Jarzynski

Jarzynski, 1997a] Jarzynski, C. Equilibrium free-energy differences from nonequilibrium measurements A master equation approach. Phys. Rev. E. 56 (1997a) 5018-5035... 

Jarzynski, 1997b] Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78 (1997b) 2690-2693... 

The work theorem in the case of mechanical work was first given by Bochkov and Kuzovlev (for the case of a long cyclic trajectory) [60] and later by Jarzynski [55]. Both derivations invoked an adiabatic trajectory, in contrast to the present result that is valid for a system that can exchange with a reservoir during the performance of the work. 

Keywords QM/MM methods, Jarzynski approximation, Enzyme mechanisms... 

Assuming an infinite number of realizations of the process, equilibrium properties can be obtained from the non-equilibrium dynamics by the Jarzynski equality,... 

Figure 1-5. Free energy profile for the reaction from chorismate (RC 1.75) to prephenate (RC — 1.75), obtained using MSMD and Jarzynski s equality and pulling speeds of 2.0 A/ps (red) and 1.0 A/ps (green), and using umbrella sampling (blue)...

One of the most important theoretical developments of the last decade is due to Chris Jarzynski, who established a remarkably simple relationship between the equilibrium free energy difference and an ensemble of properly constructed irreversible... 

Despite its apparent simplicity, this calculation is relatively difficult to perform. Gomez etal. [18] demonstrated that ABF performs much better than both slow-growth and fast-growth implementations of the nonequilibrium method of Jarzynski and Crooks (see the next chapter). 

Hendrix, D.A. Jarzynski, C., A fast growth method of computing free energy differences, J. Chem. Phys. 2001,114, 5974-5981... 

Liphardt, J. Dumont, S. Smith, S.B. Tinoco, I. Bustamante, C., Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski s equality, Science 2002, 296, 1832-1835... 

Park, S. Khalili-Araghi, F. Tajkhorshid, E. Schulten, K., Free energy calculation from steered molecular dynamics simulations using Jarzynski s equality, J. Chem. Phys. 

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... 

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. 

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