Temporal Coherence Decay in Granular Flow Simulations: Modeling Non-Equilibrium Transport Across Dimensional Crossovers

Introduction

The macroscopic behavior of granular flows, ranging from sandpile avalanches to fluidized bed dynamics, is notoriously difficult to model accurately due to inherent microscopic heterogeneity and strong non-linearity. Continuum models often fail when the system operates far from local thermodynamic equilibrium, particularly during dimensional crossovers where the effective dimensionality of particle interactions shifts abruptly [1]. This research targets the transient dynamics associated with this crossover—specifically, how the initial temporal correlation structure of the particle movement degrades as the system attempts to establish a statistically steady state in a constrained geometry.

We hypothesize that the decay of temporal coherence, $\mathcal{C}(t)$, is not monotonic but exhibits a predictable scaling law dictated by the geometry of the confinement and the driving energy input. This coherence measures the persistence of correlated momentum transfer events across the particle ensemble.

Theoretical Framework

We extend the standard DEM framework by augmenting the state vector for each particle $i$ with a localized kinetic history vector $\mathbf{H}_i(t)$, tracking the time-averaged velocity correlation over a short preceding interval $\Delta t$. The instantaneous coherence $\mathcal{C}(t)$ is then defined as the ensemble average of the product of normalized velocity fluctuations:

$$\mathcal{C}(t) = \frac{1}{N} \sum_{i=1}^{N} \left\langle \mathbf{v}_i(t) \cdot \mathbf{v}_i(t-\Delta t) \right\rangle_{t-\Delta t}^{t} \cdot e^{-\lambda (t-\Delta t)}$$

Where $\lambda$ is the fitted decay constant, and the exponential term accounts for the rapid decorrelation into the gaseous phase beyond the measurement window. In the quasi-2D regime, where collisions are primarily face-to-face and velocity projections onto the plane dominate, the decay constant $\lambda_{2D}$ is found to be strongly dependent on the confining aspect ratio $\alpha$:

$$\lambda_{2D} \propto \frac{1}{\alpha} \left( 1 - \frac{\rho_c}{\rho} \right)^{-1} \quad \text{for } \rho \approx \rho_c$$

This functional dependence captures the bottleneck effect as the system approaches jamming density $\rho_c$. [2]

Computational Methodology

Simulations were conducted using a high-performance computing cluster implementing the Hertz-Mindlin contact model with restitution coefficients $e=0.85$. A cylindrical chute geometry was employed, allowing systematic control over the imposed aspect ratio $\alpha = L_z/L_{xy}$ (axial to planar length ratio). Particles consisted of identical spheres of diameter $d=1.0$ unit length.

Data collection involved monitoring the time series of the macroscopic strain rate tensor $\dot{\mathbf{\epsilon}}(t)$ sampled at $10^4$ time steps during the transient phase. The critical measurement was the autocorrelation function of the volume fraction fluctuation $\delta\phi(t)$, which serves as a direct proxy for the loss of momentum memory in the system.

Table 1. Transient Dynamics Across Dimensional Crossover

[3]

Results and Analysis

The simulation data robustly confirms the scaling predicted by the extended model. As $\alpha$ was increased from 2.0 to 10.0 (approaching quasi-2D), the measured decay time constant $\tau_c$ exhibited a clear divergence, consistent with the theoretical prediction $\tau_c \propto \alpha$. Specifically, the transient phase where the system exhibits memory of initial conditions spans $50-150$ characteristic collision times in the 3D regime but extends beyond 500 collision times when confined to the quasi-2D boundary conditions.

Figure 1 (omitted in this text block) displayed the evolution of $\mathcal{C}(t)$ versus time. In the 2D limit, the coherence decayed via a stretched exponential form, $\mathcal{C}(t) \propto \exp[-(t/\tau_0)^\beta]$, with $\beta \ll 1$, indicative of trapped microstates contributing to long-term memory loss [4].

Discussion

The observed transition from $\beta \approx 1$ (3D) to $\beta \approx 0.3$ (2D) signals a fundamental shift in the dominant transport mechanism—from viscous-like momentum diffusion to inertial jamming dynamics. The presence of low-$\beta$ decay tails in the 2D simulations strongly suggests the formation of persistent, quasi-stable clusters whose movement dictates the macroscopic transport timescale. These clusters behave as emergent topological defects in the flow field, slowing the overall relaxation process.

This work provides a quantitative bridge between microscopic collision mechanics and macroscopic rheological behavior during dynamic dimensionality reduction. Understanding this timescale $\tau_c$ is crucial for designing reliable filtration membranes or granular computation devices where predictable transient response is paramount [5].

Conclusion

We successfully modeled the temporal coherence decay in granular flows crossing dimensional boundaries. The coherence time $\tau_c$ scales predictably with the confinement aspect ratio $\alpha$, accurately predicting the onset of long-lived memory effects characteristic of quasi-2D flow states. Future work will explore the influence of particle size polydispersity on the critical exponent $\beta$.