Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included. CONTENTS

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...

We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying "equilibrium glass transition". After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, aging and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.

The interactions of tiny objects with their environment are dominated by thermal fluctuations. Guided by theory and assisted by micromanipulation tools, scientists have begun to study such interactions in detail.Comment: PDF file, 13 pages. Long version of the paper published in Physics Toda

Abstract. This review reports on the research done during the past years on violations of the fluctuation-dissipation theorem (FDT) in glassy systems. It is focused on the existence of a quasi-fluctuation-dissipation theorem (QFDT) in glassy systems and the currently supporting knowledge gained from numerical simulation studies.It covers a broad range of non-stationary aging and stationary driven systems such as structural-glasses, spin-glasses, coarsening systems, ferromagnetic models at criticality, trap models, models with entropy barriers, kinetically constrained models, sheared systems and granular media. The review is divided into four main parts: 1) An introductory section explaining basic notions related to the existence of the FDT in equilibrium and its possible extension to the glassy regime (QFDT), 2) A description of the basic analytical tools and results derived in the framework of some exactly solvable models, 3) A detailed report of the current evidence in favour of the QFDT and 4) A brief digression on the experimental evidence in its favour. This review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the QFDT as well as for the more expert readers who may be interested in more specific results.

Abstract. I review single-molecule experiments (SME) in biological physics. Recent technological developments have provided the tools to design and build scientific instruments of high enough sensitivity and precision to manipulate and visualize individual molecules and measure microscopic forces. Using SME it is possible to: manipulate molecules one at a time and measure distributions describing molecular properties; characterize the kinetics of biomolecular reactions and; detect molecular intermediates. SME provide the additional information about thermodynamics and kinetics of biomolecular processes. This complements information obtained in traditional bulk assays. In SME it is also possible to measure small energies and detect large Brownian deviations in biomolecular reactions, thereby offering new methods and systems to scrutinize the basic foundations of statistical mechanics. This review is written at a very introductory level emphasizing the importance of SME to scientists interested in knowing the common playground of ideas and the interdisciplinary topics accessible by these techniques.

In 1997, Jarzynski proved a remarkable equality that allows one to compute the equilibrium free-energy difference ⌬F between two states from the probability distribution of the nonequilibrium work W done on the system to switch between the states, e ؊⌬F͞kT ‫؍‬ ͗e ؊W͞kT ͘, [Jarzynski, C. (1997) Phys. Rev. Lett. 87, 2690 -2693]. The Jarzynski equality provides a powerful free-energy difference estimator from a set of N irreversible experiments and is closely related to free-energy perturbation, a common computational technique for estimating free-energy differences. Despite the many applications of the Jarzynski estimator, its behavior is only poorly understood. In this article we derive the large N limit for the Jarzynski estimator bias, variance, and mean square error that is correct for arbitrary perturbations. We then analyze the properties of the Jarzynski estimator for all N when the probability distribution of work values is Gaussian, as occurs, for example, in the near-equilibrium regime. This allows us to quantitatively compare it to two other free-energy difference estimators: the mean work estimator and the fluctuation-dissipation theorem estimator. We show that, for near-equilibrium switching, the Jarzynski estimator is always superior to the mean work estimator and is even superior to the fluctuation-dissipation estimator for small N. The Jarzynski-estimator bias is shown to be the dominant source of error in many cases. Our expression for the bias is used to develop a bias-corrected Jarzynski free-energy difference estimator in the nearequilibrium regime.A ccurate measurement and calculation of free-energy differences is central to our understanding of biological, chemical, and physical molecular processes. A common method of estimating the free-energy difference ⌬F ϭ F B Ϫ F A between two states, A and B, of a classical system in contact with a heat reservoir is to perturb the system to induce a transition between these states. The average work done in such a perturbation satisfies ͗W͘ Ն ⌬F, where equality holds if and only if the perturbation is infinitely slow. An average of the work values obtained by any finite time experiment or simulation therefore overestimates the true ⌬F. Until recently, recovering the equilibrium free energy from trajectories arbitrarily far from equilibrium was therefore thought to be impossible.In 1997, however, Jarzynski (1, 2) proved an identity that relates the probability distribution of nonequilibrium work values with the equilibrium free-energy difference between the two states:where ␤ ϭ (kT) Ϫ1 . On the left-hand side of this equation is an exponential of the equilibrium free-energy difference, and the right-hand side is an exponentially weighted average over an infinite number of nonequilibrium work trajectories, all started from the same initial equilibrium state. Although the second law of thermodynamics requires that the average work over all possible trajectories be greater than the free-energy difference, the work for an individual trajectory will occasional...

Accurate knowledge of the thermodynamic properties of nucleic acids is crucial to predicting their structure and stability. To date most measurements of base-pair free energies in DNA are obtained in thermal denaturation experiments, which depend on several assumptions. Here we report measurements of the DNA base-pair free energies based on a simplified system, the mechanical unzipping of single DNA molecules. By combining experimental data with a physical model and an optimization algorithm for analysis, we measure the 10 unique nearest-neighbor base-pair free energies with 0.1 kcal mol −1 precision over two orders of magnitude of monovalent salt concentration. We find an improved set of standard energy values compared with Unified Oligonucleotide energies and a unique set of 10 base-pair-specific salt-correction values. The latter are found to be strongest for AA/TT and weakest for CC/GG. Our unique energy values and salt corrections improve predictions of DNA unzipping forces and are fully compatible with melting temperatures for oligos. The method should make it possible to obtain free energies, enthalpies, and entropies in conditions not accessible by bulk methodologies.DNA thermodynamics | DNA unzipping | nearest-neighbor model | optical tweezers T he nearest-neighbor (NN) model (1-4) for DNA thermodynamics has been successfully applied to predict the free energy of formation of secondary structures in nucleic acids. The model estimates the free-energy change to form a double helix from independent strands as a sum over all of resulting bp and adjacent-bp stacks, depending on the constituent four bases of the stack, by using 10 nearest-neighbor base-pair (NNBP) energies. These energies themselves contain contributions from stacking, hydrogen-bonding, and electrostatic interactions as well as configurational entropy loss. Accurately predicting free energies has many applications in biological science: to predict self-assembled structures in DNA origami (5, 6); achievement of high selectivity in the hybridization of synthetic DNAs (7); antigene targeting and siRNA design (8); characterization of translocating motion of enzymes that mechanically disrupt nucleic acids (9); prediction of nonnative states (e.g., RNA misfolding) (10); and DNA guided crystallization of colloids (11).Some of the most reliable estimates of the NNBP energies to date have been obtained from thermal denaturation studies of DNA oligos and polymers (2). Although early studies showed large discrepancies in the NNBP values, nowadays they are remarkably consistent among several groups. In these studies it is assumed that duplexes melt in a two-state fashion. However this assumption is not often the case and a discrepancy between the values obtained using oligomers vs. polymers remains a persistent problem that has been attributed to many factors such as the slow dissociation kinetics induced by a population of transient nondenatured intermediates that develop during thermal denaturation experiments (12). Single-molecule techniques (13) circ...