THEORY OF EXTREMAL DYNAMICS WITH QUENCHED DISORDER - INVASION PERCOLATION AND RELATED MODELS

The study of phenomena such as capillary displacement in porous media, fracture propagation, and interface dynamics in quenched random media has attracted a great deal of interest in the last few years. This cl ass of problems does not seem to be treatable with the standard theore tical methods, and the only analytical results come from scaling theor y or mapping, for some of their properties, to other solvable models. In this paper a recently proposed approach to problems with extremal d ynamics in quenched disordered media, named run time statistics (RTS) or quenched-stochastic transformation, is described in detail. This me thod allows is to map a quenched dynamics such as invasion percolation onto a stochastic annealed process with cognitive memory. By combinin g RTS with the fixed scale transformation approach, we develop a gener al and systematic theoretical method to compute analytically the criti cal exponents of invasion percolation, with and without trapping, and directed invasion percolation. In addition we can also understand and describe quantitatively the self-organized nature of the process.