Free energy nonequilibrium

Ben]amin I, Barbara P F, Gertner B J and Hynes J T 1995 Nonequilibrium free energy functions, recombination dynamics, and vibrational relaxation of tjin acetonitrile molecular dynamics of charge flow in the electronically adiabatic limit J. Phys. Chem. 99 7557-67... 

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... 

As a system moves from a nonequilibrium to an equilibrium position, AG must change from its initial value to zero. At the same time, the species involved in the reaction undergo a change in their concentrations. The Gibb s free energy, therefore, must be a function of the concentrations of reactants and products. 

In vivo, under steady-state conditions, there is a net flux from left to right because there is a continuous supply of A and removal of D. In practice, there are invariably one or more nonequilibrium reactions in a metabolic pathway, where the reactants are present in concentrations that are far from equilibrium. In attempting to reach equilibrium, large losses of free energy occur as heat, making this type of reaction essentially irreversible, eg. 

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.)... 

Many solvents do not possess the simple structure that allows their effects to be modeled by the Langevin equation or generalized Langevin equation used earlier to calculate the TS trajectory [58, 111, 112]. Instead, they must be described in atomistic detail if their effects on the effective free energies (i.e., the time-independent properties) and the solvent response (i.e., the nonequilibrium or time-dependent properties) associated with the... 

The expression in Eq. (10) for the exponent in Eq. (9) is quite similar to that for the activation free energy in electron transfer reactions derived by Marcus using the methods of nonequilibrium classical thermodynamics8 ... 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

The activation factor in the first case is determined by the free energy of the system in the transitional configuration Fa, whereas in the second case it involves the energy of the reactive oscillator U(q ) = (l/2)fi(oq 2 in the transitional configuration. The contrast due to the fact that in the first case the transition probability is determined by the equilibrium probability of finding the system in the transitional configuration, whereas in the second case the process is essentially a nonequilibrium one, and a Newtonian motion of the reactive oscillator in the field of external random forces in the potential U(q) from the point q = 0 to the point q takes place. The result in Eqs. (171) and (172) corresponds to that obtained from Kramers theory73 in the case of small friction (T 0) but differs from the latter in the initial conditions. 

As we will see further in the book, almost all methods for calculating free energies in chemical and biological problems by means of computer simulations of equilibrium systems rely on one of the three approaches that we have just outlined, or on their possible combination. These methods can be applied not only in the context of the canonical ensemble, but also in other ensembles. As will be discussed in Chap. 5, AA can be also estimated from nonequilibrium simulations, to such extent that FEP and TI methods can be considered as limiting cases of this approach. 

Hummer, G. Szabo, A., Free energy reconstruction from nonequilibrium singlemolecule pulling experiments, Proc. Natl Acad. Sci. USA 2001, 98, 3658-3661... 

Ytreberg, F. M. Zuckerman, D. M., Single-ensemble nonequilibrium pathsampling estimates of free energy differences, J. Chem. Phys. 2004, 120, 10876-10879... 

Crooks, G.E., Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems, J. Stat. Phys. 1998, 90, 1481-1487... 

Nonequilibrium Methods for Equilibrium Free Energy Calculations... 

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]. 

These bounds are the nonequilibrium equivalents of the Gibbs-Bogoliubov bounds discussed in Chap. 2. Having the free energy now bounded from above and below already demonstrates the power of using both forward and backward transformations. Moreover, as was shown by Crooks [18, 19], the distribution of work values from forward and backward paths satisfies a relation that is central to histogram methods in free energy calculations... 

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