Langevin equation inertia

We form the average of Eqs. (5.20)-(5.22) noting that (L(r)) will vanish throughout because, in the inertia corrected Langevin equation, M is statistically independent of the white noise field h(r). This is not, however, true of the noninertial Langevin equation where the multiplicative noise term L(f) contributes a noise induced drift term to the average (see Section VI). The averages so formed are... 

Here I = 7.5 X 1015 g-cm2/mol is the moment of inertia of the ring about the torsional axis, 7/3 is the friction constant, k is the harmonic restoring force constant, and/(f) represents the random torques acting on the ring due to fluctuations in its environment. In using the Langevin equation, we implicitly assume that variations in/(f) occur on a much shorter time scale than do... 

If the evolution of the system over times At j>l/ is of interest (thermalization of velocities in each time step— high friction limit ), the friction term is dominant and the inertia term on the left hand side of the Langevin equation can be neglected. Then the equation becomes first order ... 

The Zimm model rests upon the Langevin equation for over-damped motion of the monomers, i.e., it applies for times larger than the Brownian time scale Tb 2> OTm/where is Stokes friction coefficient [12]. On such time scales, velocity correlation functions have decayed to zero and the monomer momenta are in equilibrium with the solvent Moreover, hydrodynamic interactions between the various parts of the polymer are assumed to propagate instantaneously. This is not the case in our simulations. First of all, the monomer inertia term is taken into account, which implies non-zero velocity autocorrelation functions. Secondly, the hydrodynamic interactions build up gradually. The center-of-mass velocity autocorrelation function displayed in Fig. 9 reflects these aspects. The correlation function exhibits a long-time tail, which decays as (vcm(t)vcm(O)) on larger time scales. The... 

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