Fluctuation Theorems

Fluctuation theorems, which have been developed over the past 15 years, have resulted in fundamental breakthroughs in our understanding of how irreversibility emerges from reversible dynamics, and have provided new statistical mechanical relationships for free energy changes. They describe the statistical fluctuations in time-averaged properties of many-particle systems such as fluids driven to nonequilibrium states, and provide some of the very few analytical expressions that describe nonequilibrium states. Quantitative predictions on fluctuations in small systems that are monitored over short periods can also be made, and therefore the fluctuation theorems allow thermodynamic concepts to be extended to apply to finite systems. For this reason, fluctuation theorems are anticipated to play an important role in the design of nanotechnological devices and in understanding biological processes. These theorems, their physical significance and results for experimental and model systems are discussed.Comment: A review, submitted to Annual Reviews in Physical Chemistry, July 2007 Acknowledgements corrected in revisio

Equilibrium binding energies from fluctuation theorems and force spectroscopy simulations

Brownian dynamics simulations are used to study the detachment of a particle from a substrate. Although the model is simple and generic, we attempt to map its energy, length and time scales onto a specific experimental system, namely a bead that is weakly bound to a cell and then removed by an optical tweezer. The external driving force arises from the combined optical tweezer and substrate potentials, and thermal fluctuations are taken into account by a Brownian force. The Jarzynski equality and Crooks' fluctuation theorem are applied to obtain the equilibrium free energy difference between the final and initial states. To this end, we sample non--equilibrium work trajectories for various tweezer pulling rates. We argue that this methodology should also be feasible experimentally for the envisioned system. Furthermore, we outline how the measurement of a whole free energy profile would allow the experimentalist to retrieve the unknown substrate potential by means of a suitable deconvolution. The influence of the pulling rate on the accuracy of the results is investigated, and umbrella sampling is used to obtain the equilibrium probability of particle escape for a variety of trap potentials.Comment: 21 pages, 11 figures, To appear in Soft Matte

Non-equilibrium umbrella sampling applied to force spectroscopy of soft matter

Physical systems often respond on a timescale which is longer than that of the measurement. This is particularly true in soft matter where direct experimental measurement, for example in force spectroscopy, drives the soft system out of equilibrium and provides a non-equilibrium measure. Here we demonstrate experimentally for the first time that equilibrium physical quantities (such as the mean square displacement) can be obtained from non-equilibrium measurements via umbrella sampling. Our model experimental system is a bead fluctuating in a time-varying optical trap. We also show this for simulated force spectroscopy on a complex soft molecule--a piston-rotaxane

Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales

We experimentally demonstrate the fluctuation theorem, which predicts appreciable and measurable violations of the second law of thermodynamics for small systems over short time scales, by following the trajectory of a colloidal particle captured in an optical trap that is translated relative to surrounding water molecules. From each particle trajectory, we calculate the entropy production/consumption over the duration of the trajectory and determine the fraction of second law–defying trajectories. Our results show entropy consumption can occur over colloidal length and time scales

A Polymer End-Tethered to a Potential Stripe: A Simple Example of an Escape Transition

ABSTRACT: We study the problem of a single ideal polymer chain tethered to a surface at the midpoint of a repulsive potential stripe. If the potential is very weak, the chain remains unperturbed. However, as the potential is increased, the chain conformation undergoes a sudden change. The chain forms a tether to the edge of the stripe and moves most of the monomers off to the region of lower potential. This is a simple example of an escape transition previously discussed for compression of polymer chains. We show how these two systems are analogous and clear up some controversy regarding the exact form of the force versus height curve for the compressive system

Crumpling a Thin Sheet

Crumpled sheets have a surprisingly large resistance to further compression. We have studied the crumpling of thin sheets of Mylar under different loading conditions. When placed under a fixed compressive force, the size of a crumpled material decreases logarithmically in time for periods up to three weeks. We also find hysteretic behavior when measuring the compression as a function of applied force. By using a pre-treating protocol, we control this hysteresis and find reproducible scaling behavior for the size of the crumpled material as a function of the applied force.Comment: revtex 4 pages, 6 eps figures submitted to Phys Rev. let

Reversibility in nonequilibrium trajectories of an optically trapped particle

The measure of irreversibility as the dissipation function that serves as the quantitative argument in the fluctuation theorem (FT) was investigated. The FT describes the system's thermodynamic irreversibility developed in time from a completely thermodynamically reversibble system at short times to a thermodynamically irreversible one at infinitely long times. It was observed that the ensemble average of ωt was positive definite irrespective of the system for which it was constructed. It was found that the different expressions for ωt can arise in stochastic and deterministic systems

Comparison of work fluctuation relations

We compare two predictions regarding the microscopic fluctuations of a system that is driven away from equilibrium: one due to Crooks [J. Stat. Phys. 90, 1481 (1998)] which has gained recent attention in the context of nonequilibrium work and fluctuation theorems, and an earlier, analogous result obtained by Bochkov and Kuzovlev [Zh. Eksp. Teor. Fiz. 72(1), 238247 (1977)]. Both results quantify irreversible behavior by comparing probabilities of observing particular microscopic trajectories during thermodynamic processes related by time-reversal, and both are expressed in terms of the work performed when driving the system away from equilibrium. By deriving these two predictions within a single, Hamiltonian framework, we clarify the precise relationship between them, and discuss how the different definitions of work used by the two sets of authors gives rise to different physical interpretations. We then obtain a extended fluctuation relation that contains both the Crooks and the Bochkov-Kuzovlev results as special cases.Comment: 14 pages with 1 figure, accepted for publication in the Journal of Statistical Mechanic