content_image/381418.jpg content_image/381433.jpg content_image/381431.jpg content_image/381443.jpg content_image/381428.jpg content_image/381429.jpg content_image/381442.jpg content_image/381430.jpg
<img src='content_image/381418.jpg'>

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 

<img src='content_image/381433.jpg'>

and at site j with b$_{j}$ = 0 it is 

<img src='content_image/381431.jpg'>

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) 

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 

<img src='content_image/381443.jpg'>

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. 

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. 

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. 

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 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 

<img src='content_image/381428.jpg'>

<img src='content_image/381429.jpg'>

## B. Order parameter φ 2 

<img src='content_image/381442.jpg'>

<img src='content_image/381430.jpg'>

