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 distribution for equations of this kind is examined. With multiplicative white noise it is shown that under some symmetry conditions, the probability distribution of a single component has power law tails, with the exponent independent of the strength of additive noise, but dependent on the strength of the multiplicative noise. The classification of these systems into two regimes appears to be possible in the same manner as with just one degree of freedom. A physical system, that of a turbulent fluid undergoing a chemical reaction is predicted to show a transition from exponential to power law tails, as the reaction rate is increased. A variety of systems are studied numerically. A replication algorithm is used to obtain the Lyapunov exponents for high moments, which would be inaccessible by more conventional approaches.