Probability distributions of the work in the 2D-Ising model

Probability distributions of the magnetic work are computed for the 2D Ising model by means of Monte Carlo simulations. The system is first prepared at equilibrium for three temperatures below, at and above the critical point. A magnetic field is then applied and grown linearly at different rates. Probability distributions of the work are stored and free energy differences computed using the Jarzynski equality. Consistency is checked and the dynamics of the system is analyzed. Free energies and dissipated works are reproduced with simple models. The critical exponent $\delta$ is estimated in an usual manner.Comment: 12 pages, 6 figures. Comments are welcom

Work distribution for the driven harmonic oscillator with time-dependent strength: Exact solution and slow driving

We study the work distribution of a single particle moving in a harmonic oscillator with time-dependent strength. This simple system has a non-Gaussian work distribution with exponential tails. The time evolution of the corresponding moment generating function is given by two coupled ordinary differential equations that are solved numerically. Based on this result we study the behavior of the work distribution in the limit of slow but finite driving and show that it approaches a Gaussian distribution arbitrarily well

Fluctuation relations for heat engines in time-periodic steady states

A fluctuation relation for heat engines (FRHE) has been derived recently. In the beginning, the system is in contact with the cooler bath. The system is then coupled to the hotter bath and external parameters are changed cyclically, eventually bringing the system back to its initial state, once the coupling with the hot bath is switched off. In this work, we lift the condition of initial thermal equilibrium and derive a new fluctuation relation for the central system (heat engine) being in a time-periodic steady state (TPSS). Carnot's inequality for classical thermodynamics follows as a direct consequence of this fluctuation theorem even in TPSS. For the special cases of the absence of hot bath and no extraction of work, we obtain the integral fluctuation theorem for total entropy and the generalized exchange fluctuation theorem, respectively. Recently microsized heat engines have been realized experimentally in the TPSS. We numerically simulate the same model and verify our proposed theorems.Comment: 9 page

Estimate of the free energy difference in mechanical systems from work fluctuations: experiments and models

The work fluctuations of an oscillator in contact with a heat reservoir and driven out of equilibrium by an external force are studied experimentally. The oscillator dynamics is modeled by a Langevin equation. We find both experimentally and theoretically that, if the driving force does not change the equilibrium properties of the thermal fluctuations of this mechanical system, the free energy difference $\Delta F$ between two equilibrium states can be exactly computed using the Jarzynski equality (JE) and the Crooks relation (CR) \cite{jarzynski1, crooks1, jarzynski2}, independently of the time scale and amplitude of the driving force. The applicability limits for the JE and CR at very large driving forces are discussed. Finally, when the work fluctuations are Gaussian, we propose an alternative empirical method to compute $\Delta F$ which can be safely applied, even in cases where the JE and CR might not hold. The results of this paper are useful to compute $\Delta F$ in complex systems such as the biological ones.Comment: submitted to Journal of Statistical Mechanics: Theory and experimen

Feynman's ratchet and pawl: an exactly solvable model

We introduce a simple, discrete model of Feynman's ratchet and pawl, operating between two heat reservoirs. We solve exactly for the steady-state directed motion and heat flows produced, first in the absence and then in the presence of an external load. We show that the model can act both as a heat engine and as a refrigerator. We finally investigate the behavior of the system near equilibrium, and use our model to confirm general predictions based on linear response theory.Comment: 19 pages + 10 figures; somewhat tighter presentatio

Work and heat fluctuations in two-state systems: a trajectory thermodynamics formalism

Two-state models provide phenomenological descriptions of many different systems, ranging from physics to chemistry and biology. We investigate work fluctuations in an ensemble of two-state systems driven out of equilibrium under the action of an external perturbation. We calculate the probability density P(W) that a work equal to W is exerted upon the system along a given non-equilibrium trajectory and introduce a trajectory thermodynamics formalism to quantify work fluctuations in the large-size limit. We then define a trajectory entropy S(W) that counts the number of non-equilibrium trajectories P(W)=exp(S(W)/kT) with work equal to W. A trajectory free-energy F(W) can also be defined, which has a minimum at a value of the work that has to be efficiently sampled to quantitatively test the Jarzynski equality. Within this formalism a Lagrange multiplier is also introduced, the inverse of which plays the role of a trajectory temperature. Our solution for P(W) exactly satisfies the fluctuation theorem by Crooks and allows us to investigate heat-fluctuations for a protocol that is invariant under time reversal. The heat distribution is then characterized by a Gaussian component (describing small and frequent heat exchange events) and exponential tails (describing the statistics of large deviations and rare events). For the latter, the width of the exponential tails is related to the aforementioned trajectory temperature. Finite-size effects to the large-N theory and the recovery of work distributions for finite N are also discussed. Finally, we pay particular attention to the case of magnetic nanoparticle systems under the action of a magnetic field H where work and heat fluctuations are predicted to be observable in ramping experiments in micro-SQUIDs.Comment: 28 pages, 14 figures (Latex

Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms

The fluctuations in nonequilibrium systems are under intense theoretical and experimental investigation. Topical ``fluctuation relations'' describe symmetries of the statistical properties of certain observables, in a variety of models and phenomena. They have been derived in deterministic and, later, in stochastic frameworks. Other results first obtained for stochastic processes, and later considered in deterministic dynamics, describe the temporal evolution of fluctuations. The field has grown beyond expectation: research works and different perspectives are proposed at an ever faster pace. Indeed, understanding fluctuations is important for the emerging theory of nonequilibrium phenomena, as well as for applications, such as those of nanotechnological and biophysical interest. However, the links among the different approaches and the limitations of these approaches are not fully understood. We focus on these issues, providing: a) analysis of the theoretical models; b) discussion of the rigorous mathematical results; c) identification of the physical mechanisms underlying the validity of the theoretical predictions, for a wide range of phenomena.Comment: 44 pages, 2 figures. To appear in Nonlinearity (2007

Single-molecule experiments in biological physics: methods and applications

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. The review discusses SME from an experimental perspective, first exposing the most common experimental methodologies and later presenting various molecular systems where such techniques have been applied. I briefly discuss experimental techniques such as atomic-force microscopy (AFM), laser optical tweezers (LOT), magnetic tweezers (MT), biomembrane force probe (BFP) and single-molecule fluorescence (SMF). I then present several applications of SME to the study of nucleic acids (DNA, RNA and DNA condensation), proteins (protein-protein interactions, protein folding and molecular motors). Finally, I discuss applications of SME to the study of the nonequilibrium thermodynamics of small systems and the experimental verification of fluctuation theorems. I conclude with a discussion of open questions and future perspectives.Comment: Latex, 60 pages, 12 figures, Topical Review for J. Phys. C (Cond. Matt