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As shown by Jarzynski, free energy differences between equilibrium states can be expressed in terms of the statistics of work carried out on a system during non-equilibrium transformations. This exact result, as well as the related Crooks fluctuation theorem, provide the basis for the computation of free energy differences from fast switching molecular dynamics simulations, in which an external parameter is changed at a finite rate, driving the system away from equilibrium. In this article, we first briefly review the Jarzynski identity and the Crooks fluctuation theorem and then survey various algorithms building on these relations. We pay particular attention to the statistical efficiency of these methods and discuss practical issues arising in their implementation and the analysis of the results.

The calculation of free energies from atomistic simulations is of great importance in many applications, ranging from the prediction of the phase behavior of a certain substance to the calculation of ligand affinities in drug design. Since the computation of free energies (or, more precisely, of free energy differences) involves the determination of entropic contributions and, hence, the estimation of phase space volumes [

According to the maximum work theorem, a consequence of the second law of thermodynamics, the amount of work

Equivalently, the amount of work that can be extracted from a system is bounded from above by the free energy difference. In the above equation, the equal sign holds only if the transformation is carried out reversibly, maintaining equilibrium at all times. The angular brackets on the left-hand side of the maximum work theorem indicate an average over many realizations of the non-equilibrium process. If one considers a macroscopic system, for instance, a piston compressing a gas enclosed in a cylinder, the average is not necessary, because every realization of the process yields, for all practical purposes, the same amount of work

As shown by Jarzynski in 1997 [_{B}_{B} is the Boltzmann constant. Remarkably, this result, now commonly referred to as Jarzynski equation or Jarzynski non-equilibrium work theorem, relates the statistics of irreversible work carried out on the system, while it is driven away from equilibrium, to an equilibrium free energy difference. A closely connected result is the Crooks fluctuation theorem [

In general, processes during which work is performed on or by the system drive the system away from equilibrium, such that the phase space distribution obtained at the end of the process may differ strongly from the equilibrium distribution to which the system relaxes after the external perturbation has been stopped. For instance, a piston pushed quickly into a gas-filled cylinder generates non-equilibrium states with strong flows markedly different from the static equilibrium state to which the gas eventually relaxes after the piston has reached its final state. At first sight, it is therefore surprising that equilibrium properties, such as free energy differences, can be extracted from non-equilibrium trajectories. As discussed in the following sections of this paper, a closer analysis reveals that averaging over the work exponential is equivalent to removing the bias introduced during the driving process. It is this unbiasing that ultimately permits the extraction of equilibrium properties (as we will discuss in Section 5, in principle, one can determine the entire equilibrium distribution and not only the free energy) from non-equilibrium trajectories. Thus, the non-equilibrium work theorem can be viewed as a prescription of how to compensate for the effects of manipulations that drive the system into non-equilibrium rather than a tool that illuminates the nature of non-equilibrium processes. Nevertheless, it is remarkable that the bias has a very simple exponential form and can be expressed in terms of the work only.

The Jarzynski non-equilibrium work theorem, as well as the Crooks fluctuation theorem provide the framework for the interpretation of single-molecule pulling experiments [

To set the notation, consider a classical system with energy _{A}_{B}_{A}_{A}_{B}_{B}_{A}_{B}

Instead of changing the control parameter _{A}_{B}_{A}_{A}

Jarzynski has shown [

The Jarzynski equation is an exact result that holds under very general conditions. The requirements are that initially, the system must be in equilibrium and that for a fixed control parameter, the dynamics conserves the equilibrium distribution corresponding to that value of the control parameter. The latter condition is satisfied by most types of dynamics usually used in computer simulations, including Newtonian, thermostated, Langevin and Monte Carlo dynamics. It is worth pointing out that it is not necessary that the system be in an equilibrium state at the end of the transformation process or relax towards equilibrium after the control parameter switching is completed. Furthermore, it is interesting that the Jarzynski equation holds, even if the switching is carried out according to different (though prescribed) protocols provided that _{A}_{B}_{A}_{B}

In the limiting cases of infinitely fast switching and infinitely slow switching, the Jarzynski equality reduced to two well-known results. For instantaneous switching, _{A}_{λ}

As mentioned in the introduction, the Jarzynski equation can be viewed as a way to remove the bias introduced by the switching process into the phase space distribution obtained at the end of the process. Following similar considerations as those used to derive the Jarzynski equality, one can prove that for any phase space function _{B}_{B}_{B}

If the dynamics of the system not only conserves the equilibrium distribution for a fixed control parameter, but is also microscopically reversible, _{R}_{R}_{B}_{A}_{R}_{R}

Jarzynski’s non-equilibrium work theorem and the Crooks fluctuation theorem suggest interesting algorithms for the calculation of free energy differences. The power of these algorithms derives from the fact that all quantities appearing in these relations can be easily determined. The simplest of these algorithms consists in the following steps. First, one needs to prepare initial conditions distributed according to the Boltzmann–Gibbs distribution. This can be achieved using a variety of methods, for instance, canonical Monte Carlo simulation, possibly combined with enhanced sampling methods, such as parallel replica sampling, or with thermostated molecular dynamics. To improve the efficiency of the free energy calculation, it is important to make sure that these initial conditions are sufficiently decorrelated.

From these initial conditions, one then starts trajectories of the desired length that are integrated, while, at the same time, changing the control parameter according to the protocol _{t}_{t}

An important choice one has to make in the context of fast switching free energy computations is how to allocate computing time. In particular, one has to decide whether to generate many short trajectories with a large switching rate or fewer and longer trajectories along which the system is driven more gently. Without enhanced sampling schemes, as those discussed in subsequent sections, one generally expects the slow switching regime to give more accurate free energy estimates for a given amount of computing time [_{B}

As discussed above, the statistical error of a free energy computed via fast switching strongly depends on the rate at which the system is driven out of equilibrium. However, while the switching rate is certainly the most important parameter, also the particular shape of the protocol

While, in general, protocols in which the dissipated work is small are expected to yield a more accurate free energy estimate, there is no simple relation between the average work and the statistical error in the free energy. Hence, a protocol optimized with respect to the work does not necessarily minimize the statistical error. However, numerical protocol optimizations conducted for various models indicate that control parameter steps at the start and the end of the protocol (but never in between) are beneficial also for free energy computations [

The simplest, but also most error-prone, method to obtain free energies from one-sided non-equilibrium simulations is a direct evaluation of the exponential estimator:
_{i}_{B}^{2}, then the estimate will tend to be dominated by only a few trajectories [

More accurate and asymptotically unbiased free energy estimates can be obtained from two-sided simulations by using the Crooks relation. By exploiting the analogy between equilibrium perturbation theory and non-equilibrium simulations, one can adapt Bennett’s acceptance ratio as the estimator [_{i}_{i}_{f}_{b}_{i}

The analogy to the equilibrium method also allows us to adapt two-sided cumulant estimators [_{R}

Potentials of mean force (PMF)

In many practical applications, the biasing potentials ^{2}

Fast switching simulations carried out at large switching rates typically generate work distributions that lead to large statistical uncertainties in the free energy estimate. As discussed earlier, the reason is that trajectories with typical work values contribute little to the exponential average of the Jarzynski equation, while trajectories with work values dominating the average are very rare. As a consequence, the convergence of the computed free energy is impractically slow for overly fast switching. A solution to this problem consists in favoring the generation of trajectories with important work values. In this section, we discuss how path sampling techniques can be used for this purpose.

To introduce computational methods for realizing this idea, we rewrite the exponential work average as an explicit sum over pathways:
_{0}. As suggested by Ytreberg and Zuckerman [_{π}_{π}

Since the bias function should enhance the sampling of important, but rare, work values, a bias function depending on the path

Another way of realizing work biased path sampling of fast-switching trajectories for the computation of free energies was suggested by Sun [_{α}_{α}_{α}

One can show that in the limit of infinitely short trajectories, Sun’s method reduces to conventional thermodynamic integration. This result raises the question of which trajectory length leads to the most efficient free energy calculations and, in particular, if work biased path sampling algorithms perform better then conventional methods, such as thermodynamic integration or umbrella sampling. Extensive calculations carried out for various models indicate [

Molecular dynamics simulations are usually carried out with time steps that are a compromise between accuracy (often assessed in terms of energy conservation) and computing speed. Small time steps yield accurate trajectories with good energy conservation, but require a larger computational effort, because the cost of a trajectory of a given length is proportional to the number of steps and, hence, inversely proportional to the size of the time step. Larger time steps reduce the computing time, but corrupt the accuracy, resulting in poor energy conservation. In general, using such low-accuracy trajectories for free energy computations introduces a systematic error into the free energy estimate. It is, however, possible to devise exact expressions akin to the Jarzynski equation to compute free energy differences from crude trajectories calculated with large time steps [

As mentioned earlier, in the limit of instantaneous switching, the Jarzynski equation reduces to the perturbation identity of Equation _{A}_{B}^{−1}, with the corresponding work definition, one can also use this mapping approach together with the Crooks fluctuation theorem.

_{ϕ}_{ϕ}

One possible strategy to exploit _{i}_{i}_{+1} along the molecular dynamics trajectory. Hence, a sequence of _{0} into the final point _{n}_{ϕ}

The large time step formalism can also be used for the calculation of potentials of mean force [

Arguably the most important practical application of non-equilibrium work theorems has been to experiments. Almost immediately after the connection between non-equilibrium single-molecule pulling experiments and Jarzynski’s identity was rigorously established [

Applications to pulling have been mirrored on the simulation side. Simulated pulling methods mimicking experiments have been developed, initially to probe mechanical perturbations on biomolecules [

The Jarzynski non-equilibrium work theorem and the Crooks fluctuation theorem are fundamental exact relations that link the irreversible work carried out on a system during a non-equilibrium transformation to the system’s equilibrium statistics. To date, the most significant application of these relations lies in the interpretation of single-molecule pulling experiments, in which forces exerted by atomic force microscopes or optical tweezers are used to probe the properties of individual molecules. Due to technological limitations, such experiments are necessarily carried out at a finite pulling rate, leading to non-equilibrium effects that cannot be neglected. The theorems of Jarzynski and Crooks provide a practical tool for the interpretation of such single-molecule pulling experiments and permit one to extract equilibrium information, such as potentials of mean force, from data obtained under inherently non-equilibrium conditions [

From a computational point of view, the Jarzynski and Crooks theorems have provided a new and powerful framework for the calculation of free energies using computer simulations. Apart from putting earlier slow-growth free energy simulations on a firm theoretical footing, these results have spawned the development of several new free energy algorithms based on non-equilibrium, fast-switching trajectories.

Depending on the rate at which the system is driven away from equilibrium, fast switching free energy computations can be plagued by large statistical errors. For strong driving,

One aspect of fast switching simulations that has not been fully exploited is the freedom in choosing the transformation protocol. While the optimization of the time dependence of the driving parameter has been the subject of previous numerical and analytical studies [

We acknowledge the financial support of the Austrian Science Fund (FWF, Fonds zur Förderung der Wissenschaftlichen Forschung) within the SFB ViCoM (Spezialforschungsbereich Vienna Computational Materials Laboratory), Grant F41, as well as Project P24681-N20 (C.D.) and the Max Planck Society (G.H.).

The authors declare no conflict of interest.