Density relaxation in conserved Manna sandpiles

Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry

We study the Oslo model, a paradigm for absorbing-phase transition, on a one-dimensional ring of L sites with a fixed global density ρ[over ¯]; we consider the system strictly above critical density ρ_{c}. Notably, microscopic dynamics conserve both mass and center of mass (CoM), but lack time-reversal symmetry. We show that, despite having highly constrained dynamics due to CoM conservation, the system exhibits diffusive relaxation away from criticality and superdiffusive relaxation near criticality. Furthermore, the CoM conservation severely restricts particle movement, causing the mobility-a transport coefficient analogous to the conductivity for charged particles-to vanish exactly. Indeed, the steady-state temporal growth of current fluctuation is qualitatively different from that observed in diffusive systems with a single conservation law. Remarkably, far from criticality where the relative density Δ=ρ[over ¯]-ρ_{c}≫ρ_{c}, the second cumulant, or the variance, 〈Q_{i}^{2}(T,Δ)〉_{c}, of current Q_{i} across the ith bond up to time T in the steady-state saturates as 〈Q_{i}^{2}〉_{c}≃Σ_{Q}^{2}(Δ)-constT^{-1/2}; near criticality, it grows subdiffusively as 〈Q_{i}^{2}〉_{c}∼T^{α}, with 0<α<1/2, and eventually saturates to Σ_{Q}^{2}(Δ). Interestingly, the asymptotic current fluctuation Σ_{Q}^{2}(Δ) is a nonmonotonic function of Δ: It diverges as Σ_{Q}^{2}(Δ)∼Δ^{2} for Δ≫ρ_{c} and Σ_{Q}^{2}(Δ)∼Δ^{-δ}, with δ>0, for Δ→0^{+}. Using a mass-conservation principle, we exactly determine the exponents δ=2(1-1/ν_{⊥})/ν_{⊥} and α=δ/zν_{⊥} via the correlation-length and dynamic exponents, ν_{⊥} and z, respectively. Finally, we show that in the steady state the self-diffusion coefficient D_{s}(ρ[over ¯]) of tagged particles is connected to activity through the relation D_{s}(ρ[over ¯])=a(ρ[over ¯])/ρ[over ¯].

Dynamic correlations in the conserved Manna sandpile

We study dynamic correlations for current and mass, as well as the associated power spectra, in the one-dimensional conserved Manna sandpile. We show that, in the thermodynamic limit, the variance of cumulative bond current up to time T grows subdiffusively as T^{1/2-μ} with the exponent μ≥0 depending on the density regimes considered and, likewise, the power spectra of current and mass at low frequency f varies as f^{1/2+μ} and f^{-3/2+μ}, respectively. Our theory predicts that, far from criticality, μ=0 and, near criticality, μ=(β+1)/2ν_{⊥}z>0 with β, ν_{⊥}, and z being the order parameter, correlation length, and dynamic exponents, respectively. The anomalous suppression of fluctuations near criticality signifies a "dynamic hyperuniformity," characterized by a set of fluctuation relations, in which current, mass, and tagged-particle displacement fluctuations are shown to have a precise quantitative relationship with the density-dependent activity (or its derivative). In particular, the relation, D_{s}(ρ[over ¯])=a(ρ[over ¯])/ρ[over ¯], between the self-diffusion coefficient D_{s}(ρ[over ¯]), activity a(ρ[over ¯]) and density ρ[over ¯] explains a previous simulation observation [Eur. Phys. J. B 72, 441 (2009)10.1140/epjb/e2009-00367-0] that, near criticality, the self-diffusion coefficient in the Manna sandpile has the same scaling behavior as the activity.