Multiplicative stochastic processes in nonlinear systems. II. Canonical and noncanonical effects

We study a physical system consisting of a low-frequency nonlinear oscillator interacting both with a thermal bath at the temperature $T_1$ and a high-frequency linear oscillator which, in turn, interacts with a thermal bath at the temperature $T_2$ ($T_2$≥$T_1$). The interaction between slow and fast oscillator is nonlinear, thereby influencing the motion of the slow oscillator via fluctuations of a multiplicative nature. By means of a suitable procedure of elimination of the fast variables, a contracted description is obtained, which, at $T_1$=$T_2$, exhibits precisely the same structure as that recently derived by Lindenberg and Seshadri [Physica (Utrecht) 109A, 483 (1981)] from the Zwanzig Hamiltonian. Instead of the transition from the overdamped to the inertial case revealed in our earlier paper [S. Faetti et al., Phys. Rev. A 30, 3252 (1984)] in this series, it is shown that precisely the reverse effect takes place. This is confirmed by computer calculations and the reliability of the computer calculation, in turn, is confirmed in the inertial regime via analog simulation. The theory enabling us to explore the noise-induced transition to the overdamped regime is based on an improvement of the techniques of elimination of fast variables, which produces automatic resummation over infinite perturbation terms. At $T_2$>$T_1$, a space-dependent diffusion term with the same structure as that involved with the multiplicative fluctuation of the ‘‘external’’ kind is proven to be added to the canonical multiplicative diffusion term exhibited by the case $T_1$=$T_2$. Canonical and noncanonical effects, which have so far been the subject of separate investigations, may thus be described via one single picture. It is also shown that, in the purely canonical case, the ubiquitous character of the noncanonical diffusional form, ranging from Ito^-like to Stratonovich-like structure, is lost and a unique form of diffusional equation occurs.