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Extensions of statistical mechanics are routinely being used to infer free energies from the work performed over single-molecule nonequilibrium trajectories. A key element of this approach is the ubiquitous expression dW/dt=partial H(x,t)/ partial t which connects the microscopic work W performed by a time-dependent force on the coordinate x with the corresponding Hamiltonian H(x,t) at time t. Here we show that this connection, as pivotal as it is, cannot be used to estimate free energy changes. We discuss the implications of this result for single-molecule experiments and atomistic molecular simulations and point out possible avenues to overcome these limitations.
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The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to non-equilibrium measurements of the work. These relations extend to single-molecule experiments that have probed the finite-time thermodynamics of proteins and nucleic acids. The effects of experimental error and instrument noise have not previously been considered. Here, we present a Bayesian formalism for estimating free-energy changes from non-equilibrium work measurements that compensates for instrument noise and combines data from multiple driving protocols. We reanalyze a recent set of experiments in which a single RNA hairpin is unfolded and refolded using optical tweezers at three different rates. Interestingly, the fastest and farthest-from-equilibrium measurements contain the least instrumental noise, and therefore provide a more accurate estimate of the free energies than a few slow, more noisy, near-equilibrium measurements. The methods we propose here will extend the scope of single-molecule experiments; they can be used in the analysis of data from measurements with AFM, optical, and magnetic tweezers.
A compressed knotted ring polymer in a confining cavity is modelled by a knotted lattice polygon confined in a cube in ${mathbb Z}^3$. The GAS algorithm [17] is used to sample lattice polygons of fixed knot type in a confining cube and to estimate the free energy of confined lattice knots. Lattice polygons of knot types the unknot, the trefoil knot, and the figure eight knot, are sampled and the free energies are estimated as functions of the concentration of monomers in the confining cube. The data show that the free energy is a function of knot type at low concentrations, and (mean-field) Flory-Huggins theory [12,15] is used to model the free energy as a function of monomer concentration. The Flory interaction parameter of knotted lattice polygons in ${mathbb Z}^3$ is also estimated.
We experimentally realize protocols that allow to extract work beyond the free energy difference from a single electron transistor at the single thermodynamic trajectory level. With two carefully designed out-of-equilibrium driving cycles featuring kicks of the control parameter, we demonstrate work extraction up to large fractions of $k_BT$ or with probabilities substantially greater than 1/2, despite zero free energy difference over the cycle. Our results are explained in the framework of nonequilibrium fluctuation relations. We thus show that irreversibility can be used as a resource for optimal work extraction even in the absence of feedback from an external operator.
We introduce a novel and powerful method for exploring the properties of the multidimensional free energy surfaces of complex many-body systems by means of a coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates.A characteristic feature of this dynamics is the presence of a history-dependent potential term that, in time, fills the minima in the free energy surface, allowing the efficient exploration and accurate determination of the free energy surface as a function of the collective coordinates. We demonstrate the usefulness of this approach in the case of the dissociation of a NaCl molecule in water and in the study of the conformational changes of a dialanine in solution.
We introduce and implement a Monte Carlo scheme to study the equilibrium statistics of polymers in the globular phase. It is based on a model of interacting elastic lattice polymers and allows a sufficiently good sampling of long and compact configurations, an essential prerequisite to study the scaling behaviour of free energies. By simulating interacting self-avoiding rings at several temperatures in the collapsed phase, we estimate both the bulk and the surface free energy. Moreover from the corresponding estimate of the entropic exponent $alpha-2$ we provide evidence that, unlike for swollen and $Theta$-point rings, the hyperscaling relation is not satisfied for globular rings.
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