Friction constant

Langevin dynamics a technique to reduce the total number of equations of motion that are solved. Utilize the Coupled Heat Bath, wherein the method models the solvent effect by incorporating a friction constant into the overall expression for the force. 

The diffusion constant should be small enough to damp out inertial motion. In the presence of a force the diffusion is biased in the direction of the force. When the friction constant is very high, the diffusion constant is very small and the force bias is attenuated— the motion of the system is strongly overdamped. The distance that a particle moves in a short time 8t is proportional to... 

Variations on these include running simulations at a variety of temperatures or running Langevin dynamics with a low friction constant to increase the ability to cross barriers at a given temperature (Pastor, W. R. Karplus, M., to be published.). 

The friction constant is then the infinite-time value of this function ... 

Figure 9b shows the friction constant as a function of a. For large a the friction coefficient varies linearly with ct in accord with the prediction of the Stokes formula. The figure also shows a plot of (slip boundary conditions) versus ct. It lies close to the simulation value for large ct but overestimates the friction for small ct. For small ct, microscopic contributions dominate the friction coefficient as can be seen in the plot of (m. The approximate expression 1 = + 071 interpolates between the two limiting forms. Cluster friction... 

Fig. 8.3. Histogram of work values for Jarzynski s identity applied to the double-well potential, V(x) = x2(x — a)2 + x, with harmonic guide Vpun(x, t) = k(x — vt)2/2, pulled with velocity v. Using skewed momenta, we can alter the work distribution to include more low-work trajectories. Langevin dynamics on Vtot(x(t),t) = V(x(t)) + Upuii(x(t)yt) with JcbT = 1, k = 100, was run with step size At = 0.001, and friction constant 7 = 0.2 (in arbitrary units). We choose v = 4 and a = 4, so that the barrier height is many times feT and the pulling speed far from reversible. Trajectories were run for a duration t = 1000. Work histograms for 10,000 trajectories, for both equilibrium (Maxwell) initial momenta, with zero average and unit variance, and a skewed distribution with zero average and a variance of 16.0...

A fully realistic picture of solvation would recognize that there is a distribution of solvent relaxation times (for several reasons, in particular because a second dispersion is often observable in the macroscopic dielectric loss spectra [353-355], because the friction constant for various types or modes of solute motion may be quite different, and because there is a fast electronic component to the solvent response along with the slower components due to vibration and reorientation of solvent molecules) and a distribution of solute electronic relaxation times (in the orbital picture, we recognize different lowest excitation energies for different orbitals). Nevertheless we can elucidate the essential physical issues by considering the three time scales Xp, xs, and Xelec-... 

In this nonadiabatic limit, the transmission coefficient is determined, via (2.8) by the ratio of the nonadiabatic and equilibrium barrier frequencies, and is in full agreement with the MD results [5a-5c]. (By contrast, the Kramers theory prediction based on the zero frequency friction constant is far too low. Recall that we emphasized for example the importance of the tail to the full time area of the SN2 (t). In the language of (3.14), the solvation time xs is not directly relevant in determining... 

In this generalized oscillator equation, the frequency is related to the restoring force acting on a particle and Q is a friction constant. The key quantity of the theory is the memory kernel mq(l — t ), which involves higher order correlation functions and hence needs to be approximated. The memory kernel is expanded as a power series in terms of S(q, t)... 

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

The radius R is related to the friction constant F in sedimentation by Stokes s equation ... 

The friction constant is consequently related to the time dependence of the random force in the equilibrium system. 

Thus if a functional form is chosen for K ooi), K"n((o) and Ku(t) can be determined from the Kramers-Kronig relations. Moreover, the parameters in the functional form, Af j co), can be related to the moments p2 , in addition to the friction constant H(0), so that these parameters can thereby be determined. 

For most proteins vp is about 0.75 ml/g, so its value does not present much of a problem. The frictional coefficient, however, is a sensitive function of the shape, varying over a wide range, and we must usually know its value if we need a serious estimate of the molecular weight. The value of/is usually found by working with the diffusion constant D, which is related to the frictional constant by... 

The FFPE (19) contains the generalized friction constant r a and the generalized diffusion constant Ka, of dimensions [r]a] = s -2 and [i a] = cm2 s . The physical origin of these fractional dimensions will be explained in the next section. In what follows, we assume natural boundary conditions, that is, lim i-nx) W(x, t) = 0. The FFPE (19) describes a physical problem, where the system is prepared at to = 0 in the state W(x, 0). 

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