Skewed momenta

Skewing Momenta Distributions to Enhance Free Energy Calculations from Trajectory Space Methods... 

We start with some background on existing methods that alter the initial distributions in the reactant basin, focusing in particular on the puddle jumping method of Tully and coworkers [69, 110], which is the inspiration for the skewed momenta method developed in the following section. We continue with a description of the skewed momenta method, as applied to (8.48), with numerical examples for each case. We end with a concluding discussion. 

Unlike the methods mentioned above, the skewed momenta method involves accentuating the dynamics only along particularly relevant directions. Because the... 

Fig. 8.2. Maxwell (left) and skewed momenta (right) distributions in two dimensions. If a slow direction is identified, the probability can be skewed along that direction such that it is more likely to kick the system along it exact kinetics is recovered by reweighting...

Methods such as skewed momenta are expected to have an additional advantage in high-dimensional systems. Puddle jumping is efficient in such systems only if the puddle can be selectively applied across a few pertinent degrees of freedom. In contrast, the skewed momenta method can be applied without modification to trajectories involving concerted changes to many degrees of freedom. 

In this section we explore the use of the skewed momenta method for estimating the equilibrium free energy from fast pulling trajectories via the Jarzynski identity [104]. The end result will be that generating trajectories with skewed momenta improves the accuracy of the calculated free energy. As described in Chap. 5, Jarzynski s identity states that... 

With Jarzynski s identity in the form of (8.46), we can apply to it the skewed momenta method simply by setting A[r(f)] = exp(— /3Wt). However, we anticipate that the method will be most useful in the particular case of calculating free energy profiles from pulling experiments, for which Hummer and Szabo have provided a modified form of Jarzynski s expression [106]. 

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 detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

MacFadyen, 1. Andricioaei, I., A skewed-momenta method to efficiently generate conformational-transition trajectories, 7. Chem. Phys. 2005,123, 074107... 

A two-dimensional example of a spherical and a skewed momentum distribution is shown in Fig. 8.2 for this simple case, es = (J), and the variance along es has... 

A convenient choice of a skew tensorial set of operators consists of the components of the angular momentum operator I [Eq. (14)], which transform according to the irreducible representation fg of Rgt. When these are divided by ]j—2, one obtains a set of real operators whose elements are the irreducible products of the degree one (effective degree zero) of the sets and... 

Axiality of w is automatically achieved by the usual transformation ((c) in Rem. 4) of tensor W. Therefore the skew-symmetric tensors instead of axial vectors and outer product (see Rem. 16) may be used and we do it this way at the moment of momentum balances in the Sects. 3.3,4.3, cf. [7, 8, 14, 27]. Generalization of this Lemma to third-order tensors, made by M. Silhav, is published in Appendix of [28]. 

Postulates other than Eq. (12 9) for the distribution m momentum space have been used. For example, some kinetic theonsts have used a Maxwellian distribution about the velocity v at the center of mass of the molecule. Another possible assumption is that of a skewed distribution, m which additional empirical parameters are introduced so that the smoothed Brownian motion force may be stronger in the chain backbone direction than in the transverse directions [9,20a, 20b, 20c, 21], [DPL, Sect. 13.7]. This idea has been proposed for describing the restricted motion of polymer chains in concentrated polymer solutions and in undiluted polymers. An extreme case of this is the reptation assumption [9,14a], in which there is no Brownian force at all m the transverse directions, and the polymer cham is required on the average to slither back and forth along its backbone (DPL, Sect 19.2b). 

From the above comments the constraints (4.118) are satisfied and it remains to derive the d30iamic equations (4.119) and (4.120). We begin by considering the balance law for linear momentum in (4.119). The non-zero components of the symmetric rate of strain tensor Aij and the skew-symmetric vorticity tensor Wij can be given in terms of V2 and V3 by using the relation (5.519) to find... 

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