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[formula presented] In fact, energy is not usually added on top of already active sites. This makes the energy addiction [formula presented]. Also, the energy balance in the stationary state is changed accordingly, thus obtaining [formula presented]. The latter immediately gives the stationary density of active sites as [formula presented]. For [formula presented], we recover the first order result presented in the text. However, higher order corrections can be considered, obtaining consistency relations useful for a higher order treatment of the MF equations., [formula presented] is the average energy added considering just first order contributions in [formula presented]
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Jin=hLd is the average energy added considering just first order contributions in h and ρa. In fact, energy is not usually added on top of already active sites. This makes the energy addiction Jin=h(1-ρa)Ld. Also, the energy balance in the stationary state is changed accordingly, thus obtaining h(1-ρa)Ld=ερaLd. The latter immediately gives the stationary density of active sites as ρa=h/(h+ε). For ε≫h, we recover the first order result presented in the text. However, higher order corrections can be considered, obtaining consistency relations useful for a higher order treatment of the MF equations.
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our notation, [formula presented]
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In our notation, α≡(1-ε)/(2d).
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[formula presented]. In the MF description, where no anomalous dimension appears, [formula presented] , The pair connectedness function is given by the percolation properties of tree sites in the presence of a characteristic length defined by the dynamical process. The general form of the pair connectedness is [formula presented]
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The pair connectedness function is given by the percolation properties of tree sites in the presence of a characteristic length defined by the dynamical process. The general form of the pair connectedness is r2-d-ηocc Γ(r/ξ). In the MF description, where no anomalous dimension appears, ηocc=0.
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It is easy to recognize that, in a dissipative model, each active particle can wander only a finite number of steps. Each toppling process loses on average an amount [formula presented] of energy. The energy unit needed to sustain the activity (the wandering of the active site) is therefore dissipated after [formula presented] toppling events (wandering steps). This indeed recovers the result obtained for the average avalanche size in our MF approach
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It is easy to recognize that, in a dissipative model, each active particle can wander only a finite number of steps. Each toppling process loses on average an amount ε of energy. The energy unit needed to sustain the activity (the wandering of the active site) is therefore dissipated after ε-1 toppling events (wandering steps). This indeed recovers the result obtained for the average avalanche size in our MF approach.
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