Langevin equation time-scale separation

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

In the Kramers approach the friction models collisions between the particle and the surrounding medium, and it is assumed that the collisions occur instantaneously. There is a time-scale separation between the reactive mode and its thermal bath. The dynamics are described by the Langevin equation (4.141). The situation where the collisions do not occur instantaneously but take place on a time scale characterizing the interactions between the particle and its surrounding can be described by a generalized Langevin equation (GLE),158,187... 

The purpose of this section is to develop a systematic approach to evaluating higher-order corrections to the results discussed in Section II. Of course, this systematic approach can also be applied to physical systems different from that discussed in Section II. As a consequence of the linear nature of this system, these rules prove the exact equivalence between the Newtonian description of Eq. (2.1) and the standard Langevin equation provided that the assumption of time-scale separation is satisfied (see Section IV). 

We begin with an abstract of the physics that underlies the kinetics of bond dissociation and structural transitions in a liquid environment. Developed from Einstein s theory of Brownian motion, these well-known concepts take advantage of the huge gap in time scale that separates rapid thermal impulses in liquids (< 10 s) from slow processes in laboratory measurements (e.g. from 10 s to min in the case of force probe tests). Three equivalent formulations describe molecular kinetics in an overdamped liquid environment. The first is a microscopic perspective where molecules behave as particles with instantaneous positions or states x(t) governed by an overdamped Langevin equation of motion,... 

There are many problems that would require so much computer time that their study by the previous method would not be possible. For example polyelectrolyte solutions, or motions of particles in membranes would not be susceptible to study because of wide separations in the time scales for different dynamic processes characterizing solute and solvent or because the property of interest evolves so slowly that an excessively long trajectory would be required. The study of these systems requires a different approach. A beginning was made many years ago by Simon,who studied the melting of DNA by solving the coupled set of stochastic Langevin equations on a computer. This required an assumption about the statistical distribution of random forces. The precise values of the forces were then sampled from this distribution. 

The friction coefficient is the inverse particle s relaxation time, jS = 9py/(2pp ), where py is the fluid s dynamic viscosity. Since the Langevin equations are linear, particle velocity and position may be formally solved as functionals of the random force, and in the diffusive limit f >> i. e., for times much larger than the particle relaxation time, they allow for the analytical evaluation of ensemble averaged products of particle position and velocity and two-point correlation functions, in terms of the random-force strength q. The authors carefully justify why they use the classical (equilibrium) form of the fluctuation-dissipation theorem (FDT) in a Langevin description the time scale of the white noise is considered to be much shorter than the time scale of the imjxjsed flow. Thus, the non-equilibrium corrections would be of the order of the ratio of the fluid molecular relaxation time to the time scale of the imposed shear and may be neglected. In this case both the time scales are clearly separated and q may be determined solely from the classical form of the FDT,... 

Now we decompose the Langevin equation for each Rouse bond into longitudinal and transverse motions by projecting the bond vectors along the tube axis. This will illustrate that the cut-off p serves to separate the time scales for conformational changes into two groups. The longitudinal components of the bond vectors are defined by... 

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