PROBABILITY-DISTRIBUTIONS FOR MULTICOMPONENT SYSTEMS WITH MULTIPLICATIVE NOISE

Linear systems with many degrees of freedom containing multiplicative and additive noise are considered. The steady state probability distri bution for equations of this kind is examined. With multiplicative whi te noise and certain symmetry conditions it is shown that the probabil ity distribution of a single component has power law tails, with the e xponent independent of the strength of additive noise, but dependent o n the strength of the multiplicative noise. The classification of thes e systems into two regimes appears to be possible in the same manner a s with just one degree of freedom. A physical system, that of a turbul ent fluid undergoing a chemical reaction is predicted to show a transi tion from exponential to power law tails, as the reaction rate is incr eased. A variety of systems are studied numerically. A replication alg orithm is used to obtain the Lyapunov exponents for high moments, whic h would be inaccessible by more conventional approaches.