The overdamped case

The physical manifestation of friction is the relaxation of velocity. In the high friction limit velocity relaxes on a timescale much faster than any relevant observation time, and can therefore be removed from the dynamical equation, leading to a solvable equation in the position variable only, as discussed in Section 8.4.4. The Fokker-Planck or Kramers equation (14.41) then takes its simpler, Smoluchowski form, Eq. (8.132) 

Dynamical effects in barrier crossing by the boundary conditions. We then need to solve the equation 

As discussed above, we expect that a solution characterized by a constant nonzero flux out of the well will exist for imposed source and sink boundary conditions. In the model of Fig. 14.2 the source should be imposed in the well while the sink is imposed by requesting a solution with the property Pss (x) 

The choice of as the upper integration limit corresponds to the needed sink boundary condition,/(x oo) = 0, while assuming a time-independent solution in the presence of such sink is equivalent to imposing a source. Equations (14.49) and (14.51) lead to 

This result can be further simplified by using the high barrier assumption, i8(F(xb) — F(0)) 1, that was already recognized as a condition for meaningful 

The resulting rate is expressed as a corrected TST rate. Recall that we have considered a situation where the damping y is faster than any other characteristic rate in the system. Therefore, the correction term is smaller than unity, as expected. 

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This quantity is obviously smaller than 1 and, as in the overdamped case, does not depend on T. The associated inverse effective temperature is... 

The differential form of the Chapman—Kolmogorov equation [11]. 3That is, we consider the overdamped case. 

Figure 13 shows Gj t) for a second rank potential coupling. The effect of the second body is still negligible in the overdamped case (Fig. 13a),... 

If the time constants xPI and xP2 are equal, we have two equal poles. Therefore, noninteracting capacities always result in an overdamped or critically damped second-order system and never in an underdamped system. The response of two noninteracting capacities to a unit step change in the input will be given by eq. (11.7) for the overdamped case, or eq. (11.8) for the critically damped. Instead of eq. (11.7), we can use the following equivalent form for the response ... 

At E-wave inflection point, the first term in Equation 28.18 vanishes, leaving cv = -kv. This limit allows defining a DT in the overdamped case, as shown in Figure 28.3. 

Thus, the pressure difference on the two sides of the step is proportional to the difference of the inverse cubes of the terrace widths (neglecting possible intereactions with more distant steps). Again in the overdamped limit, the step velocity f)x/<5t is proportional to the pressure from the terrace behind the step minus the pressure from the terrace ahead of the step. Since the motion is again step diffusion, the prefactor ought to contain the same transport coefficient as that for equilibrium fluctuations, fa for EC or Ds Cs , for TD, in either case divided by keT. Alternatively, this can be described as a current produced by the gradient of achemicalpotential associated witheachstep(Rettori and Villain, 1988). 

Fig. 8 The time dependence of the probability amplitude of the transferred proton for a LD trajectory for a PTR that starts at His64 and ends at OFT in the overdamped version of model S/A of the K64H-F198D mutant of CA III. The calculations were accelerated by considering a case where the minimum at site d is raised by 1.2kcalmol 1 (taken from Ref. 64).

The curve defined by (118) is a very well-known object it is the minimum energy path (MEP) which connects two minima of V(r) via a saddle point. How, why and by whom the MEP was first introduced in the context of molecular dynamics is not clear (to the author at least). The argument above indicates that it is the relevant object that concentrates most of the probability current of the reactive trajectories in its vicinity in the case of the overdamped dynamics when the temperature is small and the potential is sufficiently smooth (otherwise, if V r) has many critical points, (118) has many solutions, none of which taken alone is relevant). It is however not clear when such situations arise, and the MEP may often prove irrelevant. 

In the second case (Figure 13.6b) we have major heat transfer film resistances inside and outside the thermowell casing. This is equivalent to two capacities in series and as we know from Chapter 11, the thermocouple reading will exhibit second-order (overdamped) behavior ... 

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... 

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. 

It predicts, as in the N= 1 case, a coherent exchange of energy between atoms and field at the frequency fiov and one has again to distinguish two regimes depending upon the magnitudes of and u/Q. The overdamped re-... 

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