R. Cafiero et al., THEORY OF EXTREMAL DYNAMICS WITH QUENCHED DISORDER - INVASION PERCOLATION AND RELATED MODELS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 54(2), 1996, pp. 1406-1425
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.