On the accurate estimation of free energies using the jarzynski equality. Issue 4 (18th December 2018) - Arrar, Mehrnoosh; Boubeta, Fernando Martín; Szretter, Maria Eugenia; Sued, Mariela; Boechi, Leonardo; Rodriguez, Daniela
Abstract : The Jarzynski equality is one of the most widely celebrated and scrutinized nonequilibrium work theorems, relating free energy to the external work performed in nonequilibrium transitions. In practice, the required ensemble average of the Boltzmann weights of infinite nonequilibrium transitions is estimated as a finite sample average, resulting in the so‐called Jarzynski estimator, Δ F ^ J . Alternatively, the second‐order approximation of the Jarzynski equality, though seldom invoked, is exact for Gaussian distributions and gives rise to the Fluctuation‐Dissipation estimator Δ F ^ FD . Here we derive the parametric maximum‐likelihood estimator (MLE) of the free energy Δ F ^ ML considering unidirectional work distributions belonging to Gaussian or Gamma families, and compare this estimator to Δ F ^ J . We further consider bidirectional work distributions belonging to the same families, and compare the corresponding bidirectional Δ F ^ ML ∗ to the Bennett acceptance ratio ( Δ F ^ BAR ) estimator. We show that, for Gaussian unidirectional work distributions, Δ F ^ FD is in fact the parametric MLE of the free energy, and as such, the most efficient estimator for this statistical family. We observe that Δ F ^ ML and Δ F ^ ML ∗ perform better than Δ F ^ J and Δ F ^ BAR, for unidirectional and bidirectional distributions, respectively. These results illustrate that the characterization of the underlying work distribution permits an optimal use of the Jarzynski equality. © 2018 Wiley Periodicals, Inc. Abstract : The Jarzynski equality is routinely used to compute equilibrium free‐energy differences from a set of non‐equilibrium work values. From a computational approach, these non‐equilibrium work values are often generated using steered Molecular Dynamics, and, under certain conditions a Gaussian distribution of nonequilibrium work values is to be anticipated. We highlight the advantage of parametric maximum‐likelihood estimators over the conventional use of the Jarzynski equality, when information about the distribution of nonequilibirum work values is available.