Absorbing phase transitions in deterministic fixed-energy sandpile models Su-Chan Park ( GLYPH<uni1107.1> GLYPH<uni1161.6> GLYPH<uni11A8.9> GLYPH<uni1109.28> GLYPH<uni116E.4> GLYPH<uni110E.2> GLYPH<uni1161.7> GLYPH<uni11AB.6> ) Department of Physics, The Catholic University of Korea, Bucheon 14662, Republic of Korea (Dated: November 9, 2018) We investigate the origin of the difference, which was noticed by Fey et al. [Phys. Rev. Lett. 104 , 145703 (2010)], between the steady state density of an Abelian sandpile model (ASM) and the transition point of its corresponding deterministic fixed-energy sandpile model (DFES). Being deterministic, the configuration space of a DFES can be divided into two disjoint classes such that every configuration in one class should evolve into one of absorbing states, whereas no configurations in the other class can reach an absorbing state. Since the two classes are separated in terms of toppling dynamics, the system can be made to exhibit an absorbing phase transition (APT) at various points that depend on the initial probability distribution of the configurations. Furthermore, we show that in general the transition point also depends on whether an infinite-size limit is taken before or after the infinite-time limit. To demonstrate, we numerically study the two-dimensional DFES with Bak-Tang-Wiesenfeld toppling rule (BTW-FES). We confirm that there are indeed many thresholds. Nonetheless, the critical phenomena at various transition points are found to be universal. We furthermore discuss a microscopic absorbing phase transition, or a so-called spreading dynamics, of the BTW-FES, to find that the phase transition in this setting is related to the dynamical isotropic percolation process rather than self-organized criticality. In particular, we argue that choosing recurrent configurations of the corresponding ASM as an initial configuration does not allow for a nontrivial APT in the DFES. PACS numbers: 05.70.Fh,05.70.Ln, 64.70.qj,64.60.ah A fixed-energy ensemble [1–4] of sandpile models [5– 8] was introduced to interpret self-organized criticality (SOC) [5, 9, 10] in the context of the absorbing phase transition (APT) (for a review, see, e.g., Refs. [11, 12]). The idea of introducing the fixed-energy sandpile model (FES) stems from the similarity between a stable config- uration of the sandpile model and an absorbing state. By an absorbing state we mean a configuration that does not change with time by the given dynamic rules. In a sense, it was believed that the FES is to the sandpile model as the canonical ensemble is to the grand canonical en- semble in equilibrium statistical mechanics. By the same token, a phase transition in a FES was expected to occur at the steady state density of the corresponding sandpile model. However, it was noticed [13] that for a certain deter- ministic fixed-energy sandpile model (DFES), the criti- cal energy density is different from the steady state en- ergy density of the corresponding Abelian sandpile model (ASM) [6]. This observation clearly raises many ques- tions. Why do some DFESs have the same critical density but some do not? What is the origin of the difference? Is the critical behavior of the FES at different transition points universal? Can we expect similar differences in the stochastic fixed-energy sandpile model (SFES)? The aim of this paper is to answer these questions. This paper is organized as follows. In Sec. II, we de- fine the DFES in order to introduce the notations we are using in this paper. In Sec. III, we discuss the APT in the DFES. We show that in general the transition points are dependent on the initial condition. Furthermore, we show that transition points can vary, according to which In this section, we define the DFES to introduce the notations in this paper. We consider a system with V sites. A configuration of the system is specified by a non-negative integer z$_{i}$ at each site i (1 ≤ i ≤ V ), which is called the energy at site i . The time evolution of the system is governed by a V × V integer matrix F , to be called a toppling matrix (TM), with the following prop- erties: limit is taken first, the infinite-size limit or the infinite- time limit. As a nontrivial example, Sec. IV numerically investigates APTs in the Bak-Tang-Wiesenfeld (BTW)- type FES (BTW-FES) in two dimensions. Indeed, the critical point of the BTW-FES is found to vary with the initial condition. We also show that the critical behavior is universal. In Sec. V, we discuss the implication of our observation to the SOC models and the SFES. In Sec. VI, we summarize the work. II. MODEL I. INTRODUCTION arXiv:1001.3359v2 [cond-mat.stat-mech] 7 Mar 2018 1. F$_{ij}$ ≤ 0, if i = j , 2. F$_{ii}$ = − ∑ j = $_{i}$F$_{ij}$ . A TM is said to be irreducible if, for any ordered pair of two different site indices ( i, j ) (1 ≤ i, j ≤ V ), there is an ordered set of different indices ( k$_{1}$, k$_{2}$, . . . , k$_{n}$$_{−}$$_{1}$ ) such that ∏ n − 1 l =0 F$_{k}$$_{l}$$_{k}$$_{l}$$_{+1}$ = 0, where k$_{0}$ ≡ i and k$_{n}$ ≡ j . Obvi- ously, F$_{ii}$ > 0 for all i if F is irreducible. In this paper, a TM is always assumed irreducible. For a given TM, we will say that site i is active if z$_{i}$ ≥ F$_{ii}$ . An active site i will topple and distribute its energy to other sites in such a manner that z$_{j}$ ↦→ z$_{j}$ − F$_{ij}$ for all j .