finite configuration space will eventually fall into an ab- sorbing state (for a proof, see, e.g., Ref. [14]). Thus, loosely speaking, a sharp phase transition can be de- fined only if there are configurations with zero probability current going into an absorbing state. For instance, in the contact process [15], which is a prototypical model of an APT, such configurations can emerge only under the infinite-size limit. Accordingly, the infinite-size limit should be taken before the infinite-time limit to study a nontrivial APT of the contact process. Actually, this order of the two limits is generally taken for a phase tran- sition of a dynamic system. 2 As we will see soon, however, the DFES is somewhat exceptional in that a finite system can have configura- tions with zero probability current going into any of ab- sorbing states. Hence, a nontrivial transition can exist in a finite system. Even in the case that a phase transition is well defined only in the infinite-size limit (see below), it can exist regardless of whether the infinite-size limit is taken before or after the infinite-time limit. However, the definition of an order parameter that is used to locate a transition point should depend on which limit is taken first. In the following, we define three different order parameters and discuss the general features of a phase transition associated with each one. Due to the deterministic nature of the PU, the fate of a finite system (to survive or not to survive) is already determined from the outset. The RSU cannot change the fate because of the Abelian property of toppling [6]. On this account, we can define, regardless of the update rules, the survival function S of configurations as where Pr$_{a}$( t | C ) is the probability that the system at time t has active sites if it starts from configuration C at t = 0 (under the PU, Pr$_{a}$ is either 1 or 0). Note that the infinite-size limit is not assumed in Eq. (5). Since the possible value of S is either one or zero, we can divide the set of all configurations by the two disjoint sets I and A , In fact, both I and A can be further divided into many disjoint classes according to the toppling invariant [16] as well as total energy. For our purpose, however, it is enough to consider only these two classes. Since any configuration with a total energy lower than min { F$_{ii}$ } cannot have an active site, I is not empty and the energy of a configuration in A cannot be lower than min { F$_{ii}$ } . Likewise, since any configuration with a total energy higher than ∑ $_{i}$( F$_{ii}$ − 1) cannot be absorbing, A is not empty and the energy of configurations in I cannot be higher than ∑ $_{i}$( F$_{ii}$ − 1). Thus, the total energy of If there is nonzero probability current going into an absorbing state from any configuration, a system with a does not change with time. When the total energy is low, it is likely that the system will eventually fall into one of the absorbing states. Meanwhile, if the total energy is high, the system may change with time indefinitely. Hence, as the total energy varies, the system can exhibit an APT. In the next section, we will discuss the general features of the APT exhibited by the DFES. where P ( C, t ) is the probability of being in configuration C at time t , N ( C ) is the number of active sites in con- figuration C , and C$_{j}$ is the configuration constructed by adding F$_{ji}$ to z$_{i}$ for all i in configuration C . Note that P ( C$_{j}$, t ) = 0 if C$_{j}$ has a site with negative energy. Due to the second property of a TM, the total energy defined as where dt is an exponentially distributed random number with mean 1 /N$_{t}$ and i is the index of the chosen active site. As in the PU, the system does not change with time if it is in an absorbing state. The RSU actually simulates the master equation In the PU, the system evolves deterministically and time takes integer values. In the RSU, on the other hand, the system evolves stochastically and time takes contin- uous values. Assume that there are N$_{t}$ active sites at time t . One of the active sites is randomly chosen with equal probability, and energy at every site j at time t + dt becomes where j$_{n}$ ’s are the indices of the active sites at time t and we have assumed that there are N active sites. If there are no active sites, the system does not change with time. In this sense, any configuration without active sites is an absorbing state. We consider two rules of time evolution. One is the parallel update rule (PU), and the other is the random sequential update rule (RSU). In the PU, all active sites at time t topple simultaneously and the configuration at time t + 1 becomes ( i = 1 , . . . , V ) z$_{i}$ ( t + 1) = z$_{i}$ ( t ) − N ∑ n =1 F$_{j}$$_{n}$$_{i}$, (1) z$_{j}$ ( t + dt ) = z$_{j}$ ( t ) − F$_{ij}$, (2) ∂ ∂t P ( C, t ) = ∑ j P ( C$_{j}$, t ) − N ( C ) P ( C, t ) , (3) S ( C ) = lim t $_{→∞}$Pr$_{a}$ ( t | C ) , (5) E ≡ ∑ i z$_{i}$ (4) I = { C | S ( C ) = 0 } , A = { C | S ( C ) = 1 } . (6) A. Order parameter φ 1 III. ABSORBING PHASE TRANSITIONS: GENERAL FEATURES