A STOCHASTIC APPROACH TO NONEQUILIBRIUM CHAIN-REACTIONS IN DISORDERED-SYSTEMS - BREAKDOWN OF EIKONAL APPROXIMATION

A stochastic description of chain reactions occurring in disordered sy stems is suggested by considering a statistical distribution of time-d ependent rate coefficients. The possibilities of constructing a thermo dynamic formalism for nonequilibrium chain reactions are investigated by testing the validity of the eikonal approximation in the thermodyna mic limit. If the fluctuations of the rate coefficient are restricted to a finite range, then for large systems the probability of concentra tion fluctuations obeys the eikonal scaling condition, which makes pos sible the development of a nonequilibrium thermodynamic formalism. For an infinite range of variation of the rate coefficient, however, the eikonal scaling does not hold anymore: the probability of concentratio n fluctuations has a long tail of the negative power-law type and the system displays statistical Fractal features. The passage from the sto chastic eikonal behavior to the fractal scaling is characterized by a change in the deterministic kinetic equations of the process: in the e ikonal regime the effective reaction order with respect to the active intermediate is 1, whereas for fractal scaling it is equal to 2. Due t o this change in the effective reaction order for fractal scaling, the reaction is much Faster than in the eikonal regime and the explosion threshold may be reached after a finite time interval.