Arthur Hipke, Sven Lubeck, Haye Hinrichsen
The conserved Manna model with a planar absorbing boundary is studied in various space dimensions. We present a heuristic argument that allows one to compute the surface critical exponent in one dimension analytically. Moreover, we discuss the mean field limit that is expected to be valid in d>4 space dimensions and demonstrate how the corresponding partial differential equations can be solved.
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Using inverse Ising inference we show that the absorbing states of the one-dimensional Manna model can be described by an equilibrium model with an emergent interaction displaying short-ranged repulsion and long-ranged attraction. As the model approaches the critical point the interaction becomes purely repulsive, decaying as $r^{-α}$ and we conjecture the exact value $α=1/2$, suggesting density fluctuations decay as $r^{-3/2}$. We present a simple Gaussian field theory for the long distance behaviour of critical absorbing states and discuss implications for the Manna universality class.
We study relaxation of long-wavelength density perturbations in one dimensional conserved Manna sandpile. Far from criticality where correlation length $ξ$ is finite, relaxation of density profiles having wave numbers $k \rightarrow 0$ is diffusive, with relaxation time $τ_R \sim k^{-2}/D$ with $D$ being the density-dependent bulk-diffusion coefficient. Near criticality with $k ξ\gsim 1$, the bulk diffusivity diverges and the transport becomes anomalous; accordingly, the relaxation time varies as $τ_R \sim k^{-z}$, with the dynamical exponent $z=2-(1-β)/ν_{\perp} < 2$, where $β$ is the critica…
Fixed-energy sandpiles with stochastic update rules are known to exhibit a nonequilibrium phase transition from an active phase into infinitely many absorbing states. Examples include the conserved Manna model, the conserved lattice gas, and the conserved threshold transfer process. It is believed that the transitions in these models belong to an autonomous universality class of nonequilibrium phase transitions, the so-called Manna class. Contrarily, the present numerical study of selected (1+1)-dimensional models in this class suggests that their critical behavior converges to directed percol…
Dynamic correlations in the conserved Manna sandpile
该研究揭示一维守恒曼纳沙堆中电流与质量涨落的亚扩散及功率谱标度行为，提出临界点附近的动态超均匀性，并建立自扩散系数与活性及密度间的精确关系，解释了先前模拟观察。
Sandpile models with conserved number of particles (also called fixed energy sandpiles) may undergo phase transitions between active and absorbing states. We generalize the Manna sandpile model with fixed number of particles, introducing a parameter $-1 \leq λ\leq 1$ related to the toppling of particles from active sites to its first neighbors. In particular, we discuss a model with height restrictions, allowing for at most two particles on a site. Sites with double occupancy are active, and their particles may be transfered to first neighbor sites, if the height restriction do allow the chang…
A recent study of conserved Manna model, with both discrete and continuous variable, indicates that absorbing phase transitions therein belong to the directed percolation (DP) universality class. In this context we revisit critical behaviour in energy exchange models with a threshold. Contrary to the previous claims [PRE 83, 061130 (2011), arXiv:1102.1631], our results indicate that both the maximal and minimal versions of this model belong to the DP class.
Hyperuniformity in the Manna Model, Conserved Directed Percolation and Depinning
本文证明了守恒有向渗流（CDP）类中的密度与脱钉界面位置之间存在精确映射关系，推导出超均匀性指数公式，纠正了早期工作并在各维度获得高精度结果。
We investigate the critical behavior of systems exhibiting a continuous absorbing phase transition in the presence of a conserved field coupled to the order parameter. The results obtained point out the existence of a new universality class of nonequilibrium phase transitions that characterizes a vast set of systems including conserved threshold transfer processes and stochastic sandpile models.
We investigate the critical behavior of a reaction-diffusion system exhibiting a continuous absorbing-state phase transition. The reaction-diffusion system strictly conserves the total density of particles, represented as a non-diffusive conserved field, and allows an infinite number of absorbing configurations. Numerical results show that it belongs to a wide universality class that also includes stochastic sandpile models. We derive microscopically the field theory representing this universality class.
We analyse numerically the critical behavior of a conserved lattice gas which was recently introduced as an example of the new universality class of absorbing phase transitions with a conserved field [Phys. Rev. Lett. 85, 1803 (2000)]. We determine the critical exponent of the order parameter as well as the critical exponent of the order parameter fluctuations in D=2,3,4,5 dimensions. A comparison of our results and those obtained from a mean-field approach and a field theory suggests that the upper critical dimension of the absorbing phase transition is four.
A conserved lattice gas with random neighbor hopping of active particles is introduced which exhibits a continuous phase transition from an active state to an absorbing non-active state. Since the randomness of the particle hopping breaks long range spatial correlations our model mimics the mean-field scaling behavior of the recently introduced new universality class of absorbing phase transitions with a conserved field. The critical exponent of the order parameter is derived within a simple approximation. The results are compared with those of simulations and field theoretical approaches.
We study absorbing state phase transition in one dimension in presence of a conserved continuous local field (CCLF) called energy. A pair of sites on a lattice is said to be active if one or both sites posses more energy than a pre-defined threshold. The active pair of sites are allowed to redistribute their energy following a stochastic rule. We show that, the CCLF model undergo a continuous absorbing state transition when energy per site is decreased below a critical value. The critical exponents are found to be different from those of DP.
Extensive simulations are performed to study the persistence behavior of a conserved lattice gas model exhibiting an absorbing phase transition from an active phase into an inactive phase. Both the global and the local persistence exponents are determined in two and higher dimensions. The local persistence exponent obeys a scaling relation involving the order parameter exponent of the absorbing phase transition. Furthermore we observe that the global persistence exponent exceeds its local counterpart in all dimensions in contrast to the known persistence behavior in reversible phase transition…
We analyze numerically three different models exhibiting an absorbing phase transition. We focus on the finite-size scaling as well as the dynamical scaling behavior. An accurate determination of several critical exponents allows to validate certain hyperscaling relations. Using this hyperscaling relations it is possible to express the avalanche exponents of a self-organized critical system in terms of the ordinary exponents of a continuous absorbing phase transition.
We consider two mean-field like models which belong to the universality class of absorbing phase transitions with a conserved field. In both cases we derive analytically the order parameter as function of the control parameter and of an external field conjugated to the order parameter. This allows us to calculate the universal scaling function of the mean-field behavior. The obtained universal function is in perfect agreement with recently obtained numerical data of the corresponding five and six dimensional models, showing that four is the upper critical dimension of this particular universal…
We present a general field-theoretic strategy to analyze three connected families of continuous phase transitions which occur in nonequilibrium steady-states. We focus on transitions taking place between an active state and one absorbing state, when there exist an infinite number of such absorbing states. In such transitions the order parameter is coupled to an auxiliary field. Three situations arise according to whether the auxiliary field is diffusive and conserved, static and conserved, or finally static and not conserved.
We study stochastic sandpile models with a height restriction in one and two dimensions. A site can topple if it has a height of two, as in Manna's model, but, in contrast to previously studied sandpiles, here the height (or number of particles per site), cannot exceed two. This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded. Two toppling rules are considered: in one, the particles are redistributed independently, while the other involves some cooperativity. We study the fixed-energy system (no input or loss of particles) us…
The recently proposed Langevin equation, aimed to capture the relevant critical features of stochastic sandpiles, and other self-organizing systems is studied numerically. This equation is similar to the Reggeon field theory, describing generic systems with absorbing states, but it is coupled linearly to a second conserved and static (non-diffusive) field. It has been claimed to represent a new universality class, including different discrete models: the Manna as well as other sandpiles, reaction-diffusion systems, etc. In order to integrate the equation, and surpass the difficulties associate…
We analyze numerically the critical behavior of an absorbing phase transition in the conserved transfer threshold process. We determined the steady state scaling behavior of the order parameter as a function of both, the control parameter and an external field, conjugated to the order parameter. The external field is realized as a spontaneous creation of active particles which drives the system away from criticality. The obtained results yields that the conserved transfers threshold process belongs to the universality class of absorbing phase transitions in a conserved field.
The properties of the absorbing states of non-equilibrium models belonging to the conserved directed percolation universality class are studied. We find that at the critical point the absorbing states are hyperuniform, exhibiting anomalously small density fluctuations. The exponent characterizing the fluctuations is measured numerically, a scaling relation to other known exponents is suggested, and a new correlation length relating to this ordering is proposed. These results may have relevance to photonic band-gap materials.
We analyse numerically the critical behavior of an absorbing phase transition in a conserved lattice gas in an external field. The external field is realized as a spontaneous creation of active particles which drives the system away from criticality. Nevertheless, the order parameter obeys certain scaling laws for sufficiently small external fields. These scaling laws are investigated and the corresponding exponents are determined in various dimensions ($D=2,3,4,5$). At the so-called upper critical dimension $D_{\text c}=4$ one has to modify the usual scaling laws by logarithmic corrections.
We study diffusion of hardcore particles on a one dimensional periodic lattice subjected to a constraint that the separation between any two consecutive particles does not increase beyond a fixed value $(n+1);$ initial separation larger than $(n+1)$ can however decrease. These models undergo an absorbing state phase transition when the conserved particle density of the system falls bellow a critical threshold $ρ_c= 1/(n+1).$ We find that $ϕ_k$s, the density of $0$-clusters ($0$ representing vacancies) of size $0\le k<n,$ vanish at the transition point along with activity density $ρ_a$. The ste…
Conserved lattice gas (CLG) models in one dimension exhibit absorbing state phase transition (APT) with simple integer exponents $β=1=ν=η$ whereas the same on a ladder belong to directed percolation (DP)universality. We conjecture that additional stochasticity in particle transfer is a relevant perturbation and its presence on a ladder force the APT to be in DP class. To substantiate this we introduce a class of restricted conserved lattice gas models on a multi-chain system ($M\times L$ square lattice with periodic boundary condition in both directions), where particles which have exactly one…
Towards a Matrix Product Ansatz in Two Dimensions
本文提出了一种二维矩阵积方案，用于解决非平衡随机过程的稳态权重问题，并通过求解受限跳跃的非守恒辅助排斥模型（NAEM）展示了该方法在一维和二维中的应用，揭示了该模型在临界密度ρ_c=1/2处的吸收相变及与硬方格点气体的映射关系。
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