Fluctuations In Nonequilibrium Systems Research Articles

Special issue on current trends in computational and experimental techniques in nonlinear dynamics

We present an introduction and review for the EPJ ST Special Issue on Current Trends in Computational and Experimental Techniques in nonlinear dynamics. This volume consists of 39 articles which consider topics of interest that arise in current research in nonlinear dynamics: complex systems, fluid flow problems, conservative systems, statistical fluctuations in non-equilibrium systems, networks of coupled oscillators, and numerical analysis of nonlinear oscillations. It covers theory, experimental studies, and applications.

Thermodynamic uncertainty for run-and-tumble–type processes

Thermodynamic uncertainty relations have emerged as universal bounds on current fluctuations in non-equilibrium systems. Here we derive a new bound for a particular class of run-and-tumble–type processes using the mathematical framework of renewal-reward theory which can be applied to both Markovian and non-Markovian systems. We demonstrate the results for selected single-particle models as well as a variant of the asymmetric simple exclusion process with collective tumbles. Our bound is relatively tight for a broad parameter regime and only requires knowledge of the statistics of run lengths and the mean entropy production rate of tumbles.

Dynamics of Janus motors with microscopically reversible kinetics

Janus motors with chemically active and inactive hemispheres can operate only under nonequilibrium conditions where detailed balance is broken by fluxes of chemical species that establish a nonequilibrium state. A microscopic model for reversible reactive collisions on a Janus motor surface is constructed and shown to satisfy detailed balance. The model is used to study Janus particle reactive dynamics in systems at equilibrium where generalized chemical rate laws that include time-dependent rate coefficients with power-law behavior are shown to describe reaction rates. While maintaining reversible reactions on the Janus catalytic hemisphere, the system is then driven into a nonequilibrium steady state by fluxes of chemical species that control the chemical affinity. The statistical properties of the self-propelled Janus motor in this nonequilibrium steady state are investigated and compared with the predictions of a fluctuating thermodynamics theory. The model has utility beyond the examples presented here, since it allows one to explore various aspects of nonequilibrium fluctuations in systems with self-diffusiophoretic motors from a microscopic perspective.

Hydrodynamic fluctuations in quasi-two dimensional diffusion

We study diffusion of colloids on a fluid–fluid interface using particle simulations and fluctuating hydrodynamics. Diffusion on a two-dimensional interface with three-dimensional hydrodynamics is known to be anomalous, with the collective diffusion coefficient diverging like the inverse of the wavenumber. This unusual collective effect arises because of the compressibility of the fluid flow in the plane of the interface, and leads to a nonlinear nonlocal convolution term in the diffusion equation for the ensemble-averaged concentration. We extend the previous hydrodynamic theory to account for a species/color labeling of the particles, as necessary to model experiments based on fluorescent techniques. We study the magnitude and dynamics of density and color density fluctuations using a novel Brownian dynamics algorithm, as well as fluctuating hydrodynamics theory and simulation. We find that hydrodynamic coupling between a single tagged particle and collective density fluctuations leads to a reduction of the long-time self-diffusion coefficient, even for an ideal gas of non-interacting particles. This unexpected finding demonstrates that density functional theories that do not account for thermal fluctuations are incomplete even for ideal systems. Using linearized fluctuating hydrodynamics theory, we show that for diffusion on a fluid–fluid interface, nonequilibrium fluctuations of the total density are small compared to the equilibrium fluctuations, but fluctuations of color density are giant and exhibit a spectrum that decays as the inverse cubed power of the wavenumber. We confirm these predictions through Brownian dynamics simulations of diffusive mixing with two indistinguishable species. We also examine nonequilibrium fluctuations in systems with two-dimensional hydrodynamics, such as thin smectic films in vacuum. We find that nonequilibrium fluctuations are colossal and comparable in magnitude to the mean, and can be accurately modeled using numerical solvers for the nonlinear equations of fluctuating hydrodynamics.

Skewness of steady-state current fluctuations in nonequilibrium systems.

A skewness of the probability for instantaneous current fluctuations, in a nonequilibrium steady state, is observed experimentally in a dusty plasma. This skewness is attributed to the spatial asymmetry, which is imminent to the nonequilibrium systems due to the external hydrodynamic gradient. Using the modern framework of the large deviation theory, we extend the Onsager-Machlup ansatz for equilibrium fluctuations to systems with a preferred spatial direction, and provide a modulated Gaussian probability distribution, which is tested by simulations. This probability distribution is also of potential interest for other statistical disciplines. Connections with the principles of statistical mechanics, due to Boltzmann and Gibbs, are discussed as well.

Fluctuation relations for spin currents

The fluctuation theorem establishes general relations between transport coefficients and fluctuations in nonequilibrium systems. Recently there was much interest in quantum fluctuation relations for electric currents. Since charge carriers also carry spin, it is important to extend the fluctuation theorem to spin currents. We use the principle of microscopic reversibility to derive such theorem. As a consequence, we obtain a family of relations between transport coefficients and fluctuations of spin currents. We apply the relations to the spin Seebeck effect and rectification of spin currents. Our relations do not depend on a microscopic model and hence can be used to test the validity of theoretical approximations in spin-transport problems.

Cross response in non-equilibrium systems

We study the properties of fluctuations in non-equilibrium systems, focusing our attention on the cross correlations in response to external forcing. Two typical non-equilibrium systems are considered. One is a small coupled set of systems, each of which contacts a different heat bath. The other is an oscillatory system where energy input is balanced with dissipation due to nonlinear friction. Based on the Langevin equations for these systems, we investigate the non-equilibrium statistical properties such as violation of the Clausius relation in the slow change of the parameter, and fluctuations and response around the stationary state. Our main concern is the role played by the probability current.

Indications of Universal Excess Fluctuations in Nonequilibrium Systems

The fluctuation in electric current in nonequilibrium steady states is investigated by molecular dynamics simulation of macroscopically uniform conductors. At low frequencies, appropriate decomposition of the spectral intensity of current into thermal and excess fluctuations provides a simple picture of excess fluctuations behaving as shot noise. This indicates that the fluctuation-dissipation relation may be violated in a universal manner by the appearance of shot noise for a wide range of systems with particle or momentum transport.

Efficient transition path sampling for nonequilibrium stochastic dynamics.

The transition path sampling methodology is adapted to the efficient sampling of large fluctuations in nonequilibrium systems evolving according to Langevin's equations of motion. This technique is used to simulate the behavior of the bistable Maier-Stein system at noise intensities much lower than those previously possible.

Quantum Criticality

Abstract We investigate the theory of quantum fluctuations in non-equilibrium systems having large crit­ical fluctuations. This allows us to treat the limits imposed by nonlinearities to quantum squeezing and noise reduction, and also to envisage future tests of quantum theory in regions of macroscopic quantum fluctuations. A long-term objective of this research is to identify suitable physical sys­tems in which macroscopic 'Schrödinger cat'-like behaviour may be observed. We investigate two systems in particular of much current experimental interest, namely the degenerate parametric oscillator near threshold, and the evaporatively cooled (BEC). We compare the results obtained in the positive-P representation, as a fully quantum mechanical calculation, with the truncated Wigner phase space equation, also known as semi-classical theory. We show when these results agree and differ in calculations taken beyond the linearized approximation. In the region where the largest quantum fluctuations and Schrödinger cat-like behaviour might be expected, we find that the quantum predictions correspond very closely to the semi-classical theory. Nature abhors observing a Schrödinger cat. -Pacs: 03.65.Bz

Ensemble dependence of the transient fluctuation theorem

The fluctuation theorem gives an analytical expression for the probability of observing second law violating dynamical fluctuations in nonequilibrium systems. At equilibrium, statistical mechanical fluctuations are known to be ensemble dependent. In this paper we generalize the transient and steady-state fluctuation theorems to various nonequilibrium dynamical ensembles. The transient and steady-state fluctuation theorem for an isokinetic ensemble of isokinetic trajectories is tested using nonequilibrium molecular dynamics simulations of shear flow.

Large Fluctuations in a Periodically Driven Dynamical System

Fluctuations in a periodically driven overdamped oscillator are studied theoretically and experimentally in the limit of low noise intensity by investigation of their prehistory. It is shown that, for small noise intensity, fluctuations to points in coordinate space that are remote from the stable states occur along paths that form narrow tubes. The tubes are centered on the optimal paths corresponding to trajectories of an auxiliary Hamiltonian system. The optimal paths themselves, and the tubes of paths around them, are visualized through measurements of the prehistory probability distribution for an electronic model. Some general features of fluctuations in nonequilibrium systems, such as singularities in the pattern of optimal paths, the corresponding nondifferentiability of the generalized nonequilibrium potential, and the feasibility of their experimental investigation, are discussed.

Observable and hidden singular features of large fluctuations in nonequilibrium systems

We study local features, and provide a topological insight into the global structure of the probability density distribution and of the pattern of the optimal paths for large rare fluctuations away from a stable state. In contrast to extremal paths in quantum mechanics, the optimal paths do not encounter caustics. We show how this occurs, and what, instead of caustics, are the experimentally observable singularities of the pattern. We reveal the possibility for a caustic and a switching line to start at a saddle point, and discuss the consequences.

Errata: Fluctuations in nonequilibrium systems and broken supersymmetry

Errata: Fluctuations in nonequilibrium systems and broken supersymmetry

Fluctuations in nonequilibrium systems and broken supersymmetry

The fluctuation-dissipation theorem is not expected to hold for systems that either violate detailed balance or have time-dependent or nonpotential forces. Therefore the relation between response and correlation functions should have contributions due to the nonequilibrium nature. An explicit formula for such a contribution is calculated, which in the present derivation appears as a historydependent term. These relations are the Ward-Takahashi identities of a supersymmetric formulation of the Langevin models, and the new term results from a broken supersymmetry.

Finite size effects and external noise in nonequilibrium systems

We present a treatment which describes the simultaneous effect of thermal and external fluctuations in nonequilibrium systems. We show that the impact of thermal fluctuations increases when the speed of the external noise increases.

Analysis of fluctuations in chemical reactions with several steady states

The effect of fluctuations in nonequilibrium systems is treated in terms of the stochastic theory. A solution of the “fundamental” equation is analyzed using the Monte-Carlo method /1, 2/. The results are compared with the conclusions following from the equations corresponding to a phenomenological model.

On the theory of hydrodynamic fluctuations in nonequilibrium systems

Nonequilibrium fluctuating hydrodynamic equations are derived by a generalised Chapman-Enskog development applied to a previously de¬ duced nonlinear Langevin-Boltzmann equation. The long range character of the correlations in the hydrodynamic subspace is established.

Origin of the Landau-Lifshitz hydrodynamic fluctuations in nonequilibrium systems and a new method for reducing the Boltzmann equation

The Landau-Lifshitz fluctuating fluxes in fluctuating hydrodynamics are derived from the deterministic Boltzmann equation with the aid of a reduction method developed by Fujisaka and Mori. Thus it is shown that the hydrodynamic fluctuations innonequilibrium systems are generated by the reduction of variables from theΜ-space distribution function to its five momentum moments, i.e., the hydrodynamic variables. This differs from the Bixon-Zwanzig and Fox-Uhlenbeck theories, in which the Landau-Lifshitz fluctuating fluxes are derived from the molecular fluctuating force in thestochastic Boltzmann-Langevin equation, which is, however, negligible in nonequilibrium systems. Thus the present method improves the Chapman-Enskog reduction method so as to include the hydrodynamic fluctuations generated by the reduction of variables.

Long range correlations and the onset of chemical instabilities

Abstract Recent analyses of fluctuations in non-equilibrium systems by Prigogine, Malek-Mansour and Nicolis have indicated the existence of a minimal coherence length beyond which fluctuations can amplify and induce instabilities at a macroscopic scale. Conversely, this observation suggests the emergence of long range spatial correlations in systems undergoing non-equilibrium unstable transitions. In the present paper, this problem is analyzed for reaction-diffusion systems and illustrated on a specific model. The existence of diverging form factors and of long range correlations is established. The connection between the multi-variate master equation and the local equation developed recently by Prigogine, Malek-Mansour and Nicolis is discussed.