Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative Measure of Memory Loss in Complex Spatio-Temporal Systems

Miroslav Kramar, Lenka Kovalcinova, Konstantin Mischaikow, Lou Kondic

Abstract

To make progress in understanding the issue of memory loss and history dependence in evolving complex systems, we consider the mixing rate that specifies how fast the future states become independent of the initial condition. We propose a simple measure for assessing the mixing rate that can be directly applied to experimental data observed in any metric space $X$. For a compact phase space $X \subset R^M$, we prove the following statement. If the underlying dynamical system has a unique physical measure and its dynamics is strongly mixing with respect to this measure, then our method provides an upper bound of the mixing rate. We employ our method to analyze memory loss for the system of slowly sheared granular particles with a small inertial number $I$. The shear is induced by the moving walls as well as by the linear motion of the support surface that ensures approximately linear shear throughout the sample. We show that even if $I$ is kept fixed, the rate of memory loss (considered at the time scale given by the inverse shear rate) depends erratically on the shear rate. Our study suggests a presence of bifurcations at which the rate of memory loss increases with the shear rate while it decreases away from these points. We also find that the memory loss is not a smooth process. Its rate is closely related to frequency of the sudden transitions of the force network. The loss of memory, quantified by observing evolution of force networks, is found to be correlated with the loss of correlation of shear stress measured on the system scale. Thus, we have established a direct link between the evolution of force networks and macroscopic properties of the considered system.

Quantitative Measure of Memory Loss in Complex Spatio-Temporal Systems

Abstract

To make progress in understanding the issue of memory loss and history dependence in evolving complex systems, we consider the mixing rate that specifies how fast the future states become independent of the initial condition. We propose a simple measure for assessing the mixing rate that can be directly applied to experimental data observed in any metric space . For a compact phase space , we prove the following statement. If the underlying dynamical system has a unique physical measure and its dynamics is strongly mixing with respect to this measure, then our method provides an upper bound of the mixing rate. We employ our method to analyze memory loss for the system of slowly sheared granular particles with a small inertial number . The shear is induced by the moving walls as well as by the linear motion of the support surface that ensures approximately linear shear throughout the sample. We show that even if is kept fixed, the rate of memory loss (considered at the time scale given by the inverse shear rate) depends erratically on the shear rate. Our study suggests a presence of bifurcations at which the rate of memory loss increases with the shear rate while it decreases away from these points. We also find that the memory loss is not a smooth process. Its rate is closely related to frequency of the sudden transitions of the force network. The loss of memory, quantified by observing evolution of force networks, is found to be correlated with the loss of correlation of shear stress measured on the system scale. Thus, we have established a direct link between the evolution of force networks and macroscopic properties of the considered system.

Paper Structure

This paper contains 12 sections, 3 theorems, 30 equations, 9 figures.

Table of Contents

  1. Introduction.
  2. Mixing rate.
  3. Continuous dynamics.
  4. Linearly sheared systems.
  5. Memory loss of linearly sheared systems.
  6. Conclusion.
  7. Appendix
  8. Particle interactions and simulation protocol
  9. Forces and Effective Friction
  10. Strongly mixing systems
  11. Tent map
  12. Logistic map

Key Result

Theorem \oldthetheorem

Let $X\subset {\mathbb{R}}^M$ be compact and suppose that $f\colon X\to X$ has a unique invariant measure, $\mu$, whose Radon-Nikodym derivative is continuous with respect to the Lebesgue measure on ${\mathbb{R}}^M$. Let $\left\{{\varepsilon_m}\right\}_{m=1}^{\infty}$ be a sequence of positive numbe then there exits a constant $C > 0$ such that

Figures (9)

  • Figure 1: (a) ${\hbox{CDF}}$s $F^N_m$ of $\left\{{d(x_n, x_{n+m})}\right\}_{n=0}^{N-m}$ for the orbit of the ten map starting at $x=0.1$ and $N = 5 \times 10^6$. (b) Rapid decay of $|| F^N_m - F ||_\infty$ shows that $F^N_m$ gets close to $F$ as $m$ increases.
  • Figure 2: $F^N_{2^i}[u]$ for the system (a) $S_1$ and (b) $S_5$. The value of $i$ is indicated by the color bar.
  • Figure 3: (a) Estimated limiting distributions $F[u]$, given by $F^N_{2^{15}}[u]$, for different systems. (b) Value of $|| F^N_{\Delta t}[u] - F[u]||_\infty$ as a function of $\Delta t$, where $\Delta t = 1$ corresponds to $2\tau_c$.
  • Figure 4: (a) Value of $|| F^N_{\Delta t_u} - F||_\infty$ as a function of $\Delta t_u$ for different systems. Time $t_u$ is scaled by the wall speed. (b) Value of $|| F^N_{\Delta t^*_u} - F||_\infty$ as a function of $\Delta t^*_u$ for different systems. Time $t^*_u$ is scaled by the number of transitions.
  • Figure 5: Values of (a) $D_1(t_1)$ and (b) $D_5(t_5)$. The detected transitions are marked by red boxes.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem \oldthetheorem
  • Lemma 1
  • proof
  • Theorem \oldthetheorem
  • proof