Universal Aging Dynamics and Scaling Laws in Three-Dimensional Driven Granular Gases

Abstract

We establish universal scaling laws and quantify aging in three-dimensional uniformly heated hard sphere granular gases through large-scale event-driven molecular dynamics ($N=500{,}000$). We report three primary quantitative discoveries: (i) The characteristic energy decay time exhibits a universal inverse scaling $τ_0 \propto ε^{-1.03 \pm 0.02}$ with the dissipation parameter $ε= 1 - e^2$. (ii) The steady-state temperature follows a precise power-law $T_{\mathrm{steady}} \propto ε^{-1.51 \pm 0.03}$, reflecting the non-linear balance between thermostat heating and collisional dissipation. (iii) The velocity autocorrelation function $\bar{A}(τ_w, τ)$ demonstrates pronounced aging, with decay rates $λ$ following a power-law slowing down $λ(τ_w) \propto τ_w^{-0.82 \pm 0.05}$. These results establish the first 3D quantitative benchmarks for aging in driven dissipative gases, where near-Gaussian statistics persist despite extreme structural clustering.

Figures (14)

• Figure 1: Time dependence of the granular temperature in $d = 3$, shown on a semilog scale. We plot the normalized granular temperature $T(\tau)/T(0)$ vs $\tau$ for $e = 0.80, 0.85, 0.90$, and $0.95$. The solid lines denote Haff's law.
• Figure 2: Time dependence of the granular temperature in $d = 3$, shown on a log-log scale. It can be clearly seen that the temperature for each value of $e$ settles into a constant steady-state value at large times.
• Figure 5: Scaling of characteristic decay time $\tau_0$ with dissipation strength. (a) $\tau_0$ vs $e$ shows monotonic increase as systems become more elastic. (b) Log-log plot of $\tau_0$ vs $\epsilon$ reveals a power-law relationship $\tau_0 \propto \epsilon^{-1.03}$ (excellent agreement with theory).
• Figure 6: Scaling of steady-state temperature $T_{\mathrm{steady}}$ with dissipation parameter. (a) $T_{\mathrm{steady}}$ vs $e$ shows that more inelastic systems settle at lower temperatures. (b) Log-log plot reveals a precise power-law $T_{\mathrm{steady}} \propto \epsilon^{-1.51 \pm 0.03}$, reflecting the balance between stochastic heating and collisional dissipation.
• Figure 7: Quantitative aging results. Extracted decay rates $\lambda$ versus waiting time $\tau_w$ for different $e$, demonstrating power-law slowing $\lambda(\tau_w) \propto \tau_w^{-0.82 \pm 0.05}$. This relationship holds across all levels of inelasticity.