Linear Langevin Equation Research Articles

Symmetry-based determination of space-time functions in nonequilibrium growth processes

We study the space-time correlation and response functions in nonequilibrium growth processes described by linear stochastic Langevin equations. Exploiting exclusively the existence of space- and time-dependent symmetries of the noiseless part of these equations, we derive expressions for the universal scaling functions of two-time quantities which are found to agree with the exact expressions obtained from the stochastic equations of motion. The usefulness of the space-time functions is illustrated through the investigation of two atomistic growth models, the Family model and the restricted Family model, which are shown to belong to a unique universality class in 1+1 and 2+1 space dimensions. This corrects earlier studies which claimed that in 2+1 dimensions the two models belong to different universality classes.

Delay and memory-like Langevin equations

By using an exact functional formalism we characterize general stationary properties of linear memory-like Langevin equations, including the case of delay equations driven by arbitrary noises. Spectral properties and stationary distributions are analyzed in detail.

Power-Law Tails from Multiplicative Noise

We show that the well-known linear Langevin equation, modeling the Brownian motion and leading to a Gaussian stationary distribution of the corresponding Fokker-Planck equation, is changed by the smallest multiplicative noise. This leads to a power-law tail of the distribution for sufficiently large momenta. At finite ratio of the correlation strength for the multiplicative and the additive noises the stationary energy distribution becomes exactly the Tsallis distribution.

Squeezing spectra from s -ordered quasiprobability distributions: application to dispersive optical bistability

It is well known that the squeezing spectrum of the field exiting a nonlinear cavity can be directly obtained from the fluctuation spectrum of normally ordered products of creation and annihilation operators of the cavity mode. In this article we show that the output field squeezing spectrum can be derived also by combining the fluctuation spectra of any pair of s -ordered products of creation and annihilation operators. The interesting result is that the spectrum obtained in this way from the linearized Langevin equations is exact, and this occurs in spite of the fact that no s -ordered quasiprobability distribution verifies a true Fokker–Planck equation; that is, the Langevin equations used for deriving the squeezing spectrum are not exact. The (linearized) intracavity squeezing obtained from any s -ordered distribution is also exact. These results are exemplified in the problem of dispersive optical bistability.

Enhancement of Critical Slowing Down in Chiral Phase Transition

The linear Langevin equation that describes the dynamics of the chiral phase transition above the critical temperature is derived by applying the projection operator method to the Nambu-Jona-Lasinio model at finite temperature and density. The relaxation time of the critical fluctuations increases as the temperature approaches toward the critical temperature because of the critical slowing down. The critical slowing down is enhanced in low temperature and large chemical potential region and around the tricritical point.

Functional characterization of linear delay Langevin equations

We present an exact functional characterization of linear delay Langevin equations driven by any noise structure defined through its characteristic functional. This method relies on the possibility of finding an explicitly analytical expression for each realization of the delayed stochastic process in terms of those of the driving noise. General properties of the transient dissipative dynamics are analyzed. The corresponding interplay with a color Gaussian noise is presented. As a full application of our functional method we study a model for population growth with non-Gaussian fluctuations: the Gompertz model driven by multiplicative white shot noise.

Augmented moment method for stochastic ensembles with delayed couplings. I. Langevin model

By employing a semianalytical dynamical mean-field approximation theory previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 041903 (2003)], we have developed an augmented moment method (AMM) in order to discuss dynamics of an N -unit ensemble described by Langevin equations with delays. In an AMM, original N -dimensional stochastic delay differential equations (SDDEs) are transformed to infinite-dimensional deterministic DEs for means and correlations of local as well as global variables. Infinite-order DEs arising from the non-Markovian property of SDDE, are terminated at the finite level m in the level-m AMM (AMMm), which yields (3+m)-dimensional deterministic DEs. Model calculations have been made for linear and nonlinear Langevin models. The stationary solution of AMM for the linear Langevin model with N=1 is nicely compared to the exact result. In the nonlinear Langevin ensemble, the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states. Results calculated by AMM6 are in good agreement with those obtained by direct simulations.

Enhancement of critical slowing down in chiral phase transition—Langevin dynamics approach

We derive the linear Langevin equation that describes the behavior of the fluctuations of the order parameter of the chiral phase transition above the critical temperature by applying the projection operator method to the Nambu–Jona-Lasinio model at finite temperature and density. The Langevin equation relaxes exhibiting oscillation, reveals thermalization and converges to the equilibrium state consistent with the mean-field approximation as time goes on. With the help of this Langevin equation, we further investigate the relaxation of the critical fluctuations. The relaxation time of the critical fluctuations increases at speed as the temperature approaches toward the critical temperature because of the critical slowing down. The critical slowing down is enhanced as the chemical potential increases because of the Pauli blocking. Furthermore, we find another enhancement of the critical slowing down around the tricritical point.

Functional characterization of generalized Langevin equations

We present an exact functional formalism to deal with linear Langevin equations with arbitrary memory kernels and driven by any noise structure characterized through its characteristic functional. No others hypothesis are assumed over the noise, neither the fluctuation dissipation theorem. We found that the characteristic functional of the linear process can be expressed in terms of noise's functional and the Green function of the deterministic (memory-like) dissipative dynamics. This object allow us to get a procedure to calculate all the Kolmogorov hierarchy of the non-Markov process. As examples we have characterized through the 1-time probability a noise-induced interplay between the dissipative dynamics and the structure of different noises. Conditions that lead to non-Gaussian statistics and distributions with long tails are analyzed. The introduction of arbitrary fluctuations in fractional Langevin equations have also been pointed out.

Langevin equation for the Rayleigh model with finite-range interactions.

Both linear and nonlinear Langevin equations are derived directly from the Liouville equation for an exactly solvable model consisting of a Brownian particle of mass M interacting with ideal gas molecules of mass m via a quadratic repulsive potential. Explicit microscopic expressions for all kinetic coefficients appearing in these equations are presented. It is shown that the range of applicability of the Langevin equation, as well as statistical properties of random force, may depend not only on the mass ratio m/M but also on the parameter Nm/M, involving the average number N of molecules in the interaction zone around the particle. For the case of a short-ranged potential, when N<<1, analysis of the Langevin equations yields previously obtained results for a hard-wall potential in which only binary collisions are considered. For the finite-ranged potential, when multiple collisions are important (N>>1), the model describes nontrivial dynamics on time scales that are on the order of the collision time, a regime that is usually beyond the scope of more phenomenological models.

Diffusion and memory effects for stochastic processes and fractional Langevin equations

We consider the diffusion processes defined by stochastic differential equations when the noise is correlated. A functional method based on the Dyson expansion for the evolution operator, associated to the stochastic continuity equation, is proposed to obtain the Fokker–Planck equation, after averaging over the stochastic process. In the white noise limit the standard result, corresponding to the Stratonovich interpretation of the non-linear Langevin equation, is recovered. When the noise is correlated the averaged operator series cannot be summed, unless a family of time-dependent operators commutes. In the case of a linear equation, the constraints are easily worked out. The process defined by a linear Langevin equation with additive noise is Gaussian and the probability density function of its fluctuating component satisfies a Fokker–Planck equation with a time-dependent diffusion coefficient. The same result holds for a linear Langevin equation with a fractional time derivative (defined according to Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969). In the generic linear or non-linear case approximate equations for small noise amplitude are obtained. For small correlation time the evolution equations further simplify in agreement with some previous alternative derivations. The results are illustrated by the linear oscillator with coloured noise and the fractional Wiener process, where the numerical simulation for the probability density and its moments is compared with the analytical solution.

Linear Langevin equation with time-dependent drift and multiplicative noise term: exact study

We investigate a linear Langevin equation with mulitplicative noise term and time-dependent drift. This system extends the system analyzed by Lillo and Mantegna [Phys. Rev. E 61 (2000) R4675]. We calculate the average of every moments and the variance of the system. We show that the system, with a particular choice of γ(t), can describe different processes like diffusive, subdiffusive and superdiffusive. Other kinds of processes can also been obtained. We also show that the multiplicative noise term has the effect of splitting the peak of transition probability P(x,t|x′,t′), at a fixed time, into two peaks in x coordinate.

Extremal-point densities of interface fluctuations in a quenched random medium

We give a number of exact, analytical results for the stochastic dynamics of the density of local extrema (minima and maxima) of linear Langevin equations and solid-on-solid lattice growth models driven by spatially quenched random noise. Such models can describe nonequilibrium surface fluctuations in a spatially quenched random medium, diffusion in a random catalytic environment, and polymers in a random medium. In spite of the nonuniversal character for the quantities studied, their behavior against the variation of the microscopic length scale can present generic features, characteristic of the macroscopic observables of the system.

Extremal-point densities of interface fluctuations

We introduce and investigate the stochastic dynamics of the density of local extrema (minima and maxima) of nonequilibrium surface fluctuations. We give a number of analytic results for interface fluctuations described by linear Langevin equations, and for on-lattice, solid-on-solid surface-growth models. We show that, in spite of the nonuniversal character of the quantities studied, their behavior against the variation of the microscopic length scales can present generic features, characteristic of the macroscopic observables of the system. The quantities investigated here provide us with tools that give an unorthodox approach to the dynamics of surface morphologies: a statistical analysis from the short-wavelength end of the Fourier decomposition spectrum. In addition to surface-growth applications, our results can be used to solve the asymptotic scalability problem of massively parallel algorithms for discrete-event simulations, which are extensively used in Monte Carlo simulations on parallel architectures.

Models for stochastic climate prediction.

There has been a recent burst of activity in the atmosphere/ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction. Here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory. This analysis demonstrates the potential need for more sophisticated models beyond stable linear Langevin equations. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. The strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta-plane with topography and a mean flow. In this example, the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom.

Nonlinear Langevin equations and the time dependent density functional method.

To study the time dependent density functional method (TDDFM), two streaming velocity (reversible) terms are reformulated in the nonlinear Langevin equation. Mori's [Prog. Theor. Phys. 33, 423 (1965)] projection operator method shows a variety of nonlinear Langevin equations. This is because the equations depend on the choice of phase space functions employed in the projection. If phase space functions include particular functions, however, the streaming velocity term has an invariable form. The form is independent of the choice of other phase space functions. Since the invariable streaming velocity term does not introduce the TDDFM, the second viewpoint is presented. In this, the linearization of the streaming velocity term agrees with the frequency term in the linear Langevin equation. Since only the second streaming velocity term introduces the TDDFM, one needs to be cautious in the derivation of the TDDFM.

Small delay approximation of stochastic delay differential equations

Delay differential equations evolve in an infinite-dimensional phase space. In this paper, we consider the effect of external fluctuations (noise) on delay differential equations involving one variable, thus leading to univariate stochastic delay differential equations (SDDE's). For small delays, a univariate nondelayed stochastic differential equation approximating such a SDDE is presented. Another approximation, complementary to the first, is also obtained using an average of the SDDE's drift term over the delayed dynamical variable, which defines a conditional average drift. This second approximation is characterized by the fact that the diffusion term is identical to that of the original SDDE. For small delays, our approach yields a steady-state probability density and a conditional average drift which are in close agreement with numerical simulations of the original SDDE. We illustrate this scheme with the delayed linear Langevin equation and a stochastic version of the delayed logistic equation. The technique can be used with any type of noise, and is easily generalized to multiple delays.

Spatial hole-burning effects in the amplified-spontaneous-emission spectrum of the nonlasing supermode in semiconductor laser arrays

The amplified spontaneous emission spectrum of the light field in the non-lasing supermode of two coupled semiconductor lasers is analyzed using linearized Langevin equations. It is shown that the interference betweeen the laser mode and the fluctuating light field in the non-lasing mode causes spatial holeburning. This effect introduces a phase sensitive coupling between the laser field and the fluctuations in the non-lasing mode. For high laser fields, this coupling splits the spectrum of the non-lasing mode into a triplet consisting of two relaxation oscillation sidebands which are in phase with the laser light and a center line at the lasing frequency with a phase shift of pi half relative to the laser light. As the laser intensity is increased close to threshold, the spectrum shows a continuous transition from the single amplified spontaneous emission line at the frequency of the laser mode to the triplet structure. An analytical expression for this transition is derived and typical features are discussed.

A langevin equation for the energy cascade in fully developed turbulence

Experimental data from a turbulent jet flow is analysed in terms of an additive, continuous stochastic process where the usual time variable is replaced by the scale. We show that the energy transfer through scales is well described by a linear Langevin equation, and discuss the statistical properties of the corresponding random force in detail. We find that the autocorrelation function of the random force decays rapidly: the process is therefore Markov for scales larger than Kolmogorov's dissipation scale η. The corresponding autocorrelation scale is identified as the elementary step of the energy cascade. However, the probability distribution function of the random force is both non-Gaussian and weakly scale-dependent.

A Langevin approach to stock market fluctuations and crashes

We propose a non linear Langevin equation as a model for stock market fluctuations and crashes. This equation is based on an identification of the different processes influencing the demand and supply, and their mathematical transcription. We emphasize the importance of feedback effects of price variations onto themselves. Risk aversion, in particular, leads to an “up-down” symmetry breaking term which is responsible for crashes, where “panic” is self reinforcing. It is also responsible for the sudden collapse of speculative bubbles. Interestingly, these crashes appear as rare, “activated” events, and have an exponentially small probability of occurence. The model leads to a specific “shape” of the falldown of the price during a crash, which we compare with the October 1987 data. The normal regime, where the stock price exhibits behavior similar to that of a random walk, however reveals non trivial correlations on different time scales, in particular on the time scale over which operators perceive a change of trend.