Confidence bounds for the Jarzynski estimator

The Jarzynski equality relates the free energy differences between the equilibrium states of a system and the measurements of non-equilibrium work performed to move the system between the states. The determination of free energy differences using this relationship plays a crucial role in the study of physical, chemical, and biological systems at small scales. The Jarzynski equality computes the free energy differences using the mean of the exponentiated work and is thus asymptotic in nature. However, in practice, work can be measured only a finite number of times, leading to errors in the estimates. Studies that quantify this error are limited in number and scope. We provide non-restrictive and rigorous confidence bounds on the free energy difference estimates in this setting. These bounds are valid for a large class of work distributions and so are applicable for a broad variety of experiments that take the system far from equilibrium. Also, the bounds depend only on the finite samples of work measured in an experiment. This enables one to specify the largest error acceptable in the estimates and get a stopping criterion on the experimental trials. We include a Python-based toolbox that implements the bounds and showcase an empirical study that validates our claims.