A diffusion equation for Brownian motion with arbitrary frictional coefficient: Application to the turnover problem

After a brief re-exposition of the procedure devised by the author in order to reobtain a diffusion equation from the equations of the motion of a mechanical system driven by a random force, this method is applied to derive a third-order diffusion equation for an anharmonic oscillator undergoing Brownian motion. This equation is exact to first-order in the parameter of anharmonicity, and is valid for arbitrary values of the frictional coefficient. The confrontation of this equation with a similar equation obtained previously by asymptotic expansion in inverse powers of the frictional coefficient, shows that although the two equations are different, nevertheless they reduce to the same equation (within the limits of validity of each approximation scheme) when they are both reduced to second order. An asymptotic formula for the mean first-passage time (MFPT) for escaping over a barrier is then proved in the low-temperature limit, which is related to an eigenvalue of the diffusion operator, and to the solution of an integral equation with Smoluchowski boundary conditions. This equation yields the correct behavior of the eigenvalue in both limits of high and extremely low friction, with interpolation between the two limits, while in the oscillatory regime yields a complex eigenvalue, whose imaginary part can be interpreted as a stochastic resonance frequency between the anharmonic well and its mirror image beyond the barrier. It is shown how the Kramers' result for moderate or strong friction fits in with the present theory, and what is the origin of the discrepancies. (C) 1999 American Institute of Physics. [S0021-9606(99)00146-4].