Nonlinear Langevin Equation Research Articles

Statistical Dynamics of the Gunn Instability near Threshold

The threshold behavior of the Gunn instability is investigated. A coupled mode equation for the fluctuating electric field is reduced to a nonlinear Langevin equation that describes the stochastic motion of a single mode. The steady-state probability density for the field fluctuations is expressed in a simple exponential form, as in the case of a single mode laser. It is found that the probability density has two peaks in a region below threshold, one of which appears at the zero field and the other at a nonzero value of the field intensity. This gives rise to a hysteresis effect for the threshold behavior of the Gunn instability. The conclusion is that the Gunn instability is an example of first-order phase transitions in systems far from thermal equilibrium.

Microscopic theory of brownian motion: II. Nonlinear Langevin equations

In this article nonlinear Langevin equations for a brownian (B) particle are derived and analyzed. Attention is focussed on the role of nonlinear B particle momentum ( P) modes (powers of P). The multimode Mori formalism is used to derive equations of motion for P(t) for different numbers n of modes included in the description. The well-known linear equation of Mori corresponds to the case n = 1. Friction kernels and random forces in these equations exhibit slow decay and mass ratio (λ) expansion anomalies due to mode coupling. The nonlinear Langevin equation obtained for a complete mode set ( n = ∞) is free of these difficulties and is used to examine the first correction [ O(λ 4)] to standard O(λ 2) results. Although no closed set of nonlinear Langevin equations exists at order λ 4, a truncated set extends standard momentum correlation function predictions.

A systematic derivation of exact generalized Brownian motion theory

We present here a simple unified derivation of the exact Fokker-Planck equation obtained earlier by Zwanzig and the exact Langevin and transport equations derived by Mori. The derivation, based on the use of a Hilbert space formulation of the dynamics, leads to substantial generalizations of these results in a straightforward manner. We obtain nonlinear Langevin equations for classical systems and discuss the extension of the theory to driven transport and to quantum dynamics based either on the use of density matrices or Γ-space densities as suggested by Wigner. Remaining limitations of the theory are pointed out.

Note on fluctuation phenomena in spin systems

The Fokker-Planck equation and the (nonlinear) Langevin equation associated with the Bloch equation are derived from the microscopic model of Wangness and Bloch. These equations are also discussed in the thermodynamic limit.

Renormalization of the diffusion coefficient in a fluctuating fluid I

The diffusion of tagged particles through a fluctuating fluid is described in terms of a nonlinear Langevin-like equation, taking into account the convection by the fluid and the dependence of the “bare” diffusion coefficient on, e.g., the local density of the fluid. Upon averaging over the fluid fluctuations a macroscopic diffusion equation is obtained with a renormalized wave vector and frequency-dependent diffusion coefficient. A closed expression for the this quantity is derived in terms of the correlation functions of the fluid and the bare diffusion propagator. To lowest order in the fluid correlations this expression is evaluated explicitly as a function of the wave vector and the time and as a function of the wave vector and the frequency. For zero wave vector the usual long-time and small-frequency dependence is found. An alternative exact expression for the renormalized diffusion coefficient containing the dressed instead of the bare diffusion propagator is derived. It is argued that the last expression evaluated to lowest order in the fluid correlations completely describes the asymptotic time behavior of the diffusion memory kernel for zero wave vector. Finally the fluctuation properties of the random current in the original nonlinear Langevin equation are discussed.

Simple derivations of generalized linear and nonlinear Langevin equations

With the aid of a single operator identity, the derivation of the Mori generalized linear Langevin equation is simplified and a new generalized nonlinear Langevin equation is obtained. The flexibility of the method is stressed which allows the derivation of various generalized nonlinear Langevin equations that can be used as bases for devising approximation schemes such as the mode coupling scheme.

Brownian Motion of a Nonlinear Oscillator

Brownian motion of a nonlinear oscillator which was studied by Bixon and Zwanzig who started with a nonlinear Langevin equation is re-examined from first principles with the use of a model system in which a nonlinear oscillator is embedded in a heat reservoir composed of a simple cubic lattice. Nonlinear theory of fluctuation recently developed by Mori and Fujisaka is used to obtain a non-linear Langevin equation and a linear non-Markoffian kinetic equation with a renormalized friction term. It is shown that this equation has the same form as that derived by Bixon and Zwanzig.

On Nonlinear Dynamics of Fluctuations

A projection-operator method is developed for renormalizing kinetic coefficients by the nonlinear interaction between macroscopic modes in a nonlinear Langevin equation which is often used in studying anomalous transport phenomena near critical points and anomalous transport coefficients in two dimension. It is shown that the renormalization term is given by the time correlation of a new fluctuating force. Other theories are derived from this expression by making approximations. A relation between the Langevin fluctuating force and the transport fluctuating force is derived to clarify the relation between the Fokker-Planck equation approach by Green and Zwanzig and the linear equation-of-motion approach by one of the authors.

Threshold learning and Brownian motion (Corresp.)

An approach to the analysis of threshold learning suggested by the classical theory of Brownian motion is presented. In particular, it is shown how a nonlinear Langevin equation represents the motion of the threshold of a trainable signal detector, and how a Fokker-Planck diffusion equation yields an estimate of the shape of the probability density of the threshold. Our results are applicable to all trainable signal detectors in which the training procedure 1) raises the threshold in response to a false alarm, lowers the threshold in response to a false rest, and keeps the threshold unchanged in response to a correct decision, and 2) adjusts the size of the threshold increment by an amount that depends only on the trial number, and such that the threshold can eventually reach any real number.

Retardation of falling spheres by hydrodynamic oscillations

Terminal velocities of gravitational fall of four sizes of precision sapphire spheres (0·794–0·4764 cm dia.) and six sizes of glass spheres (0·0208–0·505 cm dia.) have been measured in a vertically oscillating water column at conditions of constant velocity, amplitude, acceleration and frequency of vibration. Vibration conditions encompassing frequency and amplitude ranges of 5–50 c/s and 0–1·0 cm respectively have been studied. The experimental terminal velocities have been compared with oscillation-retarded asymptotic velocities predicted theoretically by the numerical solution on a digital computer of a nonlinear Langevin equation proposed by Houghton (1966) as representing a force balance on a free particle in an accelerating fluid. For the larger spheres the experimental velocities were found to be lower than those predicted theoretically using steady state drag coefficients with a negligible phase lag for the displaced fluid—these velocity differences are primarily attributed to oscillation-induced increases in the drag coefficient, with appreciable secondary influences arising from changes in phase lag, drag exponent and virtual mass. Experimental velocities for the smaller particles were found to be higher than those predicted theoretically, the differences being attributed to primary reductions in the drag exponent, with second order effects arising from oscillation-induced alterations in drag coefficient, virtual mass and phase lag.

A generalized van der Pol - equation for laser noise not limited to threshold region

A nonlinear Langevin equation for laser noise is stated. It has the same analytical advantages as the van der Pol-equation but is not limited to the threshold region for typical parameters of a gas laser.

Particle retardation in vertically oscillating fluids

AbstractAn analytic expression has been deduced for the ultimate velocity of a free particle in a vertically oscillating fluid by evaluating asymptotic forms of Floquet's solution to the Mathieu equation arising from the nonlinear Langevin equation describing the particle‐fluid interactions. Using general Mathieu theory it has been rigorously demonstrated that directional particle motion in a gravitational field can be retarded by sinusoidal fluid oscillations when the particle‐fluid drag law is of square form. The earlier Rayleigh‐Ritz approximation (Houghton 1966) for the retarded particle velocity is shown to be in good agreement with the results of Mathieu theory, as summarized in graphical form.

Particle trajectories and terminal velocities in vertically oscillating fluids

AbstractThe trajectories and terminal velocities of particles in vertically oscillating fluids have been studied by obtaining analytic and numerical solutions to the nonlinear Langevin equation representing a superposition of forces arising from particle acceleration, displaced fluid acceleration, buoyancy and particle‐fluid drag as represented by an nth‐power drag law. In vibrating fluids the directional particle velocities are found to be lower than in stationary fluids for drag exponents n > 1 and correspondingly higher for 0 < n < 1. The engineering significance of the results is discussed in relation to hold‐up, separation and transport phenomena in vertically oscillating multiparticle‐fluid systems.

The behaviour of particles in a sinusoidal velocity field

A non-linear Langevin equation has been developed that represents the general behaviour of free particles in a sinusoidal velocity field. Analysis shows that a linear drag law, in which the frictional force is directly proportional to the relative particle-fluid velocity (Stokes’s law), does not result in a stable particle motion. However, a non-linear relation in which the drag is proportional to the square of the relative velocity (Newton’s law) leads to the Mathieu equation, wiien it may be shown that stable particle trajectories may occur in certain ranges of frequency and amplitude. Stability criteria based on the Mathieu equation are applied to bubble-liquid systems in the sonic and ultrasonic regions, to liquid drop systems in pulse columns and to solid-liquid systems by using drag coefficients obtained from measurements of terminal velocities. The agreement between the theoretical and observed frequencies indicates that the velocity profile around a particle in a sinusoidal field is approximated by a quasi-steady state distribution and that transient effects may be neglected for the separated flow conditions considered in the present communication.