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Abstract
The functional integral method for the statistical solution of stochastic differential equations is extended to a broad, new class of nonlinear dynamical equations with random coefficients and initial conditions. This work encompasses previous results for classical systems with random forces and initial conditions with arbitrary statistics, and provides new results for systems with nonlinear interactions which are nonlocal in time. Closed equations of motion for the correlation and response functions are derived which have applications in the calculation of particle motion in stochastic magnetic fields and in the description of electromagnetic plasma turbulence. As an illustration of the new results for nonlocal interactions and non-Gaussian initial conditions, the electromagnetic dispersion tensor is calculated to first order in renormalized perturbation theory.
For a class of discrete, dissipative maps, which exhibit strange attractors, the path integrals in the functional integral representation of the dynamics can be performed analytically. Exact results are thereby derived for low-order statistical moments which are verified numerically.
The statistics of a more complicated nonlinear map, which models the heating of a Brownian charged particle by electrostatic waves, are also calculated analytically using a steepest descent method to evaluate the path integrals. Results are presented for the low-order statistical moments for three turbulent regimes which exhibit strange attractors corresponding to strong, intermediate, and weak collisional damping. In the dissipationless limit this map is equivalent to the conservative Chirikov-Taylor map. The turbulent behavior of the Chirikov-Taylor map is shown to be diffusive due to the intrinsic stochasticity; and the stochastic diffusion coefficient derived by Rechester and White is recovered. The statistical dynamics are significantly altered by the inclusion of damping which restricts the chaotic motion to a strange attractor which is bounded in phase space. These results provide a new description of the effects of collisional damping on the stochastic heating of plasmas by electrostatic waves.
