Exact Langevin Equation Research Articles

Langevin Equations for Reaction-Diffusion Processes.

For reaction-diffusion processes with at most bimolecular reactants, we derive well-behaved, numerically tractable, exact Langevin equations that govern a stochastic variable related to the response field in field theory. Using duality relations, we show how the particle number and other quantities of interest can be computed. Our work clarifies long-standing conceptual issues encountered in field-theoretical approaches and paves the way for systematic numerical and theoretical analyses of reaction-diffusion problems.

SYMMETRIC HEAT CONDUCTION IN HARMONIC CHAIN WITH HARMONIC ON-SITE POTENTIAL AND MASS GRADIENT

We investigate, using the exact Langevin equation, the properties of the heat conduction of an individual 1D harmonic chain with harmonic on-site potential. Each particle interacts with its own independent thermal reservoir. The results indicate that there is no rectification effect in this model even with mass gradient in the case of weak interparticle interaction. It may suggest that a system with harmonic on-site potential and asymmetric structure does not guarantee an asymmetric heat transport.

Occam’s razor on surfaces: renormalization of microscopic processes

We describe a methodology for the expression of atomistic models of fluctuating interfacesas continuum equations. We begin with formally exact Langevin equations based on theatomistic transition rules, which are regularized to produce stochastic partial differentialequations. Subsequent coarse graining is accomplished by calculating renormalization-group(RG) trajectories from initial conditions determined by the regularized equations. The RGanalysis shows that the morphological manifestation of a given atomistic relaxationmechanism can depend on the length scales and timescales considered as well as on thedimensionality of the fluctuating interface. Even complex surface processes, after amoderate degree of coarse graining, are reduced to low-order stochastic partialdifferential equations. We illustrate these ideas with a model of a growing surfaceunder the competition between the deposition of new material and the subsequentrelaxation through surface diffusion. We conclude with an augmentation of ourdifferential equations with a pinning term to account for lattice effects in theearly stages of growth, where surface electron diffraction oscillations indicatelayer-by-layer growth. The calculation of submonolayer morphology, which is composed ofseparated monolayer islands, provides an illustration of the efficacy of our method.

Aging processes in reversible reaction-diffusion systems

Reversible reaction-diffusion systems display anomalous dynamics characterized by a power-law relaxation toward stationarity. In this paper we study in the aging regime the nonequilibrium dynamical properties of some model systems with reversible reactions. Starting from the exact Langevin equations describing these models, we derive expressions for two-time correlation and autoresponse functions and obtain a simple aging behavior for these quantities. The autoresponse function is thereby found to depend on the specific nature of the chosen perturbation of the system.

Complex reaction noise in a molecular quasispecies model

We have derived exact Langevin equations for a model of quasispecies dynamics. The inherent multiplicative reaction noise is complex and its statistical properties are specified completely. The numerical simulation of the complex Langevin equations is carried out using the Cholesky decomposition for the noise covariance matrix. This internal noise, which is due to diffusion-limited reactions, produces unavoidable spatio-temporal density fluctuations about the mean field value. In two dimensions, this noise strictly vanishes only in the perfectly mixed limit, a situation difficult to attain in practice.

Langevin equations for fluctuating surfaces

Exact Langevin equations are derived for the height fluctuations of surfaces driven by the deposition of material from a molecular beam. We consider two types of model: deposition models, where growth proceeds by the deposition and instantaneous local relaxation of particles, with no subsequent movement, and models with concurrent random deposition and surface diffusion. Starting from a Chapman-Kolmogorov equation the deposition, relaxation, and hopping rules of these models are first expressed as transition rates within a master equation for the joint height probability density function. The Kramers-Moyal-van Kampen expansion of the master equation in terms of an appropriate "largeness" parameter yields, according to a limit theorem due to Kurtz [Stoch. Proc. Appl. 6, 223 (1978)], a Fokker-Planck equation that embodies the statistical properties of the original lattice model. The statistical equivalence of this Fokker-Planck equation, solved in terms of the associated Langevin equation, and solutions of the Chapman-Kolmogorov equation, as determined by kinetic Monte Carlo (KMC) simulations of the lattice transition rules, is demonstrated by comparing the surface roughness and the lateral height correlations obtained from the two formulations for the Edwards-Wilkinson [Proc. R. Soc. London Ser. A 381, 17 (1982)] and Wolf-Villain [Europhys. Lett. 13, 389 (1990)] deposition models, and for a model with random deposition and surface diffusion. In each case, as the largeness parameter is increased, the Langevin equation converges to the surface roughness and lateral height correlations produced by KMC simulations for all times, including the crossover between different scaling regimes. We conclude by examining some of the wider implications of these results, including applications to heteroepitaxial systems and the passage to the continuum limit.

Asymptotic form of the approach to equilibrium in reversible recombination reactions

The reversible reactions A+A<=>C and A+B<=>C are investigated. From the exact Langevin equations describing our model, we set up a systematic approximation scheme to compute the approach of the density of C particles to its equilibrium value. We show that for sufficiently long time t, this approach takes the form of a power law At^(-d/2), for any dimension d. The amplitude A is also computed exactly, but is expected to be model dependent. For uncorrelated initial conditions, the C density turns out to be a monotonic time function. The cases of correlated initial conditions and unequal diffusion constants are investigated as well. In the former, correlations may break the monotonicity of the density or in some special cases they may change the long time behavior. For the latter, the power law remains valid, only the amplitude changes, even in the extreme case of immobile C particles. We also consider the case of segregated initial condition for which a reaction front is observed, and confirm that its width is governed by mean-field exponent in any dimension.

Rotational Depolarization of Fluorescence of Prolate Molecules

Abstract The theory of rotational depolarization of fluorescence, taking into account the inertial effect, has been verified experimentally for two angular velocity autocorrelation functions obtained for both the approximate and the exact Langevin equation. To this end, the emission anisotropy was investigated as a function of viscosity for PPO, POPOP, and α-NOPON in liquid n-paraffins (from « = 5 to n = 11) at 304.5 K. For these three luminescent molecules with mean lifetimes of about 1 ns, and the viscosities ranging from 0.22 to 0.9 cP, no essential difference was found between the experiment and the theory for the angular velocity autocorrelation functions employed.

Exact Langevin equation in a cosmological setting.

Recently, various authors have used Langevin equations to model physical processes in the early Universe. Much of this work is based on a flat-space effective potential, neglecting potential complications associated with a curved spacetime. This paper considers a simple statistical-mechanical model of time-dependent, bilinearly coupled oscillators, which leads to an exact Langevin equation that mocks a field theory in a time-dependent background. This equation exhibits a nonlocal time-dependent change in the effective potential which vanishes identically when the oscillators are time independent.

Dynamics in nonentangled concentrated polymer solutions. I. The full dynamic multiple scattering approach to first order in concentration

The concentration dependence of flexible polymer relaxation times in concentrated nonentangled solutions is studied by a full dynamic multiple scattering approach. In the absence of macroscopic entanglements effects the screening of the hydrodynamic interaction is the dominant phenomenon governing this concentration dependence. The leading concentration contribution to the relaxation times is used to introduce the screening constant through its effect on the hydrodynamic interaction matrix. The exact Langevin equation for the normal coordinates of the polymer is derived and analyzed through leading order in concentration. It is shown that there is a concentration-dependent memory kernel which emerges in the exact Langevin equation for the polymer chain dynamics in concentrated solutions, a departure from the infinite dilution situation where the (delta function) kernel is instantaneous. This implies that the relaxation matrix is frequency dependent. The general conditions for having an effective relaxation spectrum in concentrated solution are considered, and the structure of the relaxation of the normal coordinate time correlation functions is given. In general this relaxation is not described by a single exponential decaying term, but by a combination of many exponentially decaying and oscillatory exponentially decaying terms; however, the latter oscillation rates are generally very slow compared to the overall decay rates. The accompanying paper considers model calculations of the concentration dependent relaxation times.

On the physics and modeling of small semiconductor devices—II: The very small device

In a previous paper, we attempted to lay a conceptual framework for an ultimate physics of small semiconductor devices and concentrated on the medium small device. Here we treat the very small device (VSD), characterized by an effective channel length of 250 Å. We demonstrate how such a device could conceivably be fabricated using two side processing of the wafer. In treating the transport, however, it is found that the time and distance scales are such that the Boltzmann transport equation is completely invalidated. Here we develop the appropriate quantum transport equations based upon the density matrix for the entire system, device plus boundaries plus environment. It is found that the boundaries and environment can lead to renormalization of the energy spectrum as well as long range dissipative interactions. Two special cases of the transport equations are treated. If the transport is dominantly stochastic, an exact Langevin equation is found for the various transport parameters. In a second case, a parameterized density matrix is used in analogy to the displaced Maxwellian. In this latter case, a hierarchy of moment equations can be developed to yield, e.g. energy and momentum balance equations.