Free-Energy Estimation

Another approach to the freezing transition was attempted by Torrie et Yhg Jo ugg jjjo single-occupancy model of Hoover and Ree, but 

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Torrie G M and Valleau J P 1977 Nonphysical sampling distributions In Monte Carlo free energy estimation umbrella sampling J. Comput. Phys. 23 187-99... 

Torrie, G.M., Valleau, J.P. Monte Carlo free energy estimates using non-Boltzmann sampling application to the subcritical Lennard-Jones fluid. Chera. Phys. Lett. 28 (1974) 578-581. 

To illustrate how stratification works in the context of free energy calculations, let us consider the transformation of state 0 into state 1 described by the parameter A. We further assume that these two states are separated by a high-energy barrier that corresponds to a value of A between Ao and Ai. Transitions between 0 and 1 are then rare and the free energy estimated from unstratified computer simulations would converge very slowly to its limiting value, irrespective of the initial conditions. If, however, the full range of A is partitioned into a number of smaller intervals, and... 

Torrie, G. M. Valleau, J. R, Nonphysical sampling distributions in Monte Carlo free energy estimation Umbrella sampling, J. Comput. Phys. 1977, 23, 187-199... 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

Broad work distributions have two important consequences first, the statistics will be poor and, second, a bias in the estimator of the free energy change, A(t ) — A(0) = —ft 1 ln(exp(—ftW(t))), will result in free energy estimates that deviate systematically from the correct free energy difference [10]. This will be discussed in depth in Chap. 6. Specifically, if the free energy is estimated from N work values IT) drawn at random from the work distribution p/ (IT),... 

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

Shown in Fig. 5.3d are free energies estimated from the same forward and backward simulation runs using Bennett s optimal estimator, obtained by solving (5.50) using a Newton-Raphson method. Unlike the direct exponential estimator (which... 

Fig. 5.3. Comparison of different free energy estimators. Plotted are distributions of estimated free energies using sample sizes (i.e., number of independent simulation runs) of N = 100 simulations (solid lines), as well as N = 1, 000 (long dashed) and N = 10,000 simulations short dashed lines), (a) Exponential estimator, (5.44). (b) Cumulant estimator using averages from forward and backward paths, (5.47). (c) Cumulant estimator using averages and variances from forward and backward paths, (5.48). (d) Bennett s optimal estimator, (5.50)...

In summary, using work collected from forward and backward paths greatly improves the accuracy of the estimates, and for the symmetric system studied here eliminates the bias. In our particular example, the cumulant estimators using forward and backward work data produce the most precise free energy estimates, followed by Bennett s optimal estimator. However, this somewhat poorer performance of the optimal estimator is caused in part by the high degree of symmetry of the system studied. 

Fig. 5.4. Comparison of different free energy estimators for asymmetric perturbation from A = 0 to 2 within t = 1. Shown are distributions of free energies estimated using the direct exponential average, (5.44), averaged over forward and backward perturbations (solid line), averages (5.47) from forward and backward paths (long dashed line)-, averages and variances (5.48) from forward and backward paths (short dashed line)-, and Bennett s optimal estimator, (5.50), (dotted line). In all cases, free energies were estimated from N 1,000 simulations. The vertical arrow indicates the actual free energy difference of (3AA —6.6...

Zuckerman, D. M. Woolf, T. B., Overcoming finite-sampling errors in fast-switching free-energy estimates. Extrapolative analysis of a molecular system, Chem. Phys. Lett. 2002, 351, 445 153... 

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

In principle, the forward and reverse calculations should produce identical free energy estimates. However in real simulations these estimates usually differ. Furthermore, as we explore finite sampling errors, we will see that the reliability (the error in each direction) of these estimates also differs [24, 26, 38]. 

To ensure the accuracy of the free energy estimate by sampling the most important set of trajectories, we choose the sequence of systems so that each successive state obeys a phase space subset relationship with the one that preceded it. This situation is illustrated schematically in Fig. 6.3. We say that a path following such a trajectory moves down the funnel [43]. 

Thus, the cause of the finite sampling error in free energy estimates from FEP or NEW simulations is the poor sampling of the important low-x tail of the / distribution and the high-x tail of the g distribution, for a forward and a reverse calculation, respectively. Sampling of these important tails corresponds to the sampling of the... 

In this model, the finite sampling systematic error is due to the missed sampling of the important region x < Xf. The free energy estimate given by the model is... 

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