FIG. 1. Plot of ζ c vs λ for the TM (14) with the initial condition (18) together with the distribution (23). B. Order parameter φ 2 2 . 495 2 . 5 2 . 505 2 . 51 2 . 515 2 . 52 2 . 525 0 0 . 2 0 . 4 0 . 6 0 . 8 1 ζ c λ Interestingly, if we choose λ = 0 . 5, ζ$_{c}$ becomes the SOC critical density 2 . 5 [13], regardless of the form of p$_{o}$ . Ex- cept for the case of λ = 1 / 2, ζ$_{c}$ should depend on p$_{o}$ and λ . To be specific, we solve the case with the Poisson dis- tribution which gives We depict the numerical solution ζ$_{c}$ as a function of λ in Fig. 1. Note that in Ref. [13] the above equation with λ = 0 was studied. For purposes of later discussions, we also consider an- other initial condition where ξ$_{i}$ is the same as in Eq. (18) and d$_{i}$ is a ran- dom number that takes either 2 or 3 with the restriction ∑ i d$_{i}$ = D (2 V ≤ D ≤ 3 V ). Since 〈 w 〉 is already 1 even if ξ$_{i}$ = 0 for all i , φ$_{1}$ = 1 for any ζ$_{0}$ > 0 in the infinite V limit. average ζ$_{0}$ = ∑ ∞ n $_{=0}$nP$_{bg}$ ( n ). The mean energy density of the whole system is ζ = ζ$_{0}$ + B/V . For the moment, we do not specify the form of P$_{bg}$ ( n ). As has already been discussed, w$_{i}$ defined in Eq. (15) will determine the transition point. With the above ini- tial condition, the mean value of pairs at site i with b$_{i}$ = 1 is Notice that a configuration with z$_{i}$ = 2 or 3 for all i is a recurrent configuration of the corresponding ASM with two dissipative boundary sites at i = 1 and i = V [13]. This example seems to suggest that there is no nontrivial transition if an initial configuration of a DFES is con- structed by adding energy to a recurrent configuration of the corresponding ASM. Later, we will argue that this is a general feature of the DFES model. P$_{bg}$ ( ξ$_{i}$ = n ) = ζ n 0 n ! e − $^{ζ$_{0}$}$, (23) p$_{o}$ = 1 − e − 2 ζ$_{0}$ 2 = 1 − e − 2( ζ − λ ) 2 . (24) z$_{i}$ = d$_{i}$ + ξ$_{i}$, (25) 2 〈 w$_{j}$ 〉 = ∞ ∑ k =1 2 k [ P$_{bg}$ ( ξ$_{i}$ = 2 k ) + P$_{bg}$ ( ξ$_{i}$ = 2 k + 1)] = ζ$_{0}$ − ∞ ∑ k =0 P$_{bg}$ ( ξ$_{i}$ = 2 k + 1) = ζ$_{0}$ − p$_{o}$, (20) and at site j with b$_{j}$ = 0 it is 2 〈 w$_{i}$ 〉 = ∞ ∑ k =1 2 k [ P$_{bg}$ ( ξ$_{i}$ = 2 k − 1) + P$_{bg}$ ( ξ$_{i}$ = 2 k )] = ζ$_{0}$ + ∞ ∑ k =0 P$_{bg}$ ( ξ$_{i}$ = 2 k + 1) = ζ$_{0}$ + p$_{o}$, (19) Now, we will discuss a tree. Since ζ$_{m}$ = ζ$_{M}$ in the infinite V limit in this case, there is a unique transition point ζ$_{c}$ = ζ$_{m}$ if the initial energy density distribution becomes a delta function in the infinite-size limit. This was observed numerically in Ref. [13] for some trees. Although we have only used symmetric TMs for illus- tration, we would like to emphasize that the existence of many transition points does not depend on whether or not a TM is symmetric. where p$_{o}$ ≡ ∑ ∞ k =0 P$_{bg}$ (2 k + 1) is the probability that ξ$_{i}$ is an odd number. Denoting the mean density of pairs by 〈 w 〉 , we get 2 〈 w 〉 ≡ 2 λ 〈 w$_{i}$ 〉 + 2(1 − λ ) 〈 w$_{j}$ 〉 = ζ$_{0}$ − (1 − 2 λ ) p$_{o}$, (21) To define another order parameter, we consider the steady state density ρ ( C ) of active sites, where ρ is re- garded as a function of the initial configuration C . We define ρ ( C ) for the PU as where λ = B/V . Note that if P$_{bg}$ ( n ) = 0 for all n , then the probability for the total number of pairs to exceed V is nonzero for any finite ζ$_{0}$ and, in turn, a phase transition can be well defined only under the infinite-size limit. Now, we take an infinite V limit with λ and ζ$_{0}$ fixed. If the variance of P$_{bg}$ is finite, the central limit theorem ensures that the distribution of the density of pairs be- comes a delta function under this limit. Thus, the tran- sition point ζ$_{c}$ is determined by the condition 〈 w 〉 = 1, which gives where C$_{t}$ is the configuration at (integer) time t with C$_{0}$ = C and n ( C$_{t}$ ) is the number of active sites in the ρ ( C ) ≡ lim T →∞ 1 T T ∑ t =0 n ( C$_{t}$ ) V , (26) ζ$_{c}$ = 2 + λ + (1 − 2 λ ) p$_{o}$. (22) 4