On the exact and phenomenological Langevin equations for a harmonic oscillator in a fluid

We have derived, starting from the Liouville equation and using projection operators, the three-dimensional Langevin and Fokker-Planck equations for the time dependence of the momenta and position of two Brownian particles of mass M interacting with a harmonic potential (i.e., a harmonic oscillator) in a fluid of particles with mass m, with M ⪢ m. The resulting Langevin equation has a very complicated structure in that the friction coefficients, x, due to the hydrodynamic interaction between the oscillator particles, are functions of R(t), the time-dependent seperation of the oscillator particles, and the noise terms, though Gaussian, are non-stationary and of a generalized form, i.e., neither “additive” nor “multiplicative”. In addition, there is a term involving the mean force exerted by the fluid on the oscillating Brownian particle. We then investigated the various approximations which must be made to reduce this “exact” Langevin equation to the frequently used one-dimensional, phenomenological Langevin equations (and corresponding Fokker-Planck equations) with purely “additive” and “multiplicative” noise. These approximations are of three types: (a) one must neglect the term arising from the rotational motion of the oscillator in the fluid; (b) one must neglect the R(t) dependence of X[R(t)], leading to purely “additive” noise, or approximate X[R(t)] by a Taylor series expansion to quadratic order in R(t) - R0 (where R0 is the equilibrium separation of the oscillator particles), leading to “additive” and “multiplicative” noise; and (c) one must either neglect the mean force term or approximate it by a term linear in R(t) - R0. We conclude from these results that one-dimensional phenomenological Langevin equations with simple noise structure are of doubtful validity for the dynamical description of the relative momenta (or vibrational energy) of oscillating molecules in a fluid.