A stochastic approach to nonequilibrium chain reactions in disordered systems: Breakdown of eikonal approximation

A stochastic description o1" chain reactions occurring in disordered systems is suggested by considering a statistical distribution of time-dependent rate coefficients. Tile possibilities of constructing a thermodynamic ft~rmalism for nonequilibrium chain reactions are investigated by testing the validity of the eikonal approximation m tile thermodynamic limit. If tile Iluctuations of the rate coefficient are restricted to a finite range, then li/r large systems the probability of concentration Iluctuations obeys tile eikonal scaling condition, which makes possible the development of a nonequilibrium thermodynamic formalisnl. For an inlhlite range ot + variation of the rate coeMcient, however, the eikonal scaling does not hold anymore: the probability of concentration fluctuations has a long tail of the negative power-law type and tile system displays statistical fractat features. Tile passage from the stochastic eikonal behavior to the IYactal scaling is characterized by a change in tile deterministic kinetic equations of tile process: in the eikonal regime tile elTective reaction order with respect to tile active iIntermediate is I. whereas Ii+r fractal scaling it is equal to 2. Due to this change in the eflective reaction order for Iʹractal sealing+ tile reaction is much Ihster than in the eikonal regime and tile explosion threshold may be reached alier a finite time iInterwd.