Nonlinear Langevin Equation Research Articles

Langevin equation for the squeezing of light by means of a parametric oscillator.

We show that the Langevin equation for a nonlinear-optical system may be obtained directly from the Heisenberg equation of motion for the annihilation operators, provided a certain linearization procedure is valid. We apply the technique to the parametric oscillator used to generate squeezed light and compare our results to those obtained from Fokker-Planck-type equations. We argue that, only when the Wigner, as opposed to the P or Q, representation of quantum optics is used, do we get a correct description of the underlying stochastic process. We show how the linearization procedure may be carried out to describe the operation of the parametric oscillator both below threshold, where a squeezed vacuum state results, and above threshold, where we find a squeezed coherent state. In the region of the threshold a heuristic extension of the method leads to a possible description of the system by means of a nonlinear Langevin equation.

Stochastic analog of the quantum theory of Coulomb problem

The path-integral formalism is used to show that the solution to a special stochastic process described by the non-linear Langevin equation can be converted into the quantum theory of the Coulomb problem. The conditional probability density for this stochastic process is obtained by explicit functional integration.

Tooth eruption: The phase transition theory on biological formation of an orderly structure

The mechanism of the formation of an orderly structure from random elements in organ development was clarified by studying the maturation of the dental arch in the human mandible. First, an application of methods established in statistical physics to a system of organ development was made possible, and then, the mathematical procedures for quantitative study of the structure and development of the dental arch were established in relation to radiographic data. The experimental results demonstrate that a parallel arrangement of the longitudinal axes of the lateral teeth is formed co-operatively in the dental arch. As formulation of the results to mathematical relations, the regulatory process was expressed by a non-linear Langevin equation of order parameter denoting an angle between longitudinal axes of the individual teeth. It follows that the orderly structure evolves with a change in thermodynamic potential; that is, the establishment of order in a phase transition. We propose the concept that organ differentiation is a phase transition in a dissipative system, with the decrease of gene activity substituted for temperature.

A Dynamical Extension of the Desity Functional Theory

A dynamical version of the density functional theory (DFT) for classical (equilibrium) statistical mechanics is presented. A nonlinear Langevin equation with multiplicative noise is derived for a density field. The corresponding Fokker-Planck equation is shown to have the appropriate stationary distribution expressed in terms of the free energy functional.

Breakdown of the Brownian motion model in ultrafast dynamics.

When the motion of the surrounding heat bath particles is a slow coordinate, a ``system'' particle and its bath become dynamically coupled. A nonlinear Langevin (Fokker-Planck) equation is required to describe the stochastic process. In general, the steady-state distribution is not canonical. The rapid-motion effect, in association with the dynamical coupling among multimodes of the system, and the combined contributions from nonlinear and non-Markovian processes, cause a breakdown of the conventional Brownian motion theory. The discussion given in this paper puts previous computer-simulation results on a better theoretical foundation.

Fluctuations and dissipation in a fluid under shear: Linear dynamics.

A set of nonlinear Langevin equations for fluctuations of the local conserved densities in a fluid under shear is proposed. These equations are a model for the extension of hydrodynamics to very short wavelengths at liquid densities. The hydrodynamic modes associated with the linearized equations are studied as a function of wave vector and shear rate. The degeneracy of the viscous shear modes is lifted by the shear, and one of these modes combines with the heat mode to form a propagating pair. As an example of nonequilibrium fluctuations, the dynamic structure factor is calculated for several values of frequency and wave vector. At large shear rates one pair of propagating modes becomes unstable at a wavelength of the order of the particle size. This instability is suggested as a possible explanation for a shear-induced disorder-order transition seen in computer simulations. Nonlinear mode-coupling effects are studied elsewhere.

Thermal fluctuations and ac response of weakly pinned charge density waves

The Fukuyama-Lee theory for the ac response σ(ω) of weakly pinned charge density waves is extended to include thermal fluctuations. The equation of motion for the local phase includes an extrinsic damping and a distinction is made between static and dynamic parameters in it. It is split into static, thermally fluctuating and response contributions to the phase, respectively. The static problem is treated using a result from Feigel'man's theory which provides a revised value for the weak pinning constant. The impurity averaging of the response equation is performed using the simplifying statistical properties of the stochastic pinning force following Bleher's recent work. The main emphasis is on the treatment of the thermal fluctuations via a thermal field δΦth. The non-linear Langevin equation for δΦth is linearized and further simplified by an RPA type approximation which eliminates the impurity fluctuations from δΦth. The resulting equation is solved exactly. It is shown that the correlation function of the thermal field decays initially with a short time constant. This allows to treat the thermal fluctuations on an equal footing with the impurity fluctuations in the self-consistent Born approximation. The main contribution of the thermal fluctuations results in powers of a thermal factor exp(-〈δΦth2〉/2) to the first and second order self energies of the phason Green's function. Numerical results due to these modifications are given for σ(ω,T). It is found that the absorption peak in Re σ(ω) broadens and shifts to lower frequencies when the temperature is raised. The corresponding treatment for three spatial dimensional is indicated. The thermal factor is evaluated for this case and differences to Maki's result are noted. The questions of analyticity and conductivity sum rule are also dealt with.

Localized solution of a simple nonlinear quantum Langevin equation

A simple nonlinear quantum Langevin equation is introduced as phenomenological equation for quantum brownian motion. Easy calculations yield a unique localized wave function in the stationary regime. The given example may encourage more general use of nonlinear quantum Langevin equations for damped quantum systems, e.g. in measurement theory, in heavy ion physics, etc.

Characterization of chaotic, nonstationary time series and its application to early detection of anomalies in nuclear reactors

A statistical theory of characterization of chaotic, nonstationary time series is presented. The theory is based on the identification method by the stochastic non-linear models. Time-domain analysis of stochastic process of jump events and of the bispectrum are the key methods to the present identification theory. It is emphasized that (i) the methods are sensitive to extract the non-linear nature of the systems; (ii) the generalized fluctuation-dissipation theorem for the non-linear Langevin equation becomes important for the precise identification of stochastic systems; and (iii) the methods are useful to detect anomalies in nuclear reactors.

Time evolution of the onsager regression variables: A microscopic theory

We derive a general exact expression obeyed by the Onsager regression variables for an arbitrary nonlinear markovian Langevin equation. In some approximations, we get a microscopic corroboration of the ansatz introduced by Hurley and Garrod to generalize the Onsager theorem. This is done explicitly for the nonlinear Zwanzig model.

Revisit to the scaling theory of transient phenomena — Generalization to correlated noise and singular perturbation expansion up to infinite order

A scaling theory is formulated in nonlinear Langevin equations with colored noise near the instability point from which the center of the initial distribution is deviated slightly. An intuitive derivation is given as well as perturbational calculations up to the infinite order of the noise. Moments of all orders are calculated in the approximation in which only the most dominant terms are retained in the scaling limit. Hence the scaling distribution function is also obtained. The onset time t 0 is derived as a functional of the correlation function of the noise and it is shown to increase when the noise correlation length increases.

Generalized Langevin-equation approach to impurity diffusion in solids: Perturbation theory

We derive, from a microscopic viewpoint, a nonlinear and non-Markovian Langevin equation to describe impurity diffusion in solids, assuming weak interaction between an impurity and the host-lattice atoms. The memory function and the periodic potential in the Langevin equation are calculated for the isotropic Debye lattice which models the host crystalline solid. The diffusion constant is obtained in a closed form with the use of a theory for noise-induced activation processes.

Analysis of a nonlinear stochastic model of cooperative behaviour

A stochastic model of cooperative behaviour is analyzed with regard to its critical properties. A cumulant expansion to fourth order is used to truncate the infinite set of coupled evolution equations for the moments. Linear stability analysis is performed around all the permissible steady states. The method is shown to be incapable of reproducing the critical boundary and the nature of the phase transition. A linearization, which respects the symmetry of the potential, is proposed which reproduces all the basic features associated with the model. The dynamics predicted by this approximation is shown to agree well with the Monte-Carlo simulation of the nonlinear Langevin equation.

Fluctuations Inherent in Hypercycles

A general theory of fluctuations of the hypercycles is presented. Since the elementary process is inherently non-deterministic, the hypercycle is also subject to fluctuations. On the basis of the simplified model of Schuster and Sigmond, a set of coupled nonlinear Langevin equations is obtained. The analytical solution is given for the simplest case of the length n =2. The stationary distribution function is completely different from what is expected from the deterministric analysis, and it shows that the shortest hypercycle is “dead” at all.

Generalized nonlinear Langevin equation for a rotor.

Generalized nonlinear Langevin equation for a rotor.

不規則進展抵抗をもつ疲労き裂進展の確率論的考察

A stochastic approach to fatigue crack propagation is proposed in consideration of random propagation resistence. It is based on the Paris-Erdogan's propagation law of fatigue crack. By adding the Gaussian white noise to propagation resistance of the propagation law, a non-linear Langevin equation is obtained. A stochastic differential equation of Ito type is derived from the Langevin equation by using a change of variable in the equation and Wong and Zakai's theorem. By using the solution of the stochastic differential equation and its probability density, a sample path and life distribution of fatigue crack propagation are derived, respectively. These theoretical results are compared with the experimental data for high tensile strength steel APFH 60. Through the comparison, an improvement in the above approach is made.

A Nonlinear Langevin Equation for Fluid Relaxation

Some theoretical work has focused recently on the problem of thermal relaxation in fluids. The relaxational properties of fluids at equilibrium have been investigated both experimentally!) and numerically (molecular dynamics). 2) The (Liouville) equation of motion describing the deterministic details of the dynamics of an ensem­ ble of interacting particles is currently reduced to two types of Langevin Equation (LE) in the trans­ lational velocity v(t). The generalized LE 3

Monte Carlo calculation of the asymptotic decay of the spectral density in a non linear stochastic system

A numerical Monte Carlo method is applied to analyze the asymptotic region of the spectral density for a system described by a nonlinear Langevin equation. An ω -2 behavior is found independently of the strength of the nonlinearity parameter.

Time evolution of stage transformation in GICs

A stochastic model is presented for intercalation processes and stage transformation is graphite intercalation compounds. By simulating the non-linear Langevin equation it is demonstrated that two distinct stages with well-ordered structures can coexists during stage transformation if the growth processes dominates over the diffusion process. The kinetics of stage transformation and the possible appearance of fractional stages are discussed.

Dissipative processes of energetic heavy charged particles in matter

The motion of heavy charged particles incident on matter is investigated in relation to charged particle radiotherapy. As the motion is dissipative and the mass of the charged particle is not large in comparison with that of the surrounding molecules, a non-linear Langevin equation is used. The mean kinetic energy is non-relativistically defined, and the relation between stopping power and the particle energy or the penetrating depth, and that between the range and the incident energy of the particle, are calculated by use of the Langevin equation, assuming the fluctuating force to be very small. The mass dependence of the stopping power and the range are also investigated. The theory appears to be valid, except in the region of high energy in which a relativistic treatment will be necessary.