Overdamped Langevin dynamics

We have performed Brownian dynamics simulations (i.e., overdamped Langevin dynamics) on the potential surface V(x, A) with a time-dependent A. In the simulations, the position x was updated as... 

N. Grpnbech-Jensen and S. Doniach, Long-Time Overdamped Langevin Dynamics of Molecular Chains, J. Comput. Chem., 15 (1994), 997-1012. 

An interesting special case of Langevin dynamics is obtained by considering the large y limit, often referred to as the overdamped limit. In cases where the friction constant is scaled by the inverse mass (so y is replaced by yM in (6.32)-(6.33)) such as in [233], this is instead known as the zero-mass limit. In this model the inertial dynamics is assumed to be dominated by collisional effects. Let v = M p and assume that the acceleration is negligible, so that Langevin dynamics reduces to... 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

This is not always the case for first order schemes, however. For example the scheme denoted [AOABJ is consistent with the dynamics, and may be used for integrating the overdamped (y = oo) Langevin equation. 

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