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# Scaling of avalanche queues in directed dissipative sandpiles

We simulate queues of activity in a directed sandpile automaton in 1+1 dimensions by adding grains at the top row with driving rate $0 < r \leq 1$. The duration of elementary avalanches is exactly described by the distribution \$P_1(t) \sim t^{-3/2}\exp{(-1

a r X i v :c o n d -m a t /0006246v 1 [c o n d -m a t .s t a t -m e c h ] 15 J u n 2000Scaling of avalanche queues in directed dissipative sandpiles Bosiljka Tadi´c 1,?and Vyatcheslav Priezzhev 2,??1Jo¡¦z ef Stefan Institute,P.O.Box 3000,1001Ljubljana,Slovenia 2Bogolubov Laboratory of Theoretical Physics,Joint Institute of Nuclear Research,141980Dubna,Russia Using numerical simulations and analytical methods we study a two-dimensional directed sandpile automaton with nonconservative random defects (concentration c )and varying driving rate r .The automaton is driven only at the top row and driving rate is measured by the number of added particles per time step of avalanche evolution.The probability distribution of duration of elementary avalanches at zero driving rate is exactly given by P 1(t,c )=t ?3/2exp [t ln(1?c )].For driving rates in the interval 0<r ¡Ü1the avalanches are queuing one after another,making increase the periods of non-interrupted activity of the automaton.Recognizing the probability P 1as a distribution of service time of jobs arriving at a server with frequency r ,the model represents a new example of the E,1,GI/¡Þ/1 server queue in the queue theory.We study scaling properties of the busy period and dissipated energy of sequences of non-interrupted activity.In the limit c ¡ú0and varying linear system size L ?1/c we ?nd that at driving rates r ¡ÜL ?1/2the distributions of duration and energy of the avalanche queues are characterized by a multifractal scaling and we determine th

e corresponding spectral functions.For L ?1/c increasing of the driving rate somewhat compensates the energy losses at defects above the line r ?¡Ì