Crossover and thermodynamic representation in the extended eta model for fractal growth.

Abstract

The \ensuremath{\eta} model for the dielectric breakdown is extended to the case where double power laws apply. It is shown that a crossover phenomenon between the diffusion-limited aggregation (DLA) fractal and the \ensuremath{\eta} fractal occurs in the extended \ensuremath{\eta} model. Through the use of the dimensional analysis, a dimensionless parameter is found to govern the crossover. It is shown that when \ensuremath{\eta}1 the crossover from the DLA fractal to the \ensuremath{\eta} fractal occurs with increasing size, and if \ensuremath{\eta}>1 the inverse crossover from the \ensuremath{\eta} fractal to the DLA fractal appears. It is also shown that the crossover radius is controlled by changing the applied field. The global flow diagram in the two-parameter space is obtained by using a two-parameter position-space renormalization-group approach. The crossover exponent and the crossover radius are calculated. The crossover phenomenon is described in terms of a thermodynamic representation of the two-phase equilibrium.