# 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.

## References

- Schöll, E.; Klapp, S.H.L.; Hövel, P. (Eds.) Control of Self-Organizing Nonlinear Systems; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Bechhoefer, J. Feedback for physicists: A tutorial essay on control. Rev. Mod. Phys.
**2005**, 77, 783. [Google Scholar] [CrossRef][Green Version] - Schöll, E.; Schuster, H.G. (Eds.) Handbook of Chaos Control; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Német, N.; Parkins, S. Enhanced optical squeezing from a degenerate parametric amplifier via time-delayed coherent feedback. Phys. Rev. A
**2016**, 94, 023809. [Google Scholar] [CrossRef][Green Version] - Mijalkov, M.; McDaniel, A.; Wehr, J.; Volpe, G. Engineering sensorial delay to control phototaxis and emergent collective behaviors. Phys. Rev. X
**2016**, 6, 011008. [Google Scholar] [CrossRef][Green Version] - Bruot, N.; Damet, L.; Kotar, J.; Cicuta, P.; Lagomarsino, M.C. Noise and synchronization of a single active colloid. Phys. Rev. Lett.
**2011**, 107, 094101. [Google Scholar] [CrossRef][Green Version] - Masoller, C. Noise-Induced Resonance in Delayed Feedback Systems. Phys. Rev. Lett.
**2002**, 88, 034102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lai, X.; Wolkenhauer, O.; Vera, J. Understanding microRNA-mediated gene regulatory networks through mathematical modelling. Nucleic Acids Res.
**2016**, 44, 6019–6035. [Google Scholar] [CrossRef][Green Version] - Josić, K.; López, J.M.; Ott, W.; Shiau, L.; Bennett, M.R. Stochastic delay accelerates signaling in gene networks. PLoS Comput. Biol.
**2011**, 7, e1002264. [Google Scholar] [CrossRef][Green Version] - Gupta, C.; López, J.M.; Ott, W.; Josić, K.; Bennett, M.R. Transcriptional Delay Stabilizes Bistable Gene Networks. Phys. Rev. Lett.
**2013**, 111, 058104. [Google Scholar] [CrossRef][Green Version] - Rateitschak, K.; Wolkenhauer, O. Intracellular delay limits cyclic changes in gene expression. Math. Biosci.
**2007**, 205, 163–179. [Google Scholar] [CrossRef][Green Version] - Bratsun, D.; Volfson, D.; Tsimring, L.S.; Hasty, J. Delay-induced stochastic oscillations in gene regulation. Proc. Natl. Acad. Sci. USA
**2005**, 102, 14593–14598. [Google Scholar] [CrossRef][Green Version] - Parmar, K.; Blyuss, K.B.; Kyrychko, Y.N.; Hogan, S.J. Time-delayed models of gene regulatory networks. Comput. Math. Meth. M.
**2015**, 2015, 347273. [Google Scholar] [CrossRef][Green Version] - Friedman, N.; Cai, L.; Xie, X.S. Linking stochastic dynamics to population distribution: An analytical framework of gene expression. Phys. Rev. Lett.
**2006**, 97, 168302. [Google Scholar] [CrossRef][Green Version] - Schiering, C.; Wincent, E.; Metidji, A.; Iseppon, A.; Li, Y.; Potocnik, A.J.; Omenetti, S.; Henderson, C.J.; Wolf, C.R.; Nebert, D.W.; et al. Feedback control of AHR signalling regulates intestinal immunity. Nature
**2017**, 542, 242. [Google Scholar] [CrossRef][Green Version] - Pardee, A.B. Regulation of Cell Metabolism; J. and A. Churchill LTD: London, UK, 1959; p. 295. [Google Scholar]
- Ito, S.; Sagawa, T. Maxwell’s demon in biochemical signal transduction with feedback loop. Nat. Commun.
**2015**, 6, 7498. [Google Scholar] [CrossRef][Green Version] - Yi, T.M.; Huang, Y.; Simon, M.I.; Doyle, J. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc. Natl. Acad. Sci. USA
**2000**, 97, 4649–4653. [Google Scholar] [CrossRef] [PubMed][Green Version] - Micali, G.; Endres, R.G. Bacterial chemotaxis: Information processing, thermodynamics, and behavior. Curr. Opin. Microbiol.
**2016**, 30, 8–15. [Google Scholar] [CrossRef][Green Version] - Paden, B.; Čáp, M.; Yong, S.Z.; Yershov, D.; Frazzoli, E. A survey of motion planning and control techniques for self-driving urban vehicles. IEEE Trans. Intell. Veh.
**2016**, 1, 33–55. [Google Scholar] [CrossRef][Green Version] - Strasberg, P.; Schaller, G.; Brandes, T.; Esposito, M. Thermodynamics of quantum-jump-conditioned feedback control. Phys. Rev. E
**2013**, 88, 062107. [Google Scholar] [CrossRef][Green Version] - Carmele, A.; Kabuss, J.; Schulze, F.; Reitzenstein, S.; Knorr, A. Single photon delayed feedback: A way to stabilize intrinsic quantum cavity electrodynamics. Phys. Rev. Lett.
**2013**, 110, 013601. [Google Scholar] [CrossRef] [PubMed] - Jun, Y.; Bechhoefer, J. Virtual potentials for feedback traps. Phys. Rev. E
**2012**, 86, 061106. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jun, Y.; Gavrilov, M.; Bechhoefer, J. High-precision test of Landauer’s principle in a feedback trap. Phys. Rev. Lett.
**2014**, 113, 190601. [Google Scholar] [CrossRef][Green Version] - Khadka, U.; Holubec, V.; Yang, H.; Cichos, F. Active Particles Bound by Information Flows. Nat. Commun.
**2018**, 9, 3864. [Google Scholar] [CrossRef] - Lavergne, F.A.; Wendehenne, H.; Bäuerle, T.; Bechinger, C. Group formation and cohesion of active particles with visual perception—Dependent motility. Science
**2019**, 364, 70–74. [Google Scholar] [CrossRef][Green Version] - Debiossac, M.; Grass, D.; Alonso, J.J.; Lutz, E.; Kiesel, N. Thermodynamics of continuous non-Markovian feedback control. Nat. Commun.
**2020**, 11, 1–6. [Google Scholar] [CrossRef] [PubMed] - Wallin, A.E.; Ojala, H.; Hæggström, E.; Tuma, R. Stiffer optical tweezers through real-time feedback control. Appl. Phys. Lett.
**2008**, 92, 224104. [Google Scholar] [CrossRef][Green Version] - Balijepalli, A.; Gorman, J.J.; Gupta, S.K.; LeBrun, T.W. Significantly improved trapping lifetime of nanoparticles in an optical trap using feedback control. Nano Lett.
**2012**, 12, 2347–2351. [Google Scholar] [CrossRef] [PubMed] - Gupta, V.; Kadambari, K. Neuronal model with distributed delay: Analysis and simulation study for gamma distribution memory kernel. Biol. Cybern.
**2011**, 104, 369–383. [Google Scholar] - Longtin, A. Complex Time-Delay Systems: Theory and Applications; Atay, F.M., Ed.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 177–195. [Google Scholar]
- Küchler, U.; Mensch, B. Langevins stochastic differential equation extended by a time-delayed term. Stoch. Stoch. Rep.
**1992**, 40, 23–42. [Google Scholar] [CrossRef] - Driver, R.D. Ordinary and Delay Differential Equations; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012; Volume 20. [Google Scholar]
- Forgoston, E.; Schwartz, I.B. Delay-induced instabilities in self-propelling swarms. Phys. Rev. E
**2008**, 77, 035203(R). [Google Scholar] [CrossRef][Green Version] - Sen, S.; Ghosh, P.; Riaz, S.S.; Ray, D.S. Time-delay-induced instabilities in reaction-diffusion systems. Phys. Rev. E
**2009**, 80, 046212. [Google Scholar] [CrossRef] [PubMed] - Grawitter, J.; van Buel, R.; Schaaf, C.; Stark, H. Dissipative systems with nonlocal delayed feedback control. New J. Phys.
**2018**, 20, 113010. [Google Scholar] [CrossRef] - Mackey, M.C.; Tyran-Kamińska, M. How can we describe density evolution under delayed dynamics? Chaos Interdiscip. J. Nonlinear Sci.
**2021**, 31, 043114. [Google Scholar] [CrossRef] - Mackey, M.C.; Nechaeva, I.G. Noise and stability in differential delay equations. J. Dyn. Diff. Equat.
**1994**, 6, 395–426. [Google Scholar] [CrossRef] - Safonov, L.; Tomer, E.; Strygin, V.; Ashkenazy, Y.; Havlin, S. Delay-induced chaos with multifractal attractor in a traffic flow model. EPL (Europhys. Lett.)
**2002**, 57, 151. [Google Scholar] [CrossRef][Green Version] - Khrustova, N.; Veser, G.; Mikhailov, A.; Imbihl, R. Delay-induced chaos in catalytic surface reactions: No reduction on Pt (100). Phys. Rev. Lett.
**1995**, 75, 3564. [Google Scholar] [CrossRef] [PubMed] - Loos, S.A.M.; Klapp, S.H.L. Fokker-Planck equations for time-delayed systems via Markovian Embedding. J. Stat. Phys.
**2019**, 177, 95–118. [Google Scholar] [CrossRef][Green Version] - Munakata, T.; Rosinberg, M.L. Entropy production and fluctuation theorems for Langevin processes under continuous non-Markovian feedback control. Phys. Rev. Lett.
**2014**, 112, 180601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Loos, S.A.M.; Klapp, S.H.L. Heat flow due to time-delayed feedback. Sci. Rep.
**2019**, 9, 2491. [Google Scholar] [CrossRef] - Pyragas, K. Continuous control of chaos by self-controlling feedback. Phys. Lett. A
**1992**, 170, 421–428. [Google Scholar] [CrossRef] - Pyragas, K. Control of chaos via extended delay feedback. Phys. Lett. A
**1995**, 206, 323–330. [Google Scholar] [CrossRef] - Fiedler, B.; Nieto, A.L.; Rand, R.H.; Sah, S.M.; Schneider, I.; de Wolff, B. Coexistence of infinitely many large, stable, rapidly oscillating periodic solutions in time-delayed Duffing oscillators. J. Differ. Equ.
**2020**, 268, 5969–5995. [Google Scholar] [CrossRef][Green Version] - Tsimring, L.S.; Pikovsky, A. Noise-Induced Dynamics in Bistable Systems with Delay. Phys. Rev. Lett.
**2001**, 87, 250602. [Google Scholar] [CrossRef][Green Version] - Khadem, S.M.J.; Klapp, S.H. Delayed feedback control of active particles: A controlled journey towards the destination. Phys. Chem. Chem. Phys.
**2019**, 21, 13776. [Google Scholar] [CrossRef][Green Version] - Schneider, I.; Bosewitz, M. Eliminating restrictions of time-delayed feedback control using equivariance. Disc. Cont. Dyn. Syst. A
**2016**, 36, 451–467. [Google Scholar] [CrossRef][Green Version] - Rosinberg, M.L.; Munakata, T.; Tarjus, G. Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities. Phys. Rev. E
**2015**, 91, 042114. [Google Scholar] [CrossRef][Green Version] - Kim, K.H.; Qian, H. Entropy production of Brownian macromolecules with inertia. Phys. Rev. Lett.
**2004**, 93, 120602. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kim, K.H.; Qian, H. Fluctuation theorems for a molecular refrigerator. Phys. Rev. E
**2007**, 75, 022102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ito, S.; Sano, M. Effects of error on fluctuations under feedback control. Phys. Rev. E
**2011**, 84, 021123. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kundu, A. Nonequilibrium fluctuation theorem for system under discrete and continuous feedback control. Phys. Rev. E
**2012**, 86, 021107. [Google Scholar] [CrossRef] [PubMed][Green Version] - Munakata, T.; Rosinberg, M. Entropy production and fluctuation theorems under feedback control: The molecular refrigerator model revisited. J. Stat. Mech. Theory Exp.
**2012**, 2012, P05010. [Google Scholar] [CrossRef][Green Version] - Munakata, T.; Rosinberg, M. Feedback cooling, measurement errors, and entropy production. J. Stat. Mech. Theory Exp.
**2013**, 2013, P06014. [Google Scholar] [CrossRef][Green Version] - Loos, S.A.M.; Klapp, S.H.L. Irreversibility, heat and information flows induced by non-reciprocal interactions. NJP
**2020**, 123051. [Google Scholar] [CrossRef] - Melnyk, S.; Usatenko, O.; Yampol’skii, V. Memory-dependent noise-induced resonance and diffusion in non-Markovian systems. Phys. Rev.
**2021**, 103, 032139. [Google Scholar] - Kotar, J.; Leoni, M.; Bassetti, B.; Lagomarsino, M.C.; Cicuta, P. Hydrodynamic synchronization of colloidal oscillators. Proc. Natl. Acad. Sci. USA
**2010**, 107, 7669–7673. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jahnel, M.; Behrndt, M.; Jannasch, A.; Schäffer, E.; Grill, S.W. Measuring the complete force field of an optical trap. Opt. Lett.
**2011**, 36, 1260–1262. [Google Scholar] [CrossRef] - MacDonald, N.; Lags, T. Lecture Notes in Biomathematics; Springer: Berlin/Heidelberg, Germany, 1978; Volume 17, pp. 1059–1062. [Google Scholar]
- Smith, H.L. An Introduction to Delay Differential Equations with Applications to the Life Sciences; Chapter Distributed Delay Equations and the Linear Chain Trick; Springer: New York, NY, USA, 2011; Volume 57. [Google Scholar]
- Zwanzig, R. Nonlinear generalized Langevin equations. J. Stat. Phys.
**1973**, 9, 215–220. [Google Scholar] [CrossRef] - MacDonald, N.; MacDonald, N. Biological Delay Systems: Linear Stability Theory; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Boese, F. Stability conditions for the general linear difference-differential equation with constant coefficients and one constant delay. J. Math. Anal. Appl.
**1989**, 140, 136–176. [Google Scholar] [CrossRef][Green Version] - Crauste, F. Complex Time-Delay Systems: Theory and Applications, Atay, F.M., Ed.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 263–296. [Google Scholar]
- Giuggioli, L.; McKetterick, T.J.; Kenkre, V.M.; Chase, M. Fokker–Planck description for a linear delayed Langevin equation with additive Gaussian noise. J. Phys. A
**2016**, 49, 384002. [Google Scholar] [CrossRef][Green Version] - Yi, S.; Nelson, P.W.; Ulsoy, A.G. Time-Delay Systems: Analysis and Control Using the Lambert W Function; World Scientific: Singapore, 2010. [Google Scholar]
- Schöll, E.; Hövel, P.; Flunkert, V.; Dahlem, M.A. Time-delayed feedback control: From simple models to lasers and neural systems. In Complex Time-Delay Systems; Springer: Berlin/Heidelberg, Germany, 2009; pp. 85–150. [Google Scholar]
- Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys.
**2012**, 75, 126001. [Google Scholar] [CrossRef][Green Version] - Sekimoto, K. Stochastic Energetics; Springer: Berlin/Heidelberg, Germany, 2010; Volume 799. [Google Scholar]
- Crisanti, A.; Puglisi, A.; Villamaina, D. Nonequilibrium and information: The role of cross correlations. Phys. Rev. E
**2012**, 85, 061127. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rosinberg, M.L.; Tarjus, G.; Munakata, T. Stochastic thermodynamics of Langevin systems under time-delayed feedback control. II. Nonequilibrium steady-state fluctuations. Phys. Rev. E
**2017**, 95, 022123. [Google Scholar] [CrossRef][Green Version] - Lebowitz, J.L.; Spohn, H. A Gallavotti—Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys.
**1999**, 95, 333–365. [Google Scholar] [CrossRef][Green Version] - Strasberg, P.; Schaller, G.; Brandes, T.; Esposito, M. Thermodynamics of a physical model implementing a Maxwell demon. Phys. Rev. Lett.
**2013**, 110, 040601. [Google Scholar] [CrossRef][Green Version] - Vishen, A.S. Heat dissipation rate in a nonequilibrium viscoelastic medium. J. Stat. Mech. Theory Exp.
**2020**, 2020, 063201. [Google Scholar] [CrossRef] - Caprini, L.; Marconi, U.M.B.; Puglisi, A.; Vulpiani, A. The entropy production of Ornstein—Uhlenbeck active particles: A path integral method for correlations. J. Stat. Mech. Theor. Exp.
**2019**, 2019, 053203. [Google Scholar] [CrossRef][Green Version] - Sarracino, A.; Villamaina, D.; Gradenigo, G.; Puglisi, A. Irreversible dynamics of a massive intruder in dense granular fluids. EPL (Europhys. Lett.)
**2010**, 92, 34001. [Google Scholar] [CrossRef][Green Version] - Doerries, T.; Loos, S.A.M.; Klapp, S.H.L. Correlation functions of non-Markovian systems out of equilibrium: Analytical expressions beyond single-exponential memory. J. Stat. Mech. Theory Exp.
**2021**, 3, 033202. [Google Scholar] [CrossRef] - Dabelow, L.; Bo, S.; Eichhorn, R. Irreversibility in active matter systems: Fluctuation theorem and mutual information. Phys. Rev. X
**2019**, 9, 021009. [Google Scholar] [CrossRef][Green Version] - Ohkuma, T.; Ohta, T. Fluctuation theorems for non-linear generalized Langevin systems. J. Stat. Mech. Theory Exp.
**2007**, 2007, P10010. [Google Scholar] [CrossRef] - Speck, T.; Seifert, U. The Jarzynski relation, fluctuation theorems, and stochastic thermodynamics for non-Markovian processes. J. Stat. Mech. Theory Exp.
**2007**, 2007, L09002. [Google Scholar] [CrossRef] - Horowitz, J.M.; Gingrich, T.R. Thermodynamic uncertainty relations constrain non-equilibrium fluctuations. Nat. Phys.
**2020**, 16, 15–20. [Google Scholar] [CrossRef] - Barato, A.C.; Seifert, U. Thermodynamic uncertainty relation for biomolecular processes. Phys. Rev. Lett.
**2015**, 114, 158101. [Google Scholar] [CrossRef][Green Version] - Di Terlizzi, I.; Baiesi, M. A thermodynamic uncertainty relation for a system with memory. J. Phys. Math. Theor.
**2020**, 53, 474002. [Google Scholar] [CrossRef] - Rosinberg, M.; Tarjus, G. Comment on Thermodynamic uncertainty relation for time-delayed Langevin systems. arXiv
**2018**, arXiv:1810.12467. [Google Scholar] - Horowitz, J.M.; Sandberg, H. Second-law-like inequalities with information and their interpretations. New J. Phys.
**2014**, 16, 125007. [Google Scholar] [CrossRef] - Hänggi, P.; Thomas, H. Stochastic processes: Time evolution, symmetries and linear response. Phys. Rep.
**1982**, 88, 207–319. [Google Scholar] [CrossRef] - Geiss, D.; Kroy, K.; Holubec, V. Brownian molecules formed by delayed harmonic interactions. New J. Phys.
**2019**, 21, 093014. [Google Scholar] [CrossRef] - Puglisi, A.; Villamaina, D. Irreversible effects of memory. EPL
**2009**, 88, 30004. [Google Scholar] [CrossRef][Green Version]

**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