An expression is derived for the classical free energy difference between two configurations of a system, in terms of an ensemble of finite-time measurements of the work performed in parametrically switching from one configuration to the other. Two well-known equilibrium identities emerge as limiting cases of this result.

It has recently been shown that the Helmholtz free energy difference between two equilibrium configurations of a system may be obtained from an ensemble of finite-time (nonequilibrium) measurements of the work performed in switching an external parameter of the system. Here this result is established, as an identity, within the master equation formalism. Examples are discussed and numerical illustrations provided.

The reason we never observe violations of the second law of thermodynamics is in part a matter of statistics: When ∼1023 degrees of freedom are involved, the odds are overwhelmingly stacked against the possibility of seeing significant deviations away from the mean behavior. As we turn our attention to smaller systems, however, statistical fluctuations become more prominent. In recent years it has become apparent that the fluctuations of systems far from thermal equilibrium are not mere background noise, but satisfy strong, useful, and unexpected properties. In particular, a proper accounting of fluctuations allows us to rewrite familiar inequalities of macroscopic thermodynamics as equalities. This review describes some of this progress, and argues that it has refined our understanding of irreversibility and the second law.

Atomic force microscopes and optical tweezers are widely used to probe the mechanical properties of individual molecules and molecular interactions, by exerting mechanical forces that induce transitions such as unfolding or dissociation. These transitions often occur under nonequilibrium conditions and are associated with hysteresis effects-features usually taken to preclude the extraction of equilibrium information from the experimental data. But fluctuation theorems 1-5 allow us to relate the work along nonequilibrium trajectories to thermodynamic free-energy differences. They have been shown to be applicable to single-molecule force measurements 6 and have already provided information on the folding free energy of a RNA hairpin 7,8 . Here we show that the Crooks fluctuation theorem 9 can be used to determine folding free energies for folding and unfolding processes occurring in weak as well as strong nonequilibrium regimes, thereby providing a test of its validity under such conditions. We use optical tweezers 10 to measure repeatedly the mechanical work associated with the unfolding and refolding of a small RNA hairpin 11 and an RNA three-helix junction 12 . The resultant work distributions are then analysed according to the theorem and allow us to determine the difference in folding free energy between an RNA molecule and a mutant differing only by one base pair, and the thermodynamic stabilizing effect of magnesium ions on the RNA structure.The Crooks fluctuation theorem 9 (CFT) predicts a symmetry relation in the work fluctuations associated with the forward and reverse changes a system undergoes as it is driven away from thermal equilibrium by the action of an external perturbation. This theorem applies to processes that are microscopically reversible, and its experimental evaluation in small systems is crucial to understand better the foundations of nonequilibrium physics 13 . A consequence of the CFT is Jarzynski's equality 14 , which relates the equilibrium free-energy difference ΔG between two equilibrium states to an exponential average (denoted by angle brackets) of the work done on the system, W, taken over an infinite number of repeated none-quilibrium experiments, exp The equality has been developed 6 into a formalism that allows us to use nonequilibrium single-molecule pulling experiments to reconstruct free-energy profiles or potentials of mean force 15 along reaction coordinates. Experimental testing of Jarzynski's equality in single-molecule force experiments 16 used the P5ab RNA hairpin 7,8 , which can be folded and unfolded quasi-reversibly. But for processes that occur far from equilibrium, the applicability of Jarzynski's equality is hampered by large statistical uncertainties that arise from the sensitivity of the exponential average to rare events 17,18 (low values of W). Moreover, although the equality 〈W〉 = ΔG holds for processes occurring near equilibrium, spatial drift in the experimental system usually makes it difficult in practice to extract unfolding free energies using...

As access to computational resources continues to increase, free-energy calculations have emerged as a powerful tool that can play a predictive role in a wide range of research areas. Yet, the reliability of these calculations can often be improved significantly if a number of precepts, or good practices, are followed. Although the theory upon which these good practices rely has largely been known for many years, it is often overlooked or simply ignored. In other cases, the theoretical developments are too recent for their potential to be fully grasped and merged into popular platforms for the computation of free-energy differences. In this contribution, the current best practices for carrying out free-energy calculations using free energy perturbation and nonequilibrium work methods are discussed, demonstrating that at little to no additional cost, free-energy estimates could be markedly improved and bounded by meaningful error estimates. Monitoring the probability distributions that underlie the transformation between the states of interest, performing the calculation bidirectionally, stratifying the reaction pathway, and choosing the most appropriate paradigms and algorithms for transforming between states offer significant gains in both accuracy and precision.