Linear control theory for jammed particle systems

Abstract

Amorphous particulate matter constitutes a wide range of natural and synthetic materials. Despite this ubiquity, the way in which these systems' disordered microstructure couples to their often subtle and complex dynamical behavior is not yet fully understood, with profound consequences for phenomena ranging from landscape evolution to cellular unjamming during tumor metastasis. With this paper, we introduce tools from linear control theory that quantify system response to external input, and demonstrate their utility in elucidating the dynamics of jammed amorphous materials under stress. Our results indicate that average controllability, the response of a system to perturbation, strongly correlates with particle rearrangement in systems subject to quasistatic shear, implying that average controllability is an accurate predictor of rearrangement dynamics in certain contexts. Moreover, we show that the time scale over which average controllability is calculated can be tuned to optimize its predictive capacity for particle rearrangement. Values of the optimal time scale provide physical insight into the system; namely, that multiple rearranging particles participate on average in vibrational eigenmodes of lower and lower energy as the system is sheared until the rearrangement event. Broadly, our study demonstrates that linear control theory is a promising mathematical framework for predicting and designing mechanical response in disordered media.

Figures (4)

• Figure 1: Rearrangements can be quantified in a jammed disordered solid undergoing quasistatic shear.(a) Shear stress $\sigma_{xy}$, potential energy $PE$, average non-affine motion $\langle D^2_{min} \rangle$, and the eigenfrequencies $\omega$ of the dynamical matrix vary as a function of strain $\gamma (t)$. Rearrangement events are characterized by drops in $\sigma_{xy}$, $PE$, and the critical eigenfrequency of the dynamical matrix, and a spike in $\langle D^2_{min} \rangle$. By choosing the rearrangement corresponding to the largest drop in $PE$ in each simulation, indicated with a transparent black line in panel (a), we select for a set of rearrangement events for which the critical mode (panels b and c, left) can range from high correlation with the $D^2_{min}$ map of rearrangements (panel b, right) to low correlation with the $D^2_{min}$ map of rearrangements (panel c, right). (b) Snapshot of a system prior to its rearrangement event. This rearrangement event has the highest Spearman correlation between the critical mode (left) and the $D^2_{min}$ map (right). (c) Snapshot of another system prior to its rearrangement event. This rearrangement event has the lowest Spearman correlation between the critical mode (left) and the $D^2_{min}$ map (right). Each Spearman correlation is calculated only for the particles with $D^2_{min}$ values in the top 95th percentile of all values in the snapshot, in order to limit the calculation to the particle subset that moves most during rearrangement and thereby avoid noise in the correlation. In each panel, particles in the leftmost image are colored linearly according to the magnitude of their polarization vector in the critical mode, with blue corresponding to the highest magnitude. Particles in the rightmost panel are colored linearly according to $D^2_{min}$, with blue corresponding to the highest value.
• Figure 2: High-controllability particles participate in varying distributions of eigenmodes in a way that depends on the associated time horizon.(a) The prefactor $\alpha$ (defined in the main text) as a function of $\omega$, for varying values of the time horizon $T$. Values of $T$ range from 1 (the most purple curve) to 50 (the most green curve). As $T$ increases, the prefactor weights low-frequency eigenmodes more and more heavily with respect to high-frequency eigenmodes. Inset: The density of states of the system immediately prior to rearrangement. Values of $\omega$ were aggregated over all frames immediately prior to rearrangement for all state points. (b) The squared magnitude of the polarization vector ($\vert e_\omega \vert^2$) of high- (solid line) and low- (dotted line) controllability particles in all eigenmodes indexed by $\omega$, for $T=50$ (left) and $T=10$ (right). High-controllability particles at the longer time horizon $T=50$ participate more heavily in the lowest-frequency eigenmodes. Each curve is an average taken over the 10 particles with the highest or lowest average controllability for all systems at the frame immediately prior to rearrangement. Values of $\omega$ were grouped into 100 bins, and averages were taken over all polarization vectors associated with each bin. Error bars are the standard deviation of the mean. (c) Snapshots of an example system immediately prior to its rearrangement event, colored by the critical mode (left), average controllability at $T=50$ (middle), and average controllability at $T=10$ (right). At the longer time horizon $T=50$, the map of average controllability resembles the critical mode map. Particles in the left image are colored linearly according to the magnitude of their polarization vector in the critical mode, with blue corresponding to the highest magnitude. Particles in the middle and right images are colored logarithmically according to their average controllability, with blue corresponding to the maximum value in each snapshot.
• Figure 3: Average controllability at long $T$ is as successful a predictor of rearrangement as vibrality, and average controllability at short $T$ hints at the eigenmode participation of rearrangers.(a,b) Normalized rank of average controllability at $T = 50$ ($\langle r_c \rangle$, black) and vibrality ($\langle r_v \rangle$, green) as a function of $\Delta \gamma$ for the particles that participate most in the critical mode at rearrangement (panel a) and have the highest value of $D^2_{min}$ during rearrangement (panel b). $\langle r_c \rangle$ for $T=50$ is as high as $\langle r_v \rangle$ in all cases. (c,d) Normalized rank of average controllability at $T = 10$ ($\langle r_c \rangle$, black) and vibrality ($\langle r_v \rangle$, green) as a function of $\Delta \gamma$ for the particles that participate most in the critical mode at rearrangement (panel c) and have the highest value of $D^2_{min}$ during rearrangement (panel d). $\langle r_c \rangle$ for $T=10$ is lower than $\langle r_v \rangle$ for a range of $\Delta \gamma$, and approximately equal to or greater than $\langle r_v \rangle$ for $\Delta \gamma \gtrsim 0.0015$, indicating that the rearranging particles participate in higher energy modes further from the rearrangement event on average. For all curves, averages are taken over all simulations, and error bars indicate standard error of the mean.
• Figure 4: Choosing $T$ to maximize the average controllability of the rearranging particle at each system snapshot provides insight into the eigenmode participation of rearrangers.(a,b) Normalized rank of average controllability at the time horizon value $T_{optimal}$ that maximizes it at each snapshot of each system ($\langle r_c \rangle$, black) and vibrality ($\langle r_v \rangle$, green) of the rearranging particle as a function of $\Delta \gamma$. Plots are shown for two definitions of rearranging particles: those that participate most in the critical mode at rearrangement (panel a) and those that have the highest value of $D^2_{min}$ during rearrangement (panel b). $\langle r_c \rangle$ is significantly higher than $\langle r_v \rangle$ in all cases. (c,d) The value $T_{optimal}$ used to calculate each normalized rank as a function of $\Delta \gamma$ for the particles that participate most in the critical mode at rearrangement (panel c) and have the highest value of $D^2_{min}$ during rearrangement (panel d). $\langle T_{optimal} \rangle$ decreases in both cases as a function of $\Delta \gamma$, indicating that the rearranging particles participate in higher energy modes further from the rearrangement event on average. Additionally, $\langle T_{optimal} \rangle$ for the rearrangers determined according to $D^2_{min}$ is lower in general than $\langle T_{optimal} \rangle$ for the rearrangers determined according to the critical mode, indicating that the rearrangers determined according to $D^2_{min}$ participate in higher energy modes in general. For all curves, averages are taken over all simulations, and error bars are standard error of the mean.