Langevin Equation Formalism Research Articles

Intriguing properties of transport at the microscales: Langevin equation approach

We present a perspective of simple models of nonequilibrium directed transport described in terms of a Langevin equation formalism. We consider a Brownian particle under various circumstances and driven by thermal (equilibrium) and non-thermal (active) fluctuations. Three examples of startling behavior are unveiled: giant transport, multiple current reversal and negative mobility.Graphic abstract

Fast protein folding is governed by memory-dependent friction

When described by a low-dimensional reaction coordinate, the folding rates of most proteins are determined by a subtle interplay between free-energy barriers, which separate folded and unfolded states, and friction. While it is commonplace to extract free-energy profiles from molecular trajectories, a direct evaluation of friction is far more elusive and typically relies on fits of measured reaction rates to memoryless reaction-rate theories. Here, using memory-kernel extraction methods founded on a generalized Langevin equation (GLE) formalism, we directly calculate the time-dependent friction acting on the fraction of native contacts reaction coordinate Q, evaluated for eight fast-folding proteins, taken from a published set of large-scale molecular dynamics protein simulations. Our results reveal that, across the diverse range of proteins represented in this dataset, friction is more influential than free-energy barriers in determining protein folding rates. We also show that proteins fold in a regime where the finite decay time of friction significantly reduces the folding times, in some instances by as much as a factor of 10, compared to predictions based on memoryless friction.

Stochastic localization and non-Boltzmann distribution

We investigate the microscopical root and macroscopical representation of the stochastic localization, which is a nonergodic motion in oppositive to the other limit of ballistic diffusion. In order to produce such anomalous kinetics, we consider that a tagged particle is linearly coupled to a series connection bath or the terminal of a coupled-oscillator-chain. By means of generalized Langevin equation formalism, we obtain the coordinate autocorrelation function. The localization emerged of a particle at finite temperature, is due to the spectrum of driving noise diverging at zero frequency. Consequently, the steady distribution of system depends on its initial coordinate preparation.

Diffusion theory of molecular liquids in the energy representation and application to solvation dynamics.

The generalized Langevin equation (GLE) formalism is a useful theoretical fundament for analyzing dynamical phenomena rigorously. Despite the systematic formulation of dynamics theories with practical approximations, however, the applicability of GLE-based methods is still limited to simple polyatomic liquids due to the approximate treatment of molecular orientations involved in the static molecular liquid theory. Here, we propose an exact framework of dynamics based on the GLE formalism incorporating the energy representation theory of solution, an alternative static molecular liquid theory. A fundamental idea is the projection of the relative positions and orientations of solvents around a solute onto the solute-solvent interaction, namely the energy coordinate, enabling us to describe the dynamics on a one-dimensional coordinate. Introducing systematic approximations, such as the overdamped limit, leads to the molecular diffusion equation in the energy representation that is described in terms of the distribution function of solvents on the energy coordinate and the diffusion coefficients. The present theory is applied to the solvation dynamics triggered by the photoexcitation of benzonitrile. The long-time behavior of the solvation time correlation function is in good agreement with that obtained by the molecular dynamics simulation.

Vapor-liquid phase transition in fluctuating hydrodynamics: The most probable transition path and its computation

In this paper, we consider transition events between metastable states in fluid systems modeled by two-phase Navier-Stokes equations with thermal noise. The vapor-liquid phase transition which occurs during the vapor condensation is such an example. We propose a numerical method to compute the transition pathway. We first approximate the continuum hydrodynamics model by the smoothed particle hydrodynamics (SPH). In the SPH formulation, the dynamics of the system take the form of Langevin equations. This allows us to derive conditions that characterize the most probable transition pathway. Then we compute the most probable transition pathway by solving these conditions in the space of mass and momentum density fields using the string method. The force, which is needed to evolve the string in the space of the density fields, is computed by discretizing the deterministic model using the finite difference method. The numerical method is applied to study the condensation of vapor in a periodic cell and the Cassie-Wenzel transition on pillared solid substrates.

Passive Advection in a Percolation Process: Two-Loop Approximation

The paradigmatic model of the directed percolation process is studied near its second order phase transition between an absorbing and an active state. The model is first expressed in a form of Langevin equation and later rewritten into a field-theoretic formulation. The ensuing response functional is then analyzed employing Feynman diagrammatic technique and perturbative renormalization group method. Percolation process is assumed to occur in external velocity field, which has an additional effect on spreading properties. Kraichnan rapid change ensemble is used for generation of velocity fluctuations. The structure of the fixed points structure is obtained within the two-loop approximation.

Fokker-Planck representations of non-Markov Langevin equations: application to delayed systems.

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Generalized self-consistent reservoir model for normal and anomalous heat transport in quantum harmonic chains.

Fictitious stochastic reservoirs incorporate scattering and dephasing mechanisms into the system in contact with these reservoirs. The reservoir-system coupling is described by the related self-energy in terms of the nonequilibrium Green's function formalism or equivalently the quantum Langevin equation formalism. In this study, we investigate thermal transport in a finite segment of an infinitely extended quantum harmonic chain with an equal self-energy at each site by using the self-consistent reservoir approach. In this setup, the entire system is lattice translation invariant so that mismatched boundaries are excluded from the model. Solving the Landauer-Büttiker equations under the self-consistent adiabatic condition, we quantitatively elucidate a thermally induced crossover of ballistic-to-diffusive transport and its scaling relation prescribed by a temperature-dependent mean free path. It is also shown that normal transport emerges in the diffusive limit for a linear self-energy, while nonlinear higher-order ones generically lead to anomalous transport. Physical implications of these observations are discussed in terms of the persistence of a massless Goldstone mode as well as the conservation of total linear momentum.

Modelling a Bistable System Strongly Coupled to a Debye Bath: A Quasiclassical Approach Based on the Generalised Langevin Equation

Bistable systems present two degenerate metastable configurations separated by an energy barrier. Thermal or quantum fluctuations can promote the transition between the configurations at a rate which depends on the dynamical properties of the local environment (i.e., a thermal bath). In the case of classical systems, strong system-bath interaction has been successfully modelled by the Generalised Langevin Equation (GLE) formalism. Here we show that the efficient GLE algorithm introduced in Phys. Rev. B 89, 134303 (2014) can be extended to include some crucial aspects of the quantum fluctuations. In particular, the expected isotopic effect is observed along with the convergence of the quantum and classical transition rates in the strong coupling limit. Saturation of the transition rates at low temperature is also retrieved, in qualitative, yet not quantitative, agreement with the analytic predictions. The discrepancies in the tunnelling regime are due to an incorrect sampling close to the barrier top. The domain of applicability of the quasiclassical GLE is also discussed.

Universal Trend of the Non-Exponential Rouse Mode Relaxation in Glass-Forming Polymers Systems: Experimental Facts, MD-Simulation Results and a Theoretical Approach Based on a Generalized Langevin Equation

ABSTRACTNowadays there are clear evidences from both experiments and MD-simulations proving that the chain Rouse modes correlation functions are non-exponential in unentangled polymer blends and also in pure polymers at low temperature (with respect to that of the glass transition Tg) even for the long wavelengths modes where local potentials and chain stiffness should not play any role. In a recent paper [S. Arrese-Igor et al, Phys. Rev. Lett.113, 078302 (2014)] it has been proposed that this non-exponential behavior depends on the ratio between the so-called Rouse time - i.e., the characteristic time of the slowest chain mode relaxation - and the time scale of the α-relaxation. This parameter is in some way ‘universal’ in the meaning that it can encode many different experimental situations. In this paper, we show that this behavior can be quantitatively explained in the framework of a theoretical approach based on: (i) a generalized Langevin equation (GLE) formalism and (ii) a memory function which takes into account the effect of collective dynamics on the chain dynamics of a tagged chain and which was constructed taking inspirations from the original ideas of the reptation model proposed by de Gennes.

Non-exponential Rouse correlators and generalized magnitudes probing chain dynamics

Inspired by the generalized Langevin equation (GLE) formalism with some approximations, we have proposed a method to generalize the Rouse expression of the magnitudes probing chain dynamics, taking into account the non-exponential character of the Rouse mode correlators. In this way, generalized expressions for the incoherent neutron scattering function and for the End-to-End relaxation – which give rise in the frequency domain to the dielectric ‘normal mode’ relaxation – have been obtained. These magnitudes have been checked by means of fully atomistic molecular dynamics (MD) simulations corresponding to pure poly(ethylene oxide) (PEO) and to PEO in an asymmetric blend with poly(methyl methacrylate) (PMMA). Moreover, the results obtained also seem to give support to the unified scenario recently proposed for explaining the non-exponential relaxation of the long-wavelength Rouse modes in polymer systems in terms of the effect of density fluctuations (α-process) on chain dynamics.

Energy balance for a dissipative quantum system

The role of random force in maintaining equilibrium in a dissipative quantum system is studied here. We compute the instantaneous power supplied by the fluctuating (random) force, which provides information about the work done by the random force on the quantum subsystem of interest. The quantum Langevin equation formalism is used here to verify that, at equilibrium, the work done by the fluctuating force balances the energy lost by the quantum subsystem to the heat bath. The quantum subsystem we choose to couple to the heat bath is the charged oscillator in a magnetic field. We perform the calculations using the Drude regularized spectral density of bath oscillators instead of using a strict ohmic spectral density that gives memoryless damping. We also discuss the energy balance for our dissipative quantum system and in this regard it is to be understood that the physical system is the charged magneto-oscillator coupled to the heat bath, not the uncoupled charged magneto-oscillator.

A stochastic extension of the explicit algebraic subgrid-scale models

The explicit algebraic subgrid-scale (SGS) stress model (EASM) of Marstorp et al. [“Explicit algebraic subgrid stress models with application to rotating channel flow,” J. Fluid Mech. 639, 403–432 (2009)] and explicit algebraic SGS scalar flux model (EASFM) of Rasam et al. [“An explicit algebraic model for the subgrid-scale passive scalar flux,” J. Fluid Mech. 721, 541–577 (2013)] are extended with stochastic terms based on the Langevin equation formalism for the subgrid-scales by Marstorp et al. [“A stochastic subgrid model with application to turbulent flow and scalar mixing,” Phys. Fluids 19, 035107 (2007)]. The EASM and EASFM are nonlinear mixed and tensor eddy-diffusivity models, which improve large eddy simulation (LES) predictions of the mean flow, Reynolds stresses, and scalar fluxes of wall-bounded flows compared to isotropic eddy-viscosity and eddy-diffusivity SGS models, especially at coarse resolutions. The purpose of the stochastic extension of the explicit algebraic SGS models is to further improve the characteristics of the kinetic energy and scalar variance SGS dissipation, which are key quantities that govern the small-scale mixing and dispersion dynamics. LES of turbulent channel flow with passive scalar transport shows that the stochastic terms enhance SGS dissipation statistics such as length scale, variance, and probability density functions and introduce a significant amount of backscatter of energy from the subgrid to the resolved scales without causing numerical stability problems. The improvements in the SGS dissipation predictions in turn enhances the predicted resolved statistics such as the mean scalar, scalar fluxes, Reynolds stresses, and correlation lengths. Moreover, the nonalignment between the SGS stress and resolved strain-rate tensors predicted by the EASM with stochastic extension is in much closer agreement with direct numerical simulation data.

Generalized Langevin equation for tracer diffusion in atomic liquids

We derive the time-evolution equation that describes the Brownian motion of labeled individual tracer particles in a simple model atomic liquid (i.e., a system of N particles whose motion is governed by Newton’s second law, and interacting through spherically symmetric pairwise potentials). We base our derivation on the generalized Langevin equation formalism, and find that the resulting time evolution equation is formally identical to the generalized Langevin equation that describes the Brownian motion of individual tracer particles in a colloidal suspension in the absence of hydrodynamic interactions. This formal dynamic equivalence implies the long-time indistinguishability of some dynamic properties of both systems, such as their mean squared displacement, upon a well-defined time scaling. This prediction is tested here by comparing the results of molecular and Brownian dynamics simulations performed on the hard sphere system.

Drift in Diffusion Gradients.

The longstanding problem of Brownian transport in a heterogeneous quasi one-dimensional medium with space-dependent self-diffusion coefficient is addressed in the overdamped (zero mass) limit. A satisfactory mesoscopic description is obtained in the Langevin equation formalism by introducing an appropriate drift term, which depends on the system macroscopic observables, namely the diffuser concentration and current. The drift term is related to the microscopic properties of the medium. The paradoxical existence of a finite drift at zero current suggests the possibility of designing a Maxwell demon operating between two equilibrium reservoirs at the same temperature.

A Generalized Rouse Incoherent Scattering Function for Chain Dynamics of Unentangled Polymers in Dynamically Asymmetric Blends

The chain dynamics of unentangled polymers strongly deviates from the expected Rouse behavior when they form part of dynamically asymmetric polymer blends. These are miscible systems where the two components display very different segmental mobility (or glass transition). The generalized Langevin equation formalism seems to be a suitable theoretical framework for this situation. In this work, we have used an approximation based on (i) a simplified generalized Langevin equation and (ii) the phenomenological result of nonexponential decay of the Rouse model correlators in asymmetric polymer blends. In the framework of this approximation, we have deduced an incoherent scattering function for chain dynamics of unentangled polymers in asymmetric blends. This function—which in the Rouse limit reduces to the well-known de Gennes expression—is validated by molecular dynamics simulation results of a canonical asymmetric blend: poly(methyl methacrylate)/poly(ethylene oxide).

The overdamped van Hove function of atomic liquids

Using the generalized Langevin equation formalism and the process of contraction of the description we derive a general memory function equation for the thermal fluctuations of the local density of a simple atomic liquid. From the analysis of the long-time limit of this equation, a striking equivalence is suggested between the long-time dynamics of the atomic liquid and the dynamics of the corresponding Brownian liquid. This dynamic equivalence is confirmed here by comparing molecular and Brownian dynamics simulations of the self-intermediate scattering function and the long-time self-diffusion coefficient for the hard-sphere liquid.

Anomalous molecular weight dependence of chain dynamics in unentangled polymer blends with strong dynamic asymmetry

We address the general question of how the molecular weight dependence of chain dynamics in unentangled polymers is modified by blending. By dielectric spectroscopy we measure the normal mode relaxation of polyisoprene in blends with a slower component of poly(ter-butylstyrene). Unentangled polyisoprene in the blend exhibits strong deviations from Rouse scaling, approaching ‘entangled-like’ behavior at low temperatures in concomitance with the increase of the dynamic asymmetry in the blend. The obtained results are discussed in the framework of the generalized Langevin equation formalism. On this basis, a non trivial relationship between the molecular weight dependence of the longest chain relaxation time and the nonexponentiality of the corresponding Rouse correlator is found. This result is confirmed by molecular dynamics simulations.

Dynamics of Proton Transfer and Vibrational Relaxation in Dilute Hydrofluoric Acid

The molecular mechanisms in both vibrational relaxation and proton transfer (PT) associated with infrared (IR)-induced PT in a dilute hydrofluoric acid solution at ambient temperature are studied by molecular dynamics (MD) simulations with the multistate empirical valence bond model. To investigate the solvation dynamics, a collective solvent coordinate and its perpendicular bath modes are defined from the diabatic energy gap and their motions are examined by the generalized Langevin equation (GLE) formalism. The GLE analysis using the equilibrium MD simulation shows that the major solvent reorganizations in the PT are represented by the libration and hindered translation. In particular, the libration gives the stronger coupling to the solvent reorganization and the faster relaxation. The nonequilibrium MD simulation demonstrated that both the HF stretching vibration and the solvent reorganization relax on a similar time scale and thus compete in the PT. It also supported the "presolvation mechanism" for the PT in this system.

Classical theory for asymmetric in-plane atom surface scattering

In-plane atom surface scattering perturbation theory within a generalized Langevin equation formalism is proposed to account for the asymmetry found in angular distributions of heavy rare gas atoms scattered by corrugated surfaces. We show that when the surface corrugation is represented in terms of the first two (sine) Fourier components, one finds an asymmetric angular distribution. This asymmetry reflects the ratchetlike form of the effective corrugation. Adding in higher-order terms can also increase the number of rainbow scattering angles. Three rainbows are found for a second-order sine term in the corrugation, four symmetrically spaced rainbow angles are found when adding in a second-order cosine term to the corrugation. Analytic expressions for the angular distribution are derived in terms of a Morse oscillator model. The theory accounts well for the asymmetry and predicts its disappearance as the incident scattering angle is increased. It also features a decrease in the distance between the rainbow angles as the angle of incidence is increased and as the incident energy is increased. The theory is successfully applied to the experimental results of Kondo et al. [Eur. Phys. J. D 38, 129 (2006)] for the scattering of Ar on LiF(100) and the results of Amirav et al. [J. Chem. Phys. 87, 1796 (1987)] for the scattering of Xe on Ge(100) and Ar and Kr on Ag(100).