation

#### 2.2. Colored Noise

#### 2.3. Limit of Infinitely Large System

**Proof.**

#### 2.4. Alternative Choice of ${\mathcal{T}}^{\prime}$

## 3. Stability for Different n

## 4. Delay-Induced Heat Flow

## 5. The Total Entropy Production

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LE | Langevin equation |

FDR | Fluctuation-dissipation relation |

ODE | Ordinary differential equation |

DDE | Delay differential equation |

EP | Entropy production |

## Appendix A. Memory Kernel and Noise Correlations

#### Appendix A.1. Physical Motivation for the Dynamics of X j>0

## Appendix B. Analytical Solutions for Correlations

## Appendix C. Path Probabilities with Onsager–Machlup Action

## Appendix D. Limit of the Heat Flow

## Appendix E. Entropy Distributions for Different Types of Delay Distributions

**Figure A1.**Distributions of the total EP (Equation (25)) for $n=1,2$ in a stable, linear system, obtained by Brownian dynamics simulations. The linear decays in this logarithmic plots indicate exponential tails. As expected, both distributions fulfill the integral fluctuation theorem $\langle {e}^{-\Delta {s}_{\mathrm{tot}}}\rangle =1$. To calculate the distributions, $>5\xb7{10}^{6}$ steady-state trajectories of length ${10}^{4}$ were generated. All parameters and ${k}_{\mathrm{B}}$ are set to unity.

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**Figure 1.**Sketch of a particle (red disk) under the influence of a force that depends on the past trajectory (indicated by lighter red disks), for different values of a of the external potential $V=a/2\phantom{\rule{0.166667em}{0ex}}{X}_{0}^{2}$: (

**a**) the particle is in a harmonic trap ($a>0$), (

**b**) the particle is “on a parabolic mountain” ($a<0$). Here, the arrows indicate the force pertaining to $k>0$, i.e., the delay force is positive when the weighted integral of the past trajectory is negative, see Equation (1).

**Figure 2.**(

**a**) Delay distribution given in Equation (3), and (

**b**) correlations of colored noise $\nu $ given in Equation (6) for different n, pertaining to arbitrary values of $\tau $, k, ${T}_{0}$, $T\prime $ and choices of V. The delay distributions K are plotted in units of $\left|k\right|$, while ${C}_{\nu}$ is plotted in units of ${k}^{2}/\left(\tau {k}_{\mathrm{B}}{\mathcal{T}}^{\prime}\right)$.

**Figure 3.**Relation between the temporal evolutions of ${X}_{0}$ and the ${X}_{j>0}$ representing the n “memory cells of the controller” (as further explained in Appendix A.1).

**Figure 4.**Stable regions of the stochastic process with ${a}_{}>0$ (“particle in harmonic trap”, see Figure 1a), in a plane spanned by the feedback gain k given in units of $\left|a\right|$ and delay time $\tau $, given in units of $1/\left|a\right|$. White areas denote unstable behavior. Four panels of (

**a**): Stability boundaries for the cases $n=1$ (yellow), $n=2$ (blue), $n=3$ (green), and $n\to \infty $ (red). (

**b**): All stability areas from panels (

**a**) plotted on top of each other. The size of the stability areas decreases with n. For example, in the red area, all systems up to $n\to \infty $ are stable. The regions exceed the shown parameter range and continue towards smaller k and larger $\tau $.

**Figure 6.**Heat flow (Equation (21)) vs. n in the system with $a=1$ (corresponding to Figure 1a and Figure 4, for different temperature ratios ${\mathcal{T}}^{\prime}/{\mathcal{T}}_{0}$ (different colors), and different feedback gains: (

**a**) $k=-1/2$, (

**b**) $k=1/2$, and (

**c**) $k=0.999$. The parameter values in (

**c**) lie for all n just next to the right stability boundary, which is given by ${k}^{*}=a=1$ (and arbitrary $\tau $). Further, the system with $n=1$ in (

**c**) fulfills approximately FDR, see Equation (7). The dashed lines show the solution for the discrete delay case Equation (A16), which the system expectantly approaches in the limit $n\to \infty $ (as follows from the analytical reasoning in Section 2.3). ${k}_{\mathrm{B}}$ is set to unity.

**Figure 7.**(

**a**,

**b**) Heat flow of ${X}_{0}$ (Equation (23)), total entropy production rate (Equation (29)) and heat flow of ${X}_{1}$ (Equation (28)) for the system with $n=1$ and different values of a: (

**a**) $a=1$ (“particle in a trap”) and (

**b**) $a=-0.99$ (“particle on parabolic mountain”). (

**c**,

**d**) Heat flow of ${X}_{0}$ (Equations (23) and (A16)) for the cases $n=1$ (i.e., exponential memory kernel) and $n\to \infty $ (delta-distributed memory kernel) in the system with (

**a**) $a=1$ and (

**b**) $a=-0.99$. In (

**a**,

**c**), the parameter settings correspond to the parameters in Figure 6. Specifically, the red crosses mark the values of k considered in Figure 6 panel (

**a**–

**c**), respectively. In all panels, the parameter values approach from the left a stability boundary given by ${k}^{*}=a$. Further, the parameters in (

**b**) lie very close to the stability boundary ${\tau}^{*}=-1/{a}_{}$. ${k}_{\mathrm{B}}$ is set to unity. For a discussion of the dependency of the heat flow on $\tau $, which is a bit more subtle, we would like to refer the reader to [43].

**Figure 8.**Total EP (Equation (27)) vs. n (giving the number of ${X}_{j>0}$) for a system with $a=1$(corresponding to Figure 1a, Figure 4 and Figure 6), at different values of feedback gain: (

**a**) $k=-1/2$ and (

**b**) $k=0.999$. The analytical results are complemented by quadratic fits ${\dot{S}}_{\mathrm{tot}}\sim {n}^{2}$ (solid lines). The parameter values in (

**b**) lie just next to the right stability boundary ${k}^{*}=a=1$ (for all n). All other parameters, and ${k}_{\mathrm{B}}$ are set to unity.

**Figure 9.**Entropy production rate in a nonlinear system with potential $V={V}_{0}({x}_{0}^{4}-2{x}_{0}^{2})$ for different n; with $\tau =4$, $k=0.1$, ${V}_{0}=1$, ${\mathcal{T}}^{\prime}=1$, ${\mathcal{T}}_{0}=0.04$, ${k}_{\mathrm{B}}=1$. The quadratic fit is a guide to the eye. The data points stem from Brownian dynamics simulations of steady-state trajectories of Equation (5) with temporal discretization $\Delta t={10}^{-4}$, and averages over a minimum of $N=1000$ runs with different random number seeds.

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Loos, S.A.M.; Hermann, S.; Klapp, S.H.L.
Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay. *Entropy* **2021**, *23*, 696.
https://doi.org/10.3390/e23060696

**AMA Style**

Loos SAM, Hermann S, Klapp SHL.
Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay. *Entropy*. 2021; 23(6):696.
https://doi.org/10.3390/e23060696

**Chicago/Turabian Style**

Loos, Sarah A. M., Simon Hermann, and Sabine H. L. Klapp.
2021. "Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay" *Entropy* 23, no. 6: 696.
https://doi.org/10.3390/e23060696