Temporal asymmetry of fluctuations in nonequilibrium steady states

Temporal asymmetries of fluctuation paths in nonequilibrium microscopic shearing systems are observed for the first time. Inspired by theories that predict asymmetry of fluctuation paths in stochastic dynamics, we focus on deterministic reversible particle models, which represent a small part of a macroscopic system. We have monitored and measured the asymmetry of the fluctuation paths of various observables as they go away from and towards the mean. The understanding of such asymmetries may scatter light on how irreversibility emerges from the microscopic reversible dynamics and on the behavior of mesoscopic (nanoscale) systems.     

Kinetic paths, time scale, and underlying landscapes: a path integral framework to study global natures of nonequilibrium systems and networks

We developed a general framework to quantify three key ingredients for dynamics of nonequilibrium systems through path integrals in length space. First, we identify dominant kinetic paths as the ones with optimal weights, leading to effective reduction of dimensionality or degrees of freedom from exponential to polynomial so large systems can be treated. Second, we uncover the underlying nonequilibrium potential landscapes from the explorations of the state space through kinetic paths. We apply our framework to a specific example of nonequilibrium network system: lambda phage genetic switch. Two distinct basins of attractions emerge. The dominant kinetic paths from one basin to another are irreversible and do not follow the usual steepest descent or gradient path along the landscape. It reflects the fact that the dynamics of nonequilibrium systems is not just determined by potential gradient but also the residual curl flux force, suggesting experiments to test theoretical predictions. Third, we have calculated dynamic transition time scales from one basin to another critical for stability of the system through instantons. Theoretical predictions are in good agreements with wild type and mutant experiments. We further uncover the correlations between the kinetic transition time scales and the underlying landscape topography: the barrier heights along the dominant paths. We found that both the dominant paths and the landscape are relatively robust against the influences of external environmental perturbations and the system tends to dissipate less with less fluctuations. Our general framework can be applied to other nonequilibrium systems.     

Simulating rare events in equilibrium or nonequilibrium stochastic systems

We present three algorithms for calculating rate constants and sampling transition paths for rare events in simulations with stochastic dynamics. The methods do not require a priori knowledge of the phase-space density and are suitable for equilibrium or nonequilibrium systems in stationary state. All the methods use a series of interfaces in phase space, between the initial and final states, to generate transition paths as chains of connected partial paths, in a ratchetlike manner. No assumptions are made about the distribution of paths at the interfaces. The three methods differ in the way that the transition path ensemble is generated. We apply the algorithms to kinetic Monte Carlo simulations of a genetic switch and to Langevin dynamics simulations of intermittently driven polymer translocation through a pore. We find that the three methods are all of comparable efficiency, and that all the methods are much more efficient than brute-force simulation.     

How the nature of an observation affects single-trajectory entropies

Projection of a Markov process with constant rates of transition to a small number of observable aggregated states can result in complex kinetics with memory. Here, we define the entropy production along a single sequence of aggregated states and show that it obeys detailed and integral fluctuation theorems. More importantly, we prove that projection shifts the distribution of entropy production over the ensemble of paths for a nonequilibrium process toward one characteristic of a system at equilibrium. This statement represents an analog of the second law of thermodynamics for path-dependent entropies and thus a new form of constraint of irreversible systems. Numeric examples are presented to illustrate these ideas.     

Equilibrium calculations of viscosity and thermal conductivity across a solid-liquid interface using boundary fluctuations

We calculate viscosity and thermal conductivity in systems of Lennard-Jones particles consisting of coexisting solid and liquid with different interface wetting properties using the recently developed equilibrium boundary fluctuation theory. We compare the slip length and equivalent liquid length obtained from these calculations with those obtained from nonequilibrium molecular dynamics. The equilibrium and nonequilibrium calculations of the slip length and the sum of the thermal equivalent lengths are in good agreement. We conclude that for both interfacial properties, the nonequilibrium simulations were probing the linear response. The significant dependence of the intrinsic equivalence length on the interfacial temperature difference used to generate the thermal gradient is explained as a consequence of the different thermodynamic states of the two interfaces.     

Equilibrium free energy estimates based on nonequilibrium work relations and extended dynamics

Jarzynski's relation and the fluctuation theorem have established important connections between nonequilibrium statistical mechanics and equilibrium thermodynamics. In particular, an exact relationship between the equilibrium free energy and the nonequilibrium work is useful for computer simulations. In this paper, we exploit the fact that the free energy is a state function, independent of the pathway taken to change the equilibrium ensemble. We show that a generalized expression is advantageous for computer simulations of free energy differences. Several methods based on this idea are proposed. The accuracy and efficiency of the proposed methods are evaluated with a model problem.     

A proof of Jarzynski's nonequilibrium work theorem for dynamical systems that conserve the canonical distribution

We present a derivation of the Jarzynski [Phys. Rev. Lett. 78, 2690 (1997)] identity and the Crooks [J. Stat. Phys. 90, 1481 (1998)] fluctuation theorem for systems governed by deterministic dynamics that conserves the canonical distribution such as Hamiltonian dynamics, Nose-Hoover dynamics, Nose-Hoover chains, and Gaussian isokinetic dynamics. The proof is based on a relation between the heat absorbed by the system during the nonequilibrium process and the Jacobian of the phase flow generated by the dynamics.     

Escorted free energy simulations

We describe a strategy to improve the efficiency of free energy estimates by reducing dissipation in nonequilibrium Monte Carlo simulations. This strategy generalizes the targeted free energy perturbation approach [C. Jarzynski, Phys. Rev. E 65, 046122 (2002)] to nonequilibrium switching simulations, and involves generating artificial, "escorted" trajectories by coupling the evolution of the system to updates in external work parameter. Our central results are: (1) a generalized fluctuation theorem for the escorted trajectories, and (2) estimators for the free energy difference ΔF in terms of these trajectories. We illustrate the method and its effectiveness on model systems.     

A differential fluctuation theorem

We derive a nonequilibrium thermodynamics identity (the "differential fluctuation theorem") that connects forward and reverse joint probabilities of nonequilibrium work and of arbitrary generalized coordinates corresponding to states of interest. This identity allows us to estimate the free energy difference between domains of these states. Our results follow from a general symmetry relation between averages over nonequilibrium forward and backward path functions derived by Crooks [Crooks, G. E. Phys. Rev. E 2000, 61, 2361-2366]. We show how several existing nonequilibrium thermodynamic identities can be obtained directly from the differential fluctuation theorem. We devise an approach for measuring conformational free energy differences, and we demonstrate its applicability to the analysis of molecular dynamics simulations by estimating the free energy difference between two conformers of the alanine dipeptide model system. We anticipate that these developments can be applied to the analysis of laboratory experiments.     

Entropy-energy decomposition from nonequilibrium work trajectories

We derive expressions for the equilibrium entropy and energy changes in the context of the Jarzynski equality relating nonequilibrium work to equilibrium free energy. The derivation is based on a stochastic path integral technique that reweights paths at different temperatures. Stochastic dynamics generated by either a Langevin equation or a Metropolis Monte Carlo scheme are treated. The approach enables the entropy-energy decomposition from trajectories evolving at a single-temperature and does not require simulations or measurements at two or more temperatures. Both finite difference and analytical formulae are derived. Testing is performed on a prototypical model system and the method is compared with existing thermodynamic integration and thermodynamic perturbation approaches for entropy-energy decomposition. The new formulae are also put in the context of more general, dynamics-independent expressions that derive from either a fluctuation theorem or the Feynman-Kac theorem.     

Extending the fluctuation theorem to describe reaction coordinates

The fluctuation theorem describes the distribution of work done on small systems which have been pushed out of equilibrium in response to an external field. The theorem has recently been a subject of much interest for describing single-molecule experiments and simulations. In this communication, it is shown how the fluctuation theorem can be extended to describe fluctuations not only in the work done on a system, but also in a reaction coordinate. The extension explored in this work allows for a generalized derivation of Hummer and Szabo's expression (G. Hummer and A. Szabo, Proc. Natl. Acad. Sci. 98, 3658 (2001)) for reconstructing the potential of mean force from nonequilibrium trajectories. The derivation demonstrates how implementation of this expression can be more easily facilitated. Atomistic simulations of a biomolecular system are presented which support these results.     

New observations regarding deterministic, time-reversible thermostats and Gauss's principle of least constraint

Deterministic thermostats are frequently employed in nonequilibrium molecular dynamics simulations in order to remove the heat produced irreversibly over the course of such simulations. The simplest thermostat is the Gaussian thermostat, which satisfies Gauss's principle of least constraint and fixes the peculiar kinetic energy. There are of course infinitely many ways to thermostat systems, e.g., by fixing sigma(i)/p(i)/mu+l. In the present paper we provide, for the first time, convincing arguments as to why the conventional Gaussian isokinetic thermostat (mu = 1) is unique in this class. We show that this thermostat minimizes the phase space compression and is the only thermostat for which the conjugate pairing rule holds. Moreover, it is shown that for finite sized systems in the absence of an applied dissipative field, all other thermostats (mu not = 1) perform work on the system in the same manner as a dissipative field while simultaneously removing the dissipative heat so generated. All other thermostats (mu not = 1) are thus autodissipative. Among all mu, thermostats, only the mu = 1 Gaussian thermostat permits an equilibrium state.     

The influence of thermostats and manostats on reverse nonequilibrium molecular dynamics calculations of fluid viscosities

Reverse nonequilibrium molecular dynamics to calculate the shear viscosity of Lennard-Jones liquids was extended to simulations at constant number of particles, constant volume, and constant pressure using a Berendsen thermostat and a Berendsen manostat. Using additional systems such as water and hexane, we also report on the performance of shear viscosity calculations of systems with electrostatic and nontrivial intramolecular interactions when a manostat is applied. We compare the shear viscosities of simulations using no coupling, only temperature coupling, and temperature and pressure coupling and characterize discrepancies, where observed. From this, we deduce guidelines for when and how manostats can be usefully applied in reverse nonequilibrium simulations.     

Statistical mechanical theory for steady state systems. VII. Nonlinear theory

The second entropy theory for nonequilibrium thermodynamics is extended to the nonlinear regime and to systems of mixed parity (even and odd functions of molecular velocities). The steady state phase space probability density is given for systems of mixed parity. The nonlinear transport matrix is obtained and it is shown to yield the analog of the linear Onsager-Casimir reciprocal relations. Its asymmetric part contributes to the flux and to the production of second entropy. The nonlinear transport matrix is not simply expressible as a Green-Kubo fluctuation equilibrium time correlation function. However, here the first nonlinear correction to the transport coefficient is given explicitly as a type of the Green-Kubo equilibrium time correlation function. The theory is illustrated by application to chemical kinetics.     

Approximating nonequilibrium processes using a collection of surrogate diffusion models

The surrogate process approximation (SPA) is applied to model the nonequilibrium dynamics of a reaction coordinate (RC) associated with the unfolding and refolding processes of a deca-alanine peptide at 300 K. The RC dynamics, which correspond to the evolution of the end-to-end distance of the polypeptide, are produced by steered molecular dynamics (SMD) simulations and approximated using overdamped diffusion models. We show that the collection of (estimated) SPA models contain structural information "orthogonal" to the RC monitored in this study. Functional data analysis ideas are used to correlate functions associated with the fitted SPA models with the work done on the system in SMD simulations. It is demonstrated that the shape of the nonequilibrium work distributions for the unfolding and refolding processes of deca-alanine can be predicted with functional data analysis ideas using a relatively small number of simulated SMD paths for calibrating the SPA diffusion models.     

Density-dependent analysis of nonequilibrium paths improves free energy estimates II. A Feynman-Kac formalism

The nonequilibrium fluctuation theorems have paved the way for estimating equilibrium thermodynamic properties, such as free energy differences, using trajectories from driven nonequilibrium processes. While many statistical estimators may be derived from these identities, some are more efficient than others. It has recently been suggested that trajectories sampled using a particular time-dependent protocol for perturbing the Hamiltonian may be analyzed with another one. Choosing an analysis protocol based on the nonequilibrium density was empirically demonstrated to reduce the variance and bias of free energy estimates. Here, we present an alternate mathematical formalism for protocol postprocessing based on the Feynmac-Kac theorem. The estimator that results from this formalism is demonstrated on a few low-dimensional model systems. It is found to have reduced bias compared to both the standard form of Jarzynski's equality and the previous protocol postprocessing formalism.     

Reciprocating motion on the nanoscale

This paper analyzes the confined motion of a Brownian particle fluctuating between two conformational states with different potential profiles and different position-dependent rate constants of the transitions, the fluctuations arising from both thermal (equilibrium) and external (nonequilibrium) noise. The model illustrates a mechanism to transduce, on the nanoscale, the energy of nonequilibrium fluctuations into mechanical energy of reciprocating motion. Expressions for the reciprocating velocity and the efficiency of energy conversion are derived. These expressions are treated in more detail in the slow-fluctuation (quasi-equilibrium) regime, by simple perturbation theory arguments, and in the fast fluctuation limit, in terms of the potential of mean force. A notable observation is that the generalized driving force of the reciprocating motion is caused by two sources: the energy contribution due to the difference between the potential profiles of the states and the entropic contribution due to the difference between the position-dependent rate constants. Two illustrative examples are presented, where one of the two sources can be ignored and an exact solution is allowed. Among other aspects, we also discuss the ways to construct a molecular motor based on the reciprocating engine.     

Nonequilibrium membrane fluctuations driven by active proteins

We extend a model for nonthermal membrane undulations driven by active (adenosine triphosphate-dependent or light-harvesting) membrane proteins [N. Gov, Phys. Rev. Lett. 93, 268104 (2004)]. The present model accounts for the fact that proteins can diffuse laterally across the membrane surface and that individual proteins are expected to exert forces preferentially in one normal direction over the other (due to their orientation within the bilayer). The addition of these effects alters the scaling of fluctuation amplitudes with system size. Additionally, theoretical arguments and dynamic simulations both suggest that, in certain regimes, the probability distribution of fluctuation amplitudes is expected to be non-Gaussian (in contrast to thermal systems).     

Path ensembles and path sampling in nonequilibrium stochastic systems

Markovian models based on the stochastic master equation are often encountered in single molecule dynamics, reaction networks, and nonequilibrium problems in chemistry, physics, and biology. An efficient and convenient method to simulate these systems is the kinetic Monte Carlo algorithm which generates continuous-time stochastic trajectories. We discuss an alternative simulation method based on sampling of stochastic paths. Utilizing known probabilities of stochastic paths, it is possible to apply Metropolis Monte Carlo in path space to generate a desired ensemble of stochastic paths. The method is a generalization of the path sampling idea to stochastic dynamics, and is especially suited for the analysis of rare paths which are not often produced in the standard kinetic Monte Carlo procedure. Two generic examples are presented to illustrate the methodology.     

A gentler approach to RNEMD: nonisotropic velocity scaling for computing thermal conductivity and shear viscosity

We present a new method for introducing stable nonequilibrium velocity and temperature gradients in molecular dynamics simulations of heterogeneous systems. This method extends earlier reverse nonequilibrium molecular dynamics (RNEMD) methods which use momentum exchange swapping moves. The standard swapping moves can create nonthermal velocity distributions and are difficult to use for interfacial calculations. By using nonisotropic velocity scaling (NIVS) on the molecules in specific regions of a system, it is possible to impose momentum or thermal flux between regions of a simulation while conserving the linear momentum and total energy of the system. To test the method, we have computed the thermal conductivity of model liquid and solid systems as well as the interfacial thermal conductivity of a metal-water interface. We find that the NIVS-RNEMD improves the problematic velocity distributions that develop in other RNEMD methods.