Jarzynski s identity

In this chapter, we will show how nonequilibrium methods can be used to calculate equilibrium free energies. This may appear contradictory at first glance. However, as was shown by Jarzynski [1, 2], nonequilibrium perturbations can be used to obtain equilibrium free energies in a formally exact way. Moreover, Jarzynski s identity also provides the basis for a quantitative analysis of experiments involving the mechanical manipulation of single molecules using, e.g., force microscopes or laser tweezers [3-6]. 

Jarzynski has shown that, even for nonequilibrium paths, the inequality (5.6) can be turned into an equality [1], Jarzynski s identity states that... 

Jarzynski s identity, (5.8), immediately leads to the second law in the form of (5.6) because of Jensen s inequality, (e x) > e. Moreover, in the limit of an infinitely fast transformation, r —> 0, we recover free energy perturbation theory. In that limit, the configurations will not relax during the transformation. The average in... 

By integrating both sides of (5.14) with respect to z, we obtain Jarzynski s identity... 

The above derivation shows that Jarzynski s identity is an immediate consequence of the Feynman-Kac theorem. This connection has not only theoretical value, but is also useful in practice. First, it forms the basis for an equilibrium thermodynamic analysis of nonequilibrium pulling experiments [3, 15]. Second, it helps in deriving a Jarzynski identity for dynamics using thermostats. Moreover, this derivation clarifies an important aspect trajectories can be thought of as mapping initial conditions (I = 0) to trajectory endpoints, and the Boltzmann factor of the accumulated work reweights that map to give the desired Boltzmann distribution. Finally, it can be easily extended to transformations between steady states [17] in which non-Boltzmann distributions are stationary. 

That is, we have recovered a Boltzmann distribution according to the Hamiltonian at time t, equivalent to (5.14). Jarzynski s identity (5.8) then follows simply by integration over phase space (p, q). 

To further illustrate the theory, we apply Jarzynski s identity to the analytically solvable example of a ID moving harmonic oscillator with Hamiltonian... 

In the following, we will show explicitly that the correct result is obtained if Jarzynski s identity is used to evaluate the free energy difference, A(t) —, 4(0) =... 

For computer simulations, (5.35) leads to accurate estimates of free energies. It is also the basis for higher-order cumulant expansions [20] and applications of Bennett s optimal estimator [21-23], We note that Jarzynski s identity (5.8) follows from (5.35) simply by integration over w because the probability densities are normalized to 1 ... 

Because of the normalization condition, the right-hand side is equal to exp( —f3AA), and Jarzynski s identity follows. 

The Crooks relation follows from an elegant derivation of Jarzynski s identity using path-sampling ideas [18], For instructive purposes, that derivation is briefly summa-rized here. Consider generating a discrete trajectory z0 - z i . .. z v, where... 

This procedure follows, in effect, the derivation of Jarzynski s identity in discrete time [2,18], as outlined in Sect. 5.5. Finally, for Hamiltonian dynamics, one can use (5.23) and calculate the work directly from the difference in total energy between trajectory start and end points. 

When calculating the potential of mean force along a fluctuating coordinate r, we can at best observe r (e.g., the instantaneous molecular extension), but we do not set it explicitly. Therefore, r is no longer an externally controlled coupling parameter, and Jarzynski s identity does not immediately apply. However, as was shown in [3], an extension produces the desired result. 

One of the major advances in the application of Jarzynski s identity to the calculation of free energies came from coupling it to path sampling [46, 47]. In a typical application with fast switching, the system is rapidly driven out of equilibrium as the coupling parameter is changed, and nearly all trajectories are essentially... 

To repeat briefly, the NEW method is related to free energy differences between systems 0 and 1 through Jarzynski s identity... 

In spite of these potential concerns, the MEHMC method is expected to be a useful tool for many applications. One task for which it might be particularly well suited is to generate a canonical ensemble of representative configurations of a bio-molecular system quickly. Such an ensemble is needed, for example, to represent the initial conditions for the ensemble of trajectories used in fast-growth free energy perturbation methods such as the one suggested by Jarzynski s identity [104] (see also Chap. 5). 

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...

As an illustrative example of this approach, consider applying Jarzynski s identity to reconstruct a one-dimensional double-well energy profile of the form V(x), assumed unknown and which is to be recovered by the method, from pulling trajectories with a harmonic guiding potential... 

---