Langevin Equation Method Research Articles

Nuclear viscosity estimated by dynamics of neck formation in the early stage of nuclear collision

The very early stage of the coalescence of two nuclei is studied and used to estimate the nuclear viscosity. The time evolution of the neck region has been simulated by the unified Langevin equation method, which is used in the analysis of heavy-ion collisions from the approaching stage to the fusion-fission stage. It is found that the transition from viscous to inertial coalescence that appeared in the neck growth of macroscopic drops can also be seen in the present simulation in nucleus–nucleus collisions. The dynamics of neck growth is analyzed in terms of the hydrodynamical formula and the viscosity coefficient of nuclear matter is estimated using the analogy of macroscopic drops.

Accessing the bath information in open quantum systems with the stochastic c -number Langevin equation method

In traditional open quantum systems, the baths are usually traced out so that only the system information is left in the equations of motion. However, recent studies reveal that using only the system degrees of freedom can be insufficient. In this work, we develop a stochastic c-number Langevin equation method which can conveniently access the bath information. In our method, the studied quantities are the expectation values of operators which can contain both system operators and bath operators. The dynamics of the operators of interest is formally divided into separate system and bath parts, with auxiliary stochastic fields. After solving the independent stochastic dynamics of the system part and the bath part, we can recombine them by taking the average over these stochastic fields to obtain the desired quantities. Several applications of the theory are highlighted, including the pure dephasing model, the spin-boson model, and an optically excited quantum dot coupled to a bath of phonons.

A new approach to modeling TiO2−x-based memristors using molecular dynamics simulation

This study aims to investigate the oxygen ion migration in defect rutile titanium dioxide (TiO2−x) in the presence of oxygen and titanium vacancies using a new approach based on molecular dynamic simulation. In this approach, the force field models along with Buckingham and Columbic potentials are used. For this purpose, the simulation is conducted in two different phases. In the first phase, the effect of temperature on the mean square displacements of oxygen ions is examined; besides, the diffusion and ionic conductivity of oxygen are studied in the absence of the electrical field. The results reveal that in the temperature range of 1100–1900 K, the oxygen vacancies tend to form clusters. These clusters consist of two oxygen vacancies. In the second phase, the memristor is applied to the model and hysteresis loops are studied in five cycles. The results of the second phase show that the new model correctly predicts the migration behavior of the oxygen ions in the memristor configuration. In conclusion, compared to the models using the Langevin equation method, the proposed model provides information which is closer to reality. Furthermore, the mean square displacements of the oxygen ions versus time are studied in the presence and absence of the electrical field at 300 K, and the results indicate greater diffusion in the presence of the electrical field.

Langevin equation for coulomb collision in non-Maxwellian plasmas

The Langevin equation method is developed for the simulation of elastic collisions in non-Maxwellian plasmas by particle methods. The properties of random processes are shown to allow splitting not only of collisions between different compounds but even splitting inside a compound. The latter drastically simplifies extension of the method for colliding plasmas. The method is also verified by obtaining characteristics of runaway electron generation under an external electric field and under an external magnetic field.

Nonequilibrium processes from generalized Langevin equations: Realistic nanoscale systems connected to two thermal baths

We extend the Generalised Langevin Equation (GLE) method [Phys. Rev. B 89, 134303 (2014)] to model a central classical region connected to two realistic thermal baths at two different temperatures. In such nonequilibrium conditions a heat flow is established, via the central system, in between the two baths. The GLE-2B (GLE two baths) scheme permits us to have a realistic description of both the dissipative central system and its surrounding baths. Following the original GLE approach, the extended Langevin dynamics scheme is modified to take into account two sets of auxiliary degrees of freedom corresponding to the mapping of the vibrational properties of each bath. These auxiliary variables are then used to solve the non-Markovian dissipative dynamics of the central region. The resulting algorithm is used to study a model of a short Al nanowire connected to two baths. The results of the simulations using the GLE-2B approach are compared to the results of other simulations that were carried out using standard thermostatting approaches (based on Markovian Langevin and Nose-Hoover thermostats). We concentrate on the steady state regime and study the establishment of a local temperature profile within the system. The conditions for obtaining a flat profile or a temperature gradient are examined in detail, in agreement with earlier studies. The results show that the GLE-2B approach is able to treat, within a single scheme, two widely different thermal transport regimes, i.e. ballistic systems, with no temperature gradient, and diffusive systems with a temperature gradient.

Applications of the generalized Langevin equation: Towards a realistic description of the baths

The Generalised Langevin Equation (GLE) method, as developed in Ref. [Phys. Rev. B 89, 134303 (2014)], is used to calculate the dissipative dynamics of systems described at the atomic level. The GLE scheme goes beyond the commonly used bilinear coupling between the central system and the bath, and permits us to have a realistic description of both the dissipative central system and its surrounding bath. We show how to obtain the vibrational properties of a realistic bath and how to convey such properties into an extended Langevin dynamics by the use of the mapping of the bath vibrational properties onto a set of auxiliary variables. Our calculations for a model of a Lennard-Jones solid show that our GLE scheme provides a stable dynamics, with the dissipative/relaxation processes properly described. The total kinetic energy of the central system always thermalises toward the expected bath temperature, with appropriate fluctuation around the mean value. More importantly, we obtain a velocity distribution for the individual atoms in the central system which follows the expected canonical distribution at the corresponding temperature. This confirms that both our GLE scheme and our mapping procedure onto an extended Langevin dynamics provide the correct thermostat. We also examined the velocity autocorrelation functions and compare our results with more conventional Langevin dynamics.

Langevin Equation Path Integral Ground State

We propose a Langevin equation path integral ground state (LePIGS) approach for the calculation of ground state (zero temperature) properties of molecular systems. The approach is based on a modification of the finite temperature path integral Langevin equation (PILE) method (J. Chem. Phys. 2010, 133, 124104) to the case of open Feynman paths. Such open paths are necessary for a ground state formulation. We illustrate the applicability of the method using model systems and the weakly bound water-parahydrogen dimer. We show that the method can lead to converged zero point energies and structural properties.

Full density matrix dynamics for large quantum systems: interactions, decoherence and inelastic effects

We develop analytical tools and numerical methods for time evolving the total density matrix of the finite-size Anderson model. The model is composed of two finite metal grains, each prepared in canonical states of differing chemical potential and connected through a single electronic level (quantum dot or impurity). Coulomb interactions are either excluded all together or allowed on the dot only. We extend this basic model to emulate decohering and inelastic scattering processes for the dot electrons with the probe technique. Three methods, originally developed to treat impurity dynamics, are augmented to yield global system dynamics: the quantum Langevin equation method, the well-known fermionic trace formula and an iterative path integral approach. The latter accommodates interactions on the dot in a numerically exact fashion. We apply the developed techniques to two open topics in nonequilibrium many-body physics. (i) We explore the role of many-body electron–electron repulsion effects on the dynamics of the system. Results, obtained using exact path integral simulations, are compared with mean-field quantum Langevin equation predictions. (ii) We analyze aspects of quantum equilibration and thermalization in large quantum systems using the probe technique, mimicking elastic-dephasing effects and inelastic interactions on the dot. Here, unitary simulations based on the fermionic trace formula are accompanied by quantum Langevin equation calculations.

Ladder Lattice Model of Soft Optical Phonon

A linear lattice model for demonstrating a soft optical mode at a zone center is proposed. The unit cell consists of only two atoms. If the optical mode is sufficiently soft, anticrossing takes place between the optical and acoustic phonon branches. By using the generalized Langevin's equation method, the spectral function is calculated analytically. The intensity of the optical mode disappears at approximately the anticrossing wave number. In general, the spectral line shape of the optical mode is asymmetric because of the interaction between the two branches.

A constructive role of internal noise on coherence resonance induced by external noise in a calcium oscillation system

A stochastic cytosolic calcium oscillation model simultaneously driven by external noise and internal noise has been investigated by using the chemical Langevin equation method. When the extracellular stimulation is subthreshold, it is found that the internal-noise coherence resonance can be suppressed by the external noise, while the external-noise coherence resonance can be enhanced by modulating the internal-noise intensity. Besides a qualitative analysis with a competitive mechanism is given for the results, a further explanation is also provided theoretically by using the small noise approximation and bifurcation analysis.

A mesoscopic stochastic mechanism of cytosolic calcium oscillations

Based on a model of intracellular calcium (Ca 2+) oscillation with self-modulation of inositol 1,4,5-trisphosphate signal, the mesoscopic stochastic differential equations for the intracellular Ca 2+ oscillations are theoretically derived by using the chemical Langevin equation method. The effects of the finite biochemical reaction molecule number on both simple and complex cytosolic Ca 2+ oscillations are numerically studied. In the case of simple intracellular Ca 2+ oscillation, it is found that, with the increase of molecule number, the coherence resonance or autonomous resonance phenomena can occur for some external stimulation parameter values. In the cases of complex cytosolic Ca 2+ oscillations, each extremum of concentration of cytosolic Ca 2+ oscillations corresponds to a peak in the histogram of Ca 2+ concentration, and the most probability appeared during the bursting plateau level for bursting, but at the largest minimum of Ca 2+ concentration for chaos. For quasi-periodicity, however, there are only two peaks in the histogram of Ca 2+ concentration, and the most probability is located at low concentration state.

Enhancement of internal-noise coherence resonance by modulation of external noise in a circadian oscillator

A circadian oscillator driven by external noise and internal noise has been studied by use of the chemical Langevin equation method. When the system is near a Hopf bifurcation and driven by internal noise only, it is found that the coherence resonance phenomenon can be induced by the internal noise. When the system is simultaneously driven by internal and external noise, it is found that external-noise coherence resonance can be suppressed by internal noise, while internal-noise coherence resonance can be enhanced by modulation of the external noise intensity in a certain range of noise intensity. Another interesting result is that the external noise can regulate the optimal system size when the internal-noise coherence resonance occurs.

Langevin equation method for the rotational Brownian motion and orientational relaxation in liquids: spherical top molecules

A solution of the infinite hierarchy of differential–recurrence relations for the statistical moments (orientational correlation functions) governing the inertial rotational Brownian motion and the dielectric relaxation of an assembly of spherical top molecules is obtained in terms of a continued fraction. It is shown analytically that the solution based on this approach agrees with that of Sack [ Proc. Phys. Soc. Lond. B 70 (1957) 414] (with a corrected misprint) and Fixman and Rider [ J. Chem. Phys. 51 (1969) 2425] obtained from the corresponding Fokker–Planck equation.

Langevin equation method for the rotational Brownian motion and orientational relaxation in liquids: II. Symmetrical top molecules

At heory of orientational relaxation for the inertial rotational Brownian motion of a symmetric top molecule is developed using the Langevin equation rather than the Fokker–Planck equation. The infinite hierarchy of differentialrecurrence relations for the orientationa lc orrelation functions for the relaxation behaviour is derived by averaging the corresponding Euler–Langevin equations. The solution of this hierarchy is obtained using matrix continued fractions allowing the calculation of the correlation times and the spectra of the orientational correlation functions for typical values of the model parameters.

Langevin equation method for the rotational Brownian motion and orientational relaxation in liquids

A theory of orientational relaxation for the inertial rotational Brownian motion of a linear molecule (rotator in space) is developed in the context of the Langevin equation method without recourse to the Fokker–Planck equation. The general term in the time-dependent infinite hierarchy of differential-recurrence relations for the orientational correlation functions describing the relaxation behaviour of the system is derived by averaging the corresponding Euler–Langevin equation. The solution of this hierarchy is obtained in terms of continued fractions. The correlation times and the spectra of the orientational correlation functions are calculated for typical values of the model parameters.

Incoherent Magnetization Model for Perpendicular Recording Media

To study the influence of thermal fluctuation in perpendicular recording media, we investigated a new incoherent magnetization model and combined it with the Langevin equation method. In our model, the energy barrier for magnetization reversal is reduced by using an incoherent reversal process. We simulated the M-H curve and the torque curve for perpendicular media with small Hc/Hk values, and calculated the time decay of Mr due to thermal fluctuation in the perpendicular media. The results roughly agreed with the experimental values of the time decay.

Transforming signs to phase distributions in quantum simulations.

A method is developed which speeds up averaging in quantum simulations where minus signs cause difficulties. A Langevin equation method in conjunction with a replication algorithm is used enabling one to average over a continuously varying complex number. Instead of ensemble averaging this number directly, the phase of the complex number is followed over time. The method is illustrated in some simple cases where the answers obtained can be compared to exact results, and also compared to conventional averaging procedures which converge orders of magnitude slower than this method. Limitations of this method are also described.

Langevin Equation Approach to Reactor Noise Analysis: Stochastic Transport Equation

The application of the Langevin equation method to the study of fluctuations in the space- and velocity-dependent neutron density as well as in the detector outputs in nuclear reactors is presented. In this case, the Langevin equation is the stochastic linear neutron transport equation with a space- and velocity-dependent random neutron source, often referred to as the noise equivalent source (NES). The power spectral densities (PSDs) of the NESs in the transport equation, as well as in the accompanying detection rate equations, are obtained, and the cross- and auto-power spectral densities of the outputs of pairs of detectors are explicitly calculated. The transport-level expression for the R([omega]) ratio measured in the [sup 252]Cf source-driven noise analysis method is also derived. Finally, the implementation of the Langevin equation approach at different levels of approximation is discussed, and the stochastic one-speed transport and one-group P[sub 1] equations are derived by first integrating the stochastic transport equation over speed and then eliminating the angular dependence by a spherical harmonics expansion. By taking the large transport rate limit in the P[sub 1] description, the stochastic diffusion equation is obtained as well as the PSD of the NES in it. This procedure also leads directlymore » to the stochastic Fick's law.« less

On the derivation of the after-effect solution for the Kerr effect relaxation from the Langevin equation for rotation in three dimensions

The Kerr effect response (KER) of an assembly of non-interacting dipolar molecules following the removal of a dc electric field is calculated by direct averaging of the Langevin equation. The Hermite polynomials in the angular velocity components (ω 1 and ω 2) and the associated Legendre functions of order 2 in the polar angle, ϑ, are chosen as variables. This leads directly, using the statistical properties of the white noise driving torques, to the differential-difference equations which govern in Kerr effect relaxation. The general term in the average of the product of Hermite polynomials and the associated Legendre function of order 2 is given. This allows us to calculate the Laplace transform of the average value of the second order Legendre polynomial; < P 2 (cos ϑ( t))>, following the removal of the a dc field, i.e. the affect-effect solution for the dynamic Kerr effect, up to any desired degree of accuracy using the Langevin equation method by converting the set of differential-difference equations into a set of ordinary differential equations and successively increasing the size of the transition matrix. The convergence of the procedure for various values of the inertial parameters is investigated by computing the Fourier transform of the response for successively increasing sizes of the transition matrix.

Foundations and Extensions of the Cahn-Hilliard-Cook Theory

Abstract In this paper we examine the foundations of the Cahn-Hilliard-Cook (CHC) theory of phase separation, which has been used routinely in recent years to interpret dynamic scattering experiments on binary polymer blends, and discuss its extension to multi-component mixtures. The CHC-theory is based on the following nonlinear Langevin equation1: where Φ(q,t) is the Fourier transform of the local volume fraction Φr,t) of one of the components of the binary mixture which is assumed to be incompressible; λ(q) is the q-dependent onsager coefficient; μ(q,t) is the Fourier transform of the local chemical potential difference μ(r,t); and η(q,t) is the random force which is added to account for the thermal fluctuations. The latter is assumed to be a stationary white noise process with zero mean and an autocovariance (η(q,t) q(-q,t′)) = λ(q) q2 δ(t - t′), which is determined from fluctuation-dissipation theorem. The use of the Langevin equation method to describe fluctuations in nonlinear macroscopic systems, in the way done in the CHC theory, was questioned by van Kampen2,3 for reasons which we discuss in this paper. The proper way of using the Langevin equation method in nonlinear systems was provided by Akcasu4 in 1977 through van Kampen's system size expansion, which we sketch next.