Jarzynskis 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. 

The Jarzynski Identity Path Sampling of Nonequilibrium Trajectories... 

The angular brackets ( ) in (7.33) denote an average over an ensemble of nonequilibrium transformation processes initiated from states z distributed according to a canonical distribution. The Jarzynski identity (7.33) is valid for nonequilibrium transformations carried out at arbitrary speed. 

The Jarzynski identity can be used to calculate the free energy difference between two states 0 and 1 with Hamiltonians J%(z) and -A (z). To do that we consider a Hamiltonian -AA iz, A) depending on the phase-space point z and the control parameter A. This Hamiltonian is defined in such a way that A0 corresponds to the Hamiltonian of the initial state, Af(z, A0) = Atfo (z), and Ai to the Hamiltonian of the final state, Ai) = Aif z). By changing A continuously from A0 to Ai the Hamiltonian of the initial state is transformed into that of the final state. The free energy difference ... 

The Jarzynski identity (7.33) is an exact result and applies to transformations of arbitrary length From a computational point of view this property seems very attractive because it implies that free energy differences can be calculated from short and therefore computationally inexpensive trajectories. However, the convergence of the exponential average from (7.33) quickly deteriorates if the transformation (or the switching) is carried out too rapidly. 

This statistical problem, which can easily offset the gain originating from the low computational cost of short trajectories, is best understood by rewriting the Jarzynski identity as an average over the work distributions P(W)... 

We will now turn our attention to the reconstruction of free energy profiles using the Jarzynski identity. This identity can be cast in terms of an equilibrium average, (8.49), as explained in Chap. 5. We can then bias the dynamics to follow the motion of the pulling potential, enhancing sampling of the low-work tail of the work distribution and thereby increasing the accuracy of the calculation. 

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... 

Schulten and coworkers coupled SMD [23] simulations with the Jarzynski identity [102] to derive the free energy profile for glycerol conduction in the facilitator GlpF, a channel that allows the selective passage of water and small, linear alcohols,... 

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... 

This identity [3, 15] between a weighted average of nonequilibrium trajectories (r.h.s.) and the equilibrium Boltzmann distribution (l.h.s.) is implicit in the work of Jarzynski [2], and is given explicitly by Crooks [16]. The average (... is over an ensemble of trajectories starting from the equilibrium distribution at / 0 and... 

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

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... 

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