Fractals in surface growth with power - law noise

We present a microscopic description of interface growth with power-law noise distriiurion in the iorm P(qjiiq!'", which exhibits non-universai roughening. For the p = d + 1 case in d + 1 dimensions, the existence of a fractal pattern in the bulk of the aggregate is explained, leading trivially to the proof of the identity a + i = 2 for the roughening and the dynamical scaling exponents a and i respectively. Investigations on the distribution of step sizes of the discretized interface and the saturated growth speed funher support our arguments. There has been a growing interest in the interfaces which are self-affine fractals [l]. In these cases, the interface width w(L, f ) is expected to be a scaling function of the substrate size L and the growth time f in the form w(L, t ) L q f ( f / L I ) , such that f ( x ) + constant for x -f m and f ( x ) xn" for x+ 0. The roughening exponent LI and the dynamical exponent z are of particular interest. Both computer simulation and theoretical analysis support the exact values n = f and z = $ in 1 + 1 dimensions in a class of models. However, recent experiments on immiscible fluid displacement gave a 30.73 [2] and 0.81 [3] respectively and a/z=0.625 [3], while a bacteria colony expansion experiment found n -0.78 [4]. There have been several suggestions [5-81 to account for this. In particular, Zhang [5] proposed that this may be due to the existence of noise with a power law distribution instead of the usually assumed Gaussian form. Specifically, the proposed noise 11 has a distribution: Simulation in 1 + 1 [5,9] and 2 + 1 dimensions [IO] shows that the interface scales with exponents which are non-trivial functions of the parameter p and the usual exponent identity [ 13