M. Battezzati, A diffusion equation for Brownian motion with arbitrary frictional coefficient: Application to the turnover problem, J CHEM PHYS, 111(22), 1999, pp. 9932-9943
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 mec
hanical system driven by a random force, this method is applied to derive a
third-order diffusion equation for an anharmonic oscillator undergoing Bro
wnian motion. This equation is exact to first-order in the parameter of anh
armonicity, and is valid for arbitrary values of the frictional coefficient
. The confrontation of this equation with a similar equation obtained previ
ously by asymptotic expansion in inverse powers of the frictional coefficie
nt, shows that although the two equations are different, nevertheless they
reduce to the same equation (within the limits of validity of each approxim
ation scheme) when they are both reduced to second order. An asymptotic for
mula 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 wit
h Smoluchowski boundary conditions. This equation yields the correct behavi
or of the eigenvalue in both limits of high and extremely low friction, wit
h interpolation between the two limits, while in the oscillatory regime yie
lds a complex eigenvalue, whose imaginary part can be interpreted as a stoc
hastic resonance frequency between the anharmonic well and its mirror image
beyond the barrier. It is shown how the Kramers' result for moderate or st
rong friction fits in with the present theory, and what is the origin of th
e discrepancies. (C) 1999 American Institute of Physics. [S0021-9606(99)001
46-4].