Cascades and self-organized criticality

We generalize the model of self-organized critical systems to cases where due to some internal degrees of freedom the local conservation law is violated. This can be realized by taking a transfer ratio different from the critical one in a sand pile model (global violation) or allowing fluctuations around the critical ratio (local violation). In the first case the deviation from the critical ratio R is a critical parameter and the characteristic avalanche size diverges as |R|^-ψ. In the second case the global conservation assures criticality; however, our numerical results indicate that the model is in a new universality class.