Coherence and saturation properties of second-harmonic generation with a nonlinear crystal inside the laser cavity

The basic equations for second-harmonic generation including noise are derived for the case that the nonlinear crystal is put inside the laser cavity. A realistic model of a (detuned) laser with two-level atoms in single-mode operation is taken using the nonlinear theory of laser noise which describes the laser saturation effects, the phase diffusion and the intensity fluctuations. The reaction of the second-harmonic field on the fundamental field is taken into account as well as the reaction of the fundamental field on the laser. The nonlinear crystal is described by microscopic anharmonic oscillator equations (without introducing nonlinear susceptibilities by perturbation theory). The saturation of the polarization of the nonlinear medium is taken into account exactly with the only assumption that the influence of third and higher harmonics should be small. The electromagnetic field is described semiclassically by stochastic equations. In all equations, the damping is introduced simultaneously with Markoffian fluctuating forces by coupling to heatbaths. The equations are solved exactly in the stationary state without noise (the time dependent solution including noise will be presented in a subsequent paper). The most important saturation effect is a frequency shift which depends on the laser intensity.     

Theory of external two-state Markov noise in the presence of internal fluctuations

A theory is presented to take into account internal fluctuations in the study of stochastically driven systems. Internal fluctuations are modeled by a master equation in which external noise is introduced. External noise is modeled by a two-state Markov process. A unified theory of internal and external fluctuations is described in terms of an effective integrodifferential master equation or its equivalent generating function representation. Two examples for which exact analytical results can be obtained are presented. A discussion of the white noise limit of the theory is also given.     

表面催化反应模型中关联噪声诱导非平衡相变

建立含有关联噪声的双分子-单分子(DM)表面催化反应延迟反馈模型,该模型能同时显示一级和二级非平衡动力学相变,即在一级和二级非平衡动力学相变之间的反应窗口展现.讨论双分子在DM延迟反馈模型中两种吸附机制,即局域和随机吸附模型.研究结果表明:1)外部噪声及两噪声关联性致使反应窗口的宽度收缩;2)内部噪声对非平衡动力学相变行为的影响依赖两噪声关联性,即当两噪声负关联,内部噪声致使反应窗口的宽度变宽;而当两噪声正关联时,内部噪声致使反应窗口的宽度收缩;3)关联噪声致使反应窗口变化对DM模型中一级和二级非平衡动力学相变研究具有重要的科学意义.     

Theory of fluctuations in quasiharmonic oscillators

An approach is developed for analyzing fluctuation effects in quasiharmonic oscillators formed by connecting a nonlinear inertialess element to a resonator. The effect of the nonlinear element on the resonator is treated as a pulsed process. It is assumed that the effect of noise on the system destroys the coherence and changes the response parameters of the resonator to pulse effects, this, ultimately, being responsible for the amplitude and frequency fluctuations. An equation is obtained that describes the fluctuations of the oscillation period. A detailed study is made of the fluctuations of the period in an electronic oscillator due to white noise.     

Dynamical correlations near instabilities in nonlinear chemical reaction systems

Projection operator techniques are applied to stochastic master equations for homogeneous chemical reaction systems to derive a continued fraction representation for dynamical correlations of the particle numbers. The formalism is applied to two simple nonlinear chemical reactions which exhibit first and second order phase transition analogies, respectively. Numerical results are obtained and various approximations are investigated to describe memory effects arising at the instability points. The method presented here provides a systematic way of investigating the dynamics of nonlinear chemically reacting systems showing unstable behaviour and enhanced fluctuations far from thermal equilibrium.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

Dynamics of nonlinear oscillators with fluctuating parameters

We derive the (integro-differential) master equation of an oscillator in a thermal environment which is driven by a non-linear randomly varying force. The thermal noise is assumed to be δ-correlated gaussian noise and the parameter fluctuations are assumed to be multiplicative white Poisson noise. For the case of a large viscosity we derive a generalized Smoluchowski equation and sketch the modification of Kramers' reaction rate. The rate is shown to contain a temperature-independent “tunneling” contribution.     

The Theory of the Convectively unstable Random Ionization Waves

A linear theory of the fluctuations growth in the positive column of a glow discharge due to ionization wave weak convective instability is given. The spectrum of the fluctuations as function of the position along the axis of the column is calculated assuming localised, white noise source. The calculated results are in some cases in a good agreement with the experimental observations.     

受迫薛罗格双匣化学反应模型的随机共振

为了在理论上揭示高斯白噪声激励的薛罗格双匣化学反应模型对弱周期扰动的线性与非线性响应 ,分四态近似和两态近似两种情形 ,基于绝热近似与速率方程方法 ,解析导出线性的和非线性的敏感性以及信噪比的表达式 ,并与数值模拟结果进行比较 ,在一次谐波的意义上得到了解析结果与数值模拟结果的定量一致性 .理论上讲 ,该模型只能表现出奇次谐波的随机共振 ,但数值模拟结果也出现了二次谐波的随机共振 ,其原因可能归结为在数值模拟中有限频率的截断引入了误差 ,也可能归结为信号的高次谐波与背景噪声难以区分所致 .     

Stochastic dynamics of a bistable reaction system

The method of stochastic dynamics is used for the study of a discrete trimolecular chemical model with internal fluctuations. Several trajectories of the phase point are simulated for the bistable Schlögl model at small volumes. The influence due to the discontinuity of the process on the stationary probability distribution function is analyzed. The transition to the bimodal distribution is shown to be a cusp catastrophe. The location of the cuspoid strongly depends on the reaction volume. Autocorrelation functions are calculated numerically for short and large time scales.     

The classical nucleation rate in two dimensions

In many systems in condensed matter physics and quantum field theory, first order phase transitions are initiated by the nucleation of bubbles of the stable phase. In homogeneous nucleation theory the nucleation rate can be written in the form of the Arrhenius law: . Here is the energy of the critical bubble, and the prefactor can be expressed in terms of the determinant of the operator of fluctuations near the critical bubble state. In general it is not possible to find explicit expressions for and . If the difference between the energies of the stable and metastable vacua is small, the constant can be determined within the leading approximation in , which is an extension of the ”thin wall approximation”. We have done this calculation for the case of a model with a real-valued order parameter in two dimensions. Received: 11 September 2002 / Revised version: 30 October 2002 / Published online: 24 January 2003 RID="a" ID="a" e-mail: munsteg@uni-muenster.de RID="b" ID="b" e-mail: rut@ifttp.bas-net.by ^* Present address: Universit?t Essen, Fachbereich 7 - Physik, Universit?tsstr. 5, 45117 Essen, Germany     

On the theory of fluctuations around non-equilibrium steady states: A generalized time-convolutionless projector formalism

A new method of finding nonlinear Langevin type equations of motion for relevant macrovariables and the corresponding master equation for systems far from thermal equilibrium is presented by generalizing the time-convolutionless formalism proposed previously for equilibrium hamiltoian systems by Tokuyama and Mori. The Langevin type equation consists of a fluctuating force, and the nonlinear drift coefficients which are always identical to those of the master equation. A simple formula which relates the drift coefficients to the time correlation of the fluctuating forces is derived. This is a generalization of the fluctuation-dissipation theorem of the second kind in equilibrium systems and is valid not only for transport phenomena due to internal fluctuations but also for transport phenomena due to externally-driven fluctuations. A new cumulant expansion of the master equation is also obtained. The conditions under which a Langevin and a Fokker-Planck equation of a generalized type for non-equilibrium open systems can be derived are clarified.The theory is illustrated by studying hydrodynamic fluctuations near the Rayleigh-Bénard instability. The effects of two kinds of fluctuations, internal fluctuations of irrelevant macrovariables and external (thermal) noises, on the convective instability are investigated. A stochastic Ginzburg-Landau type equation for the order parameter and the corresponding nonlinear Fokker-Planck equation are derived.     

Oscillations and Waves in a Nonlinear System with the 1/<Emphasis Type="Italic">f</Emphasis> Spectrum

Numerical methods are used to study a spatially distributed system of two nonlinear stochastic equations that simulate interacting phase transitions. Conditions for self-oscillations and waves are determined. The 1/f and 1/k spectra of extreme fluctuations are formed when waves emerge and move under the action of white noise. The distribution of the extreme fluctuations corresponds to the maximum entropy, which is proven by the stability of the 1/f and 1/k spectra. The formation and motion of waves under external periodic perturbation are accompanied by spatiotemporal chaotic resonance in which the domain of periodic pulsations is extended under the action of white noise.     

Unstable state dynamics: Treatment of colored noise

The diffusion of a particle set near an unstable point in a bistable potential is considered. The scaling theory of fluctuations proposed originally for onedimensional systems driven by Gaussian white noise is extended to arbitrary dimensions. The merits and drawbacks of the scaling theory are discussed by taking a model problem in one dimension. It is shown in passing that the saddle point approximation enables one to get analytic expressions for various moments of the stochastic process. The two different methods to include asymptotic fluctuations-which are absent in the usual scaling solution-are shown to be equivalent. An alternate way of including asymptotic fluctuations is attempted by solving the associated Fokker-Planck equation using the Fer formula. The reason for the failure of this method is traced. After this, it is argued that the unified scaling theory should be applicable for treatment of colored noise as well, for the scaling assumption is independent of the statistical property of the driving noise. Explicit Monte Carlo simulation of a model onedimensional system driven by exponentially correlated Gaussian noise is performed and compared with the scaling solution to bolster this point. The agreement is very good.     

Monofractal and multifractal analysis of simulated heat release fluctuations in a spark ignition heat engine

We study data from cycle-by-cycle variations in heat release for a simulated spark-ignited engine. Our analyses are based on nonlinear scaling properties of heat release fluctuations obtained from a turbulent combustion model. We apply monofractal and multifractal methods to characterize the fluctuations for several fuel-air ratio values, ?, from lean mixtures to stoichiometric situations. The monofractal approach reveals that, for lean and stoichiometric conditions, the fluctuations are characterized by the presence of weak anticorrelations, whereas for intermediate mixtures we observe complex dynamics characterized by a crossover in the scaling exponents: for short scales, the variations display positive correlations while for large scales the fluctuations are close to white noise. Moreover, a broad multifractal spectrum is observed for intermediate fuel ratio values, while for low and high ? the fluctuations lead to a narrow spectrum. Finally, we explore the origin of correlations by using the surrogate data method to compare the findings of multifractality and scaling exponents between original simulated and randomized data.