Nonequilibrium work

Helmholtz free energies divided by temperature, which is the work theorem [55]. For a cyclic process, the latter difference is zero, and hence the average is unity, as shown by Bochkov and Kuzovlev [58-60], The work theorem has been rederived in different fashions [57, 64, 65] and verified experimentally [66], What is remarkable about the work theorem is that it holds for arbitrary rates of nonequilibrium work, and there is little restriction beyond the assumption of equilibration at the beginning and end of the work and sufficiently long time interval to neglect end effects. (See Sections IVC4 and VB for details and generalizations.)... 

Crooks, G. E., Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 1999,60, 2721-2726... 

Protocol for Free Energy Estimates from Nonequilibrium Work Averages... 

Following is a schematic protocol summarizing the main steps in estimating free energies from nonequilibrium work averages. 

Atilgan, E. Sun, S. X., Equilibrium free energy estimates based on nonequilibrium work relations and extended dynamics, J. Chem. Phys. 2004,121, 10392-10400... 

Chernyak, V. Chertkov, M. Jarzynski, C., Dynamical generalization of nonequilibrium work relation, Phys. Rev. E 2005, 71, 025102... 

In this chapter, we will examine in depth the characteristic errors of two free energy techniques and present improved methods based on a better understanding of their behavior. The two techniques examined are free energy perturbation (FEP) [2] and nonequilibrium work (NEW) based on Jarzynski s equality [3-6]. These techniques are discussed in Chaps. 2 and 5. The FEP method is one of the most popular approaches for computing free energy differences in molecular simulation see, e.g., [1, 7-10]. The recently developed NEW method, which is closely related to FEP, is gaining popularity in both simulation [11-18] and experimental applications [19-21],... 

Fig. 6.3. To ensure the accuracy of a nonequilibrium work free energy calculation, the switching paths should go down the funnel. The important phase space regions for the intermediate states along the ideal funnel paths are illustrated in this plot, for the case where r0 and / are partially overlapped. Two funnel paths need to be constructed to transfer the systems from both 0 and 1 to a common intermediate M where rm is inside the r0 and J overlap region. The construction of such paths is discussed in Sect. 6.6...

Lu, N. et al., Using overlap and funnel sampling to obtain accurate free energies from nonequilibrium work measurements, Phys. Rev. E 2004... 

A closer look at different methods helps us to understand which features are responsible for their success. Let us compare, for instance, the nonequilibrium work method with the adaptive equilibrium approaches, described above. In the most common implementation of the former, the instantaneous force acting on the system along the order parameter is always equal to zero. In contrast, in the adaptive... 

Further large-deviation dynamical relationships are the so-called flucmation theorems, which concern the probability than some observable such as the work performed on the system would take positive or negative values under the effect of the nonequilibrium fluctuations. Since the early work of the flucmation theorem in the context of thermostated systems [52-54], stochastic [55-59] as well as Hamiltonian [60] versions have been derived. A flucmation theorem has also been derived for nonequilibrium chemical reactions [62]. A closely related result is the nonequilibrium work theorem [61] which can also be derived from the microscopic Hamiltonian dynamics. 

FTs are related to the so-called nonequilibrium work relations introduced by Jarzynski [31]. This fundamental relation can be seen as a consequence of the FTs [32, 33]. It represents a new result beyond classical thermodynamics that shows the possibility to recover free energy differences using irreversible processes. Several reviews have been written on the subject [3, 34—37] with specific emphasis on theory and/or experiments. In the next sections we review some of the main results. Throughout the text we will take k =... 

The nonequilibrium equality in Eq. (16) becomes the nonequilibrium work relation originally derived by Jarzynski using Hamiltonian dynamics [31],... 

Figure 12. (Upper panel) Path entropy i(w) (Middle panel) path free-energy (w) = w — Ts(w), and (lower panel) Lagrange multipher X(w) equal to the inverse of the path temperature 1/7 (m ). is the most probable work value given by y(w P) = X,(rv P) = 0 or = 1 is the value of the work that has to be sampled to recover free energies from nonequilibrium work values using the JE. This is given by y(w() = l/T or d> (w() = 0 Wrev and Wdis are the reversible and average dissipated work, respectively. (From Ref. 117.)...

J. Kurchan, Nonequilibrium work relations, J. Stat. Mech. (2007) P07005. 

J. M. Schurr and B. S. Fujimoto, Equalities for the nonequilibrium work transferred from an external potential to a molecular system analysis of single-molecule extension experiments. J. Phys. Chem. B 107, 14007-14019 (2003). 

C. Jarzyuski, Nonequilibrium work theorem for a system strougly coupled to a thermal euviroumeut. J. Stat. Mechanics (Theor. Exp.) P09005 (2004). 

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

Thanks to work by Jarzynski [3], excess heat production is known to be positive in a wide class of Markov processes. In Appendix A, we will demonstrate Jarzynski s nonequilibrium work relation. 

In Appendix A, we will demonstrate Jarzynski s nonequilibrium work relation. 

Jarzynski s nonequilibrium work relation is based on the following equality ... 

Jarzynski relation can be seen as a generalization of EXP, and Crooks relation can be used to derive the BAR equation for nonequilibrium work [28,42]. A multi-state... 

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