Dynamical and structural properties of an absorbing phase transition: a case study from granular systems

TL;DR

The study investigates absorbing phase transitions in vibrofluidized granular systems using two complementary models: a realistic quasi-2D setup and a simplified effective 2D model. It identifies that synchronization between vertical driving and grain motion crucially dictates whether the APT is continuous (CDP-like) or discontinuous (nucleation-driven), and develops a kinetic theory plus fluctuating hydrodynamics framework to capture the observed dynamical and structural properties. Near the continuous transition, the system exhibits hyperuniform density fluctuations and critical scaling compatible with CDP universality; the discontinuous transition emerges from a synchronization-enabled energy-transfer mechanism, with distinct nucleation dynamics. The fluctuating-hydrodynamics approach predicts, and simulations confirm, hyperuniformity in the active state and provides quantitative predictions for correlation functions, highlighting the role of non-equilibrium noise and synchronization in shaping non-equilibrium liquid structure.

Abstract

We investigate the dynamical and structural properties of absorbing phase transitions (APTs) within granular systems. Specifically, we examine a model for vibrofluidized systems of spherical grains, which undergo a transition from a state of purely vertical motion to one characterized by horizontal diffusion as the density increases. Numerical simulations reveal that, depending on the specific system parameters, both continuous and discontinuous transitions can occur, each associated with markedly distinct structural properties at the transition point. We explain this using a theoretical analysis based on kinetic theory applied to an effective 2D model, which elucidates the role of a synchronization effect in determining the nature of the transition. A fluctuating hydrodynamic theory, which quantitatively describes the structural and dynamical properties of the active state such as hyperuniformity is derived from the microscopic dynamics, together with an equilibrium-like assumption concerning the noises on the hydrodynamic fields. This work expands on previous studies by providing a comprehensive examination of the APT characteristics and proposing new theoretical models to interpret the observed behaviors.

Figures (14)

• Figure 1: Numerical quasi-2D geometry used in realistic simulations. A vertical displacement $z_p(t)$ is imposed to the box in order to provide external energy to the system. Because of tangential frictional forces, the grains lose horizontal energy during collisions with the top and bottom walls. This mechanism introduces an effective dissipation rate $\bar{\gamma}$ for the horizontal dynamics. Energy transfer between $z$ and $xy$ directions occurs during grain-grain non-planar collisions. This introduces an effective energy gain at collision for the $xy$-motion.
• Figure 2: Mean diffusivity (rhombi) and horizontal kinetic energy (circles) as a function of the packing fraction for $h=1.51\sigma$ and $A=0.085\sigma$. Simulations are performed with $N=10^{3}$ grains.
• Figure 3: Comparison of the realistic quasi-2D model (a) and b)) with the effective 2D model (c) and d)). a) Evolution of the $xy$ temperature as a function of the density for various amplitude and height in the realistic quasi-2D model. The values of these parameters for each curve are indicated on the inset with corresponding markers. The inset is the synchronization map of the absorbing state with varying $h$ and $A$. When the absorbing state is synchronized, the transition is discontinuous, while the opposite is observed when the absorbing state is chaotic. b) Critical packing fraction as a function of $1/\tau_s$ in the realistic quasi-2D model. The synchronization time controls the packing fraction of the transition. c) Temperature as a function of the packing fraction in the effective 2D model. Without synchronization ($1/\tau_s=0$), the transition is continuous, while at finite $\tau_s$, the transition is discontinuous (see however Sec. \ref{['sec: synchroaa']} for a discussion of the existence of a tricritical point at finite $\tau_s$). d) Critical packing fraction as a function of the synchronization time for various damping in the effective 2D model. Consistently with the realistic model, the synchronization time increases the critical packing fraction since it facilitates the transition.
• Figure 4: a, b, c) Power law behavior of observables for the realistic quasi-2D model with parameters $A = 0.085\sigma$, $h = 1.5077\sigma$ and $N = 30000$ leading to $\phi_c \simeq 0.186$. d, e, f) Power law behavior of observables for the effective 2D model with parameters $\Delta/(\sigma \gamma) = 0.15$, $\alpha = 0.95$, $N = 10^6$ leading to $\phi_c\simeq 0.1509$. The dashed lines represent the power law expected from a $2D$ system belonging to the conserved directed percolation universality class tjhung2016criticalityodor2004universality. a, d) Order parameter as a function of $\phi - \phi_c$. Expected exponent $\beta \simeq 0.64$ with $T^{ss}\sim (\phi-\phi_c)^{\beta}$. b, e) Average over 100 runs of the evolution of the order parameter at $\phi_c$. Expected exponent $a \simeq 0.42$ with $T(t)\sim t^{-a}$. c, f) Time to relax to the steady state as a function of $|\phi - \phi_c|$. Expected exponent $\gamma \simeq 1.3$ with $\tau_r\sim |\phi-\phi_c|^{-\gamma}$.
• Figure 5: Comparison of the order parameter evolution as a function of packing fraction in the realistic quasi-2D a) and effective 2D model b) for different system sizes. The main figures represent the evolution of the steady state temperature in the discontinuous case (we recall that the temperature is the $xy$-kinetic energy for the realistic quasi-2D model and the usual kinetic energy for the effective 2D model) for different system sizes and insets show the evolution of the order parameter of the last (the most dilute) active steady state ($\Delta T_{ss}$) as a function of the system size a) realistic quasi-2D model: $A = 0.0626\sigma$ and $h = 1.95\sigma$. b) Effective 2D model: $\Delta/\gamma\sigma = 1.5$, $\alpha = 0.95$, and $1/\gamma\tau_s=16.67$.