Large deviations for the stochastic derivative Ginzburg-Landau equation with multiplicative noise

This paper first proves the existence of a unique mild solution to the stochastic derivative Ginzburg-Landau equation. The fixed point theorem for the corresponding truncated equation is used as the main tool. Since we restrict our study to the one-dimensional case, it is not necessary to introduce another Banach space and thus the estimates of the stochastic convolutions in the Banach space are avoided. Secondly, we also consider large deviations for the stochastic derivative Ginzburg-Landau equation perturbed by a small noise. Since the underlying space considered is Polish, using the weak convergence approach, we establish a large deviations principle by proving a Laplace principle.     

Steady-state analysis subject to a coloured and a white additive of a bistable system multiplicative noise noise with coloured cross-correlated noises

The steady-state properties of a bistable system are investigated when both the multiplicative noise and the coupling between additive and multiplicative noises are coloured with different values of noise correlation times T1 and T2. After introducing a dimensionless parameter R（R = α/D, D is the intensity of the multiplicative noise and a is the intensity of the additive noise）, and performing the numerical computations, we find the following points： （1） For the case of R 〉 1, A （the intensity of correlation between additive and multiplicative noises）, T1 and T2 can induce the stationary probability distribution （SPD） transition from bimodal to unimodal in structure, but for the cases of R _〈 1, the bimodal structure is preserved; （2） a can also induce the SPD transition from bimodal to unimodal in structure; （3） the bimodal structure of the SPD exhibits a symmetrical structure as D increases.     

Stochastic perturbative derivation of the axial anomaly

The axial anomaly is calculated as the infinite Langevin time limit of stochastic triangle diagrams. Their regularization is insured with the help of an analytic stochastic regulator. The usual axial anomaly is recovered only when the Langevin equations used to generate the perturbative expansion are gauge covariant.     

Stochastic resonance in a bistable system with coloured correlation between additive and multiplicative noise

The phenomenon of stochastic resonance (SR) in a bistable nonlinear system with coupling between additive and multiplicative noises is investigated when the correlation between two noise terms is coloured. It is found that the signal-to-noise ratio (SNR) of the system is affected not only by the coupling strength λ between two noise terms, but also by the noise correlation time τ. The SNR is changed from a single peak, to two peaks with a dip, and then to a monotonically decreasing function with noise strength. The dependence of the SR on the initial conditions is entirely caused by the coupling strength λ between two noise terms.     

Fokker-Planck equation for a multiplicative stochastic process with a stationary gaussian noise

Starting from a Langevin equation with stationary gaussian noise of arbitrary correlation time, a corresponding Fokker-Planck equation is derived under the condition of small noise strength.     

Stochastic resonance in an asymmetric bistable system driven by coloured correlated multiplicative and additive noise

This paper investigates the stochastic resonance （SR） phenomenon in an asymmetric system with coupling between multiplicative and additive noise when the coupling between two noise terms is coloured. The approximate expression of signal-to-noise ratio has been obtained by applying the two-state theory and SR exhibits in the bistable system. Moreover, the potential asymmetry r and cross-correlation strength λ can weaken the SR phenomenon, while the cross-correlation time r can strengthen the SR phenomenon.     

On the interpretation of multiplicative white noise

The interpretation of a stochastic differential equation as an approximation to a stochastic difference equation is discussed. A criterion is presented for determining which stochastic calculus should be used.     

A derivation of the Broadwell equation

We consider a stochastic system of particles in a two dimensional lattice and prove that, under a suitable limit (i.e.N, 0,N²const, whereN is the number of particles and is the mesh of the lattice) the one-particle distribution function converges to a solution of the two-dimensional Broadwell equation for all times for which the solution (of this equation) exists. Propagation of chaos is also proven.Research partially supported by CNR-PS-MMAIT     

Rigorous derivation of the Gross-Pitaevskii equation

The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schr?dinger equation with a short-scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short-scale correlation structure in the condensate.     

Branch selection in the presence of coloured noise

We have experimentally studied the effects of exponentially correlated, gaussian, additive noise on an electronic bistable system whose instability is driven by a time dependent external control parameter. The results, which indicate that the noise correlation time has little influence on the statistics of branch selection, are in substantial agreement with the predictions of a simple, physically motivated theory outlined herein.     

Exact derivation of the synchrotron-radiation equation

The energy radiated by a charge moving along a circle or a helix is calculated exactly (i.e., without an expansion in terms of some parameter). The method used seems extremely promising for analyzing macroscopic systems for the case in which the observer may be much closer to the source than the wave zone. These equations can be used to analyze the coherent radiation of several electrons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 14–24, April, 1971.     

Associated relaxation time and intensity correlation function of a bistable system driven by cross-correlation additive and multiplicative coloured noise sources

The associated relaxation time and the intensity correlation function of a bistable system driven by an additive and a multiplicative coloured noise with coloured cross-correlation are investigated. Using the Novikov theorem and the projection operator method, the analytic expressions of the stationary probability distribution P_st(x), the relaxation time T_c, and the normalized correlation function C(s) of the system are obtained. The effects of the noise intensity, the cross-correlation strength λ and the cross-correlation time τ are discussed. By numerical computation, it is found that the cross-correlation strength |λ| and the quantum noise intensity D decrease the relaxation of the system from unstable points. The cross-correlation time τ delays relaxation of the system from unstable points. The cross-correlation strength λ and the cross-correlation time τ can alter the effects of the pump noise intensity Q. Thus, the relaxation time T_c is a stochastic resonant phenomenon, and distribution curves exhibit a single-maximum structure.     

A derivation of the compressibility equation for liquid metals

The Kirkwood and Buff compressibility equation for multicomponent systems is applied to an assembly of ions and electrons, where the Coulomb potential is replaced by the Yukawa potential. It is then shown that the resulting formula gives the Watabe-Hasegawa-Chihara compressibility equation in the limit of infinitesimal damping constant of the Yukawa potential.     

Renormalized perturbation theory for a multiplicative noise system

We study an anharmonic overdamped oscillator driven by both additive and multiplicative Gaussian white noise. Within the Martin-Siggia-Rose formalism we employ renormalized perturbation theory to calculate the stationary correlation function and the response function of the oscillator variable. Due to the absence of a simple fluctuation-dissipation theorem renormalization cannot make use of exact stationary moments, but is perormed with the help of renormalization conditions.     

Work fluctuation theorem for coloured noise driven open systems

We have studied work fluctuation behaviour in the presence of internal thermal noise as well as external coloured noise. The external coloured noise may have both Gaussian or non-Gaussian characteristics. We have investigated the dependence of position and work fluctuations on the properties of both the environments. For thermal noise driven systems, there is a maximum in the variation of mean square fluctuation of work (MSFW) as a function of damping strength at intermediate times, while at asymptotic long times MSFW monotonically increases in the same damping regime. But for external noise, MSFW monotonically decreases as a function of damping strength at intermediate times, whereas at long times it becomes almost independent of damping strength.Another interesting observation is that for the external noise driven systems, noise correlation time and damping strength have similar roles in the dynamics.     

Renormalization-group derivation of Navier-Stokes equation

The Navier-Stokes equation is proved from first principles (rotational symmetry and conservation of momentum, mass, and energy) using renormalization-group ideas. That is, we consider a system described by one (classical) conserved vector field and two conserved scalar fields, and demonstrate that on a large scale it obeys the Navier-Stokes equation. No assumptions about the physical meanings of the fields are required; in particular, no results from thermodynamics are used. The result comes about because the Euler equation is an exact fixed point of an appropriate scale-coarsening transformation, and the coefficients of the eigenvectors of the transformation with the largest (most relevant) eigenvalues include (in dimensiond>2) the thermal conductivity and the bulk and shear viscosities, leading to Navier-Stokes behavior on a large scale. Ford<2, the largest eigenvalue corresponds to a convection term, and the Navier-Stokes equation is incorrect. Our method differs from previous renormalization approaches in using time-coarsening as well as space-coarsening transformations. This allows renormalization trajectories to be determined exactly, and allows the determination of the macroscopic behavior of specific microscopic models. The Navier-Stokes equation we obtain is almost, but not exactly, the same as the conventional one; distinguishing between them experimentally would require measurement of the very small asymmetry of the Brillouin line in a simple fluid.     

Statistical properties of a single-mode laser driven by additive and multiplicative coloured noises with a coloured cross-correlation for different correlation times

We study a system for a single-mode laser driven by additive and multiplicative coloured noises with a coloured cross-correlation. The analytical expression of the stationary intensity distribution (SID) for the laser is derived in the case of three different correlation times. The influences of each stochastic parameter on the SID are discussed, the effects of the noise "colour" and the cross-correlation of noises on the mean intensity , the variance, λ_2(0), and the skewness,λ_3(0) of the single-mode laser are investigated. We find that there are colourful phase transitions for the SID above a threshold, and re-entrant transitions induced by the "colour" of the additive noises. Further research of the curves of -τ_i, λ_2(0)-τ_i, and λ_3(0)-τ_i (i=1,2,3) (where τ_i is the correlation time) shows that the output intensity not only increases with the additive noise correlation time τ_2 and the cross-correlation time τ_3, but also the quality of the output of laser beams is optimized.