Langevin System Research Articles

Current reversals of coupled driven and damped particles evolving in a tilted potential landscape

We explore the driven and damped dynamics of two coupled particles evolving in a symmetric and periodic substrate potential that is subjected to a static bias force. In addition, each particle is time-periodically driven with the same magnitude as, but out of phase to, its counterpart. It is shown that, for a certain parameter regime, the coupled particles can become self-organized and go against the direction of the bias force. This self-organization involves the particles becoming frequency locked with the driving force, and thus periodic motion ensues. We employ numerical arguments to show that running periodic states provide solutions of the system. Further, heuristic evidence is provided explaining how the two particles can travel against the bias force. In an effort to unearth coupling phenomena within the system, a detailed analysis of how the coupling strength affects the nonlinear dynamics is carried out. We show that within a range of coupling strengths the existence of periodic running solutions associated with negative mobility. To examine the robustness of our results we compare the deterministic system with the corresponding Langevin system. It is shown that, below a critical temperature, the qualitative behavior of the system remains the same.

Nonequilibrium fluctuations for linear diffusion dynamics

We present the theoretical study on nonequilibrium (NEQ) fluctuations for diffusion dynamics in high dimensions driven by a linear drift force. We consider a general situation in which NEQ is caused by two conditions: (i) drift force not derivable from a potential function, and (ii) diffusion matrix not proportional to the unit matrix, implying nonidentical and correlated multidimensional noise. The former is a well-known NEQ source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources. We develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy, work, and heat. The NEQ fluctuation theorems are reproduced successfully. We also find the time-dependent probability distribution function exactly as well as the NEQ work production distribution P(W) in terms of solutions of nonlinear differential equations. In addition, we compute low-order cumulants of the NEQ work production explicitly. In two dimensions, we carry out numerical simulations to check out our analytic results and also to get P(W). We find an interesting dynamic phase transition in the exponential tail shape of P(W), associated with a singularity found in solutions of the nonlinear differential equation. Finally, we discuss possible realizations in experiments.

Stochastic thermodynamics for delayed Langevin systems

We discuss stochastic thermodynamics (ST) for delayed Langevin systems in this paper. By using the general principles of ST, the first-law-like energy balance and trajectory-dependent entropy s(t) can be well defined in a way that is similar to that in a system without delay. Because the presence of time delay brings an additional entropy flux into the system, the conventional second law (Δs(tot))≥0 no longer holds true, where Δs(tot) denotes the total entropy change along a stochastic path and (·) stands for the average over the path ensemble. With the help of a Fokker-Planck description, we introduce a delay-averaged trajectory-dependent dissipation functional η[χ(t)] which involves the work done by a delay-averaged force F(x,t) along the path χ(t) and equals the medium entropy change Δs(m)[x(t)] in the absence of delay. We show that the total dissipation functional R=Δs+η, where Δs denotes the system entropy change along a path, obeys (R)≥0, which could be viewed as the second law in the delayed system. In addition, the integral fluctuation theorem (e(-R))=1 also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change (Δs(tot)) could indeed be negative when the delay feedback is positive. By using an inversing-mapping approach, we are able to obtain the delay-averaged force F(x,t) from the stationary distribution and then calculate the functional R as well as its distribution. The second law (R)≥0 and the fluctuation theorem are successfully validated.

Stochastic Lagrangian method for downscaling problems in computational fluid dynamics

This work aims at introducing modelization, theoretical and numerical studies related to a new downscaling technique applied to Computational Fluid Dynamics. Our method consists in building a local model, forced by large scale information computed thanks to a classical numerical weather predictor. The local model, compatible with the Navier-Stokes equations, is used for the small scale computation (downscaling) of the considered fluid. It is inspired by S.B. Pope's works on turbulence, and consists in a so-called Langevin system of stochastic differential equations. We introduce this model and exhibit its links with classical RANS models. Well-posedness, as well as mean-field interacting particle approximations and boundary condition issues are addressed. We present the numerical discretization of the stochastic downscaling method and investigate the accuracy of the proposed algorithm on simplified situations.

Extended Jarzynski equality in general Langevin system

The Speck–Seifert equality and the Hatano–Sasa equality are known to be bases to construct stationary state thermodynamics. The well known nonequilibrium relations, the Jarzynski equality and the fluctuation theorem, share the mathematical structure with the above equalities. This hidden common property motivates us to extend the Jarzynski equality to a relation applicable in the generalized dynamical system. As a result, we find several equalities which are able to be established in this generalized system described by the Langevin equation. These results easily reproduce the Speck–Seifert equality and the Hatano–Sasa equality. We hope that our formulation gives a new insight on nonequilibrium statistical physics.

Coarse-grained Langevin approximations and spatiotemporal acceleration for kinetic Monte Carlo simulations of diffusion of interacting particles

Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting particles on a surface, at a detailed atomistic level. However such algorithms are typically computationally expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-grained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.

Nonequilibrium Steady-State Fluctuations in Actively Cooled Resonators

We analyze heat and work fluctuations in the gravitational wave detector AURIGA, modeled as a macroscopic electromechanical oscillator in contact with a thermostat and cooled by an active feedback system. The oscillator is driven to a steady state by the feedback cooling, equivalent to a viscous force. The experimentally measured fluctuations are in agreement with our theoretical analysis based on a stochastically driven Langevin system. The asymmetry of the fluctuations of the absorbed heat characterizes the oscillator's nonequilibrium steady state and reveals the extent to which a feedback cooled system departs from equilibrium in a statistical mechanics perspective.

Temperature of a Hamiltonian system given as the effective temperature of a nonequilibrium steady-state Langevin thermostat

In nonequilibrium steady states (NESS) far from equilibrium, it is known that the Einstein relation is violated. Then, the ratio of the diffusion coefficient to the mobility is called an effective temperature, and the physical relevance of this effective temperature has been studied in several works. Although the physical relevance is not yet completely clear, it has been found that the role of an effective temperature in NESS is indeed analogous to that of the temperature in equilibrium systems in a number of respects. In this paper, we find further evidence establishing this analogy. We employ a nonequilibrium Langevin system as a thermostat for a Hamiltonian system and find that the kinetic temperature of this Hamiltonian system is equal to the effective temperature of the thermostat.

A moment approach to non-Gaussian colored noise

The Langevin system subjected to non-Gaussian colored noise has been discussed, by using the second-order moment approach with two kinds of models for generating the noise. We have derived the effective differential equation (DE) for a variable x, from which the stationary probability distribution P ( x ) has been calculated with the use of the Fokker–Planck equation. The result of P ( x ) calculated by the moment method is compared to several expressions obtained by different methods such as the universal colored noise approximation (UCNA) [Jung and Hänggi, Phys. Rev. A 35 (1987) 4464] and the functional-integral method. It has been shown that our P ( x ) is in good agreement with that of direct simulations (DSs). We have also discussed dynamical properties of the model with an external input, solving DEs in the moment method.

Phase Diffusion Coefficient for Oscillators Perturbed by Colored Noise

The phase diffusion coefficient and the mean frequency of a two-dimensional nonlinear oscillator perturbed by colored noise is theoretically predicted and compared with numerical simulations of the Langevin system. At high oscillator frequencies, the first-order perturbation approximation of Demir is observed to yield inaccurate results for the phase diffusion coefficient when the spectrum of the noise sources decay faster than omega-2. A novel asymptotic approach which describes the diffusion coefficient in such instances is developed

Determinism, noise, and spurious estimations in a generalised model of population growth

We study a generalised model of population growth in which the state variable is population growth rate instead of population size. Stochastic parametric perturbations, modelling phenotypic variability, lead to a Langevin system with two sources of multiplicative noise. The stationary probability distributions have two characteristic power-law scales. Numerical simulations show that noise suppresses the explosion of the growth rate which occurs in the deterministic counterpart. Instead, in different parameter regimes populations will grow with “anomalous” stochastic rates and (i) stabilise at “random carrying capacities”, or (ii) go extinct in random times. Using logistic fits to reconstruct the simulated data, we find that even highly significant estimations do not recover or reflect information about the deterministic part of the process. Therefore, the logistic interpretation is not biologically meaningful. These results have implications for distinct model-aided calculations in biological situations because these kinds of estimations could lead to spurious conclusions.

Statistics of a noise-driven Manakov soliton

We investigate the statistics of a vector Manakov soliton in the presence of additive Gaussian white noise. The adiabatic perturbation theory for a Manakov soliton yields a stochastic Langevin system which we analyse via the corresponding Fokker-Planck equation for the probability density function (PDF) for the soliton parameters. We obtain marginal PDFs for the soliton frequency and amplitude as well as soliton amplitude and polarization angle. We also derive formulae for the variances of all soliton parameters and analyse their dependence on the initial values of polarization angle and phase.

Unified treatment of the quantum fluctuation theorem and the Jarzynski equality in terms of microscopic reversibility

There are two related theorems which hold even in far from equilibrium, namely fluctuation theorem and Jarzynski equality. Fluctuation theorem states the existence of symmetry of fluctuation of entropy production, while the Jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes. On the other hand, the relationship between these theorems was investigated by Crooks [Phys. Rev. E 60, 2721 (1999)] for the classical stochastic systems. In this paper, we derive quantum analogues of fluctuation theorem and Jarzynski equality in terms of microscopic reversibility. In other words, the quantum analog of the work by Crooks is presented. Also, for the quasiclassical Langevin system, microscopically reversible condition is confirmed.

Decomposition of force fluctuations far from equilibrium

By studying a nonequilibrium Langevin system, we find that a simple condition determines the decomposition of the coarse-grained force into a dissipative force, an effective driving force and noise. From this condition, we derive a universal inequality, D > or = gamma mu2(d)T , relating the diffusion constant D , the differential mobility mu(d) , the bare friction constant gamma and the temperature T . Due to the general nature of the argument we present, we believe that our idea concerning this decomposition can be applied to a wide class of systems far from equilibrium.

Thermal diffusion of envelope solitons on anharmonic atomic chains

We study the motion of envelope solitons on anharmonic atomic chains in the presence of dissipation and thermal fluctuations. We consider the continuum limit of the discrete system and apply an adiabatic perturbation theory which yields a system of stochastic integro-differential equations for the collective variables of the ansatz for the perturbed envelope soliton. We derive the Fokker-Planck equation of this system and search for a statistically equivalent system of Langevin equations, which shares the same Fokker-Planck equation. We undertake an analytical analysis of the Langevin system and derive an expression for the variance of the soliton position Var[xs] which predicts a stronger than linear time dependence of Var[xs] (superdiffusion). We compare these results with simulations for the discrete system and find they agree well. We refer to recent studies where the diffusion of pulse solitons were found to exhibit a superdiffusive behaviour on longer time scales.

Limit cycles and stochastic resonance in a periodically driven Langevin equation subject to white noise

We investigate stochastic resonance (SR) in a periodically driven Langevin system on a circle. The bifurcation parameter of the system and the amplitude of the periodic signal are controlled. The response amplitude that characterizes the degree of the amplification of the input periodic signal and the quality factor that is a coherence measure of the spontaneous periodic motion sustained by noise are computed and plotted as functions of the noise intensity. According to different parameter values, four types of behaviour are observed, as the results of interplay between two types of stochastic resonances, the spontaneous SR and conventional SR, respectively.

Fluctuation-response relation in a rocking ratchet.

The relationship between the positional fluctuation and mechanical response for a thermal ratchet at a nonequilibrium steady state was investigated experimentally and considered theoretically. We constructed a ratchet system using a scanning optical trapping system and microbeads (0.20 microm diameter), which is interpreted as a realization of a rocking ratchet [M. O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993)]. In the experiment, an asymmetric periodic potential (2.5 microm period) was generated by optical trap scanning, which traps a bead. When the potential profile was rocked sinusoidally, diffusion of the bead was rectified in one direction. We confirmed that both the diffusion coefficient and the mobility increased with potential rocking with a positive correlation. To obtain better insight, we performed numerical and theoretical analyses of the corresponding Langevin system. Although there is a positive relationship between the diffusion coefficient and mobility, the diffusion coefficient is greater than the value given in the Einstein relation. This result means that the effective temperature of the thermal ratchet at a nonequilibrium steady state becomes greater than that of the environment. We propose that this elevation of the effective temperature causes a decrease in the energetic efficiency of the thermal ratchet through irreversible dissipation to the heat bath.

Lévy-Driven Langevin Systems: Targeted Stochasticity

Langevin dynamics driven by random Wiener noise (“white noise”), and the resulting Fokker–Planck equation and Boltzmann equilibria are fundamental to the understanding of transport and relaxation. However, there is experimental and theoretical evidence that the use of the Gaussian Wiener noise as an underlying source of randomness in continuous time systems may not always be appropriate or justified. Rather, models incorporating general Levy noises, should be adopted. In this work we study Langevin systems driven by general Levy, rather than Wiener, noises. Various issues are addressed, including: (i) the evolution of the probability density function of the system's state; (ii) the system's steady state behavior; and, (iii) the attainability of equilibria of the Boltzmann type. Moreover, the issue of reverse engineering is introduced and investigated. Namely: how to design a Langevin system, subject to a given Levy noise, that would yield a pre-specified “target” steady state behavior. Results are complemented with a multitude of examples of Levy driven Langevin systems.

Biologically Plausible Associative Memory: Continuous Unit Response + Stochastic Dynamics

A neural network model of associative memory is presented which unifies the two historically more relevant enhancements to the basic Little-Hopfield discrete model: the graded response units approach and the stochastic, Glauber-inspired model with a random field representing thermal fluctuations. This is done by casting the retrieval process of the model with graded response neurons, into the framework of a diffusive process governed by the Fokker-Plank equation, which leads to a Langevin system describing the process at a microscopic level, while the time evolution of the probability density function is governed by a multivariate Fokker Planck equation operating over the space of all possible activation patterns. The present unified approach has two notable features: (i) greater biological plausibility and (ii) ability to escape local minima of energy (associated with spurious memories), which makes it a potential tool for those complex optimization problems for which the previous models failed.

Controlling one-dimensional Langevin dynamics on the lattice

Stochastic evolutions of classical field theories have recently become popular in the study of problems such as the determination of the rates of topological transitions and the statistical mechanics of nonlinear coherent structures. To obtain high precision results from numerical calculations, a careful accounting of spacetime discreteness effects is essential, as well as the development of schemes to systematically improve convergence to the continuum. With a kink-bearing ${\ensuremath{\varphi}}^{4}$ field theory as the application arena, we present such an analysis for a (1+1)-dimensional Langevin system. Analytical predictions and results from high resolution numerical solutions are found to be in excellent agreement.