Different approaches for evaluating exponentially weighted nonequilibrium relations

Abstract

The Kawasaki identity ͑KI͒ and the Jarzynski equality ͑JE͒ are important nonequilibrium relations. Both of these relations take the form of an ensemble average of an exponential function and can exhibit convergence problems when the average of the exponent differs greatly from the log of the average of the exponential function. In this work, we re-express these relations so that only selected regions need to be evaluated in an attempt to avoid these convergence issues. In the context of measuring free energies, we compare our method to the JE and the literature standard approach, the maximum likelihood estimator ͑MLE͒, and show that in a system with asymmetric work distributions it can perform as well as the MLE. For the KI, we derive an analog to the MLE to compare with our relation and show that these two new relations improve on the KI and are complimentary to each other.

Non-Uniform Convergence in Moment Expansions of Integral Work Relations

“…The asymptotic behavior of the integral in (23) depends on n and V through the ratio α = n+1 V 2 . Several distinct regimes of moments are possible depending on the value of α. α…”

“…A large body of work was devoted to the investigation of the convergence of calculations based on Eq. ( 1), and the development of methods to improve them [18][19][20][21][22][23][24][25][26][27]. A physically intuitive interpretation of the underlying reasons for slow convergence was given by Jarzynski [28]; the rare dominant realizations that are needed to be sampled are the typical realizations of a reversed process.…”

Nonequilibrium free-energy estimation conditioned on measurement outcomes

“…(1) is likely to return very misleading results. Motivated by potential applications in physics, chemistry and computer simulations of biological molecules, several papers were devoted to the convergence of the free energy calculations, and various ways of improving it [15][16][17][18][19][20][21][22].…”

Free Energy Calculations with Reduced Potential Cutoff Radii

“…Here, the JE is used to determine the initial estimate for the iterative process, and we use an equal number of trajectories for both forward and reverse transformations, N 1→2 = N 2→1 , which is generally the most efficient balance of samples for the MLE. 8 At the same time, research has been carried out on optimizing the pathway to improve nonequilibrium, free energy calculation methods. 9 As the free energy of a system is a state function, it is possible to consider a variety of thermodynamic pathways in order to transform a system from one state to another when calculating free energy differences.…”