Correlation function of the amplitude and of the intensity fluctuation for a laser model near threshold

In extension of a preceding paper the correlation function of the amplitude and of the intensity fluctuation are calculated in the threshold region. The laser amplitude is treated as a classical random variable obeying a van der Pol equation with a noise term. In order to get correlation functions, the method of distribution functions is employed. The distribution functions are evaluated by the Fokker-Planck equation. The lowest eigensolutions of the Fokker-Planck equation are obtained approximately by a variational method.     

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.     

Quantum mechanical masterequation and Fokker-Planck equation for the damped harmonic oscillator

For the statistical operator of the damped harmonic oscillator a Masterequation is given in operator form describing both inelastic and elastic, purely phase destroying processes. By expressing the statistical operator in the diagonal representation with respect toGlauber's coherent states the Masterequation is transformed into a Fokker-Planck equation forGlauber's quasiprobability distribution function. The general solution of this Fokker-Planck equation is calculated. It is shown how the solution of a Masterequation can be used for calculating correlation functions and expressions are given for the amplitude and intensity correlation functions which are in complete formal agreement with the corresponding classical formulae.     

Distribution function for classical and quantum systems far from thermal equilibrium

We consider a system composed of many subsystems which are coupled to individual reservoirs at different temperatures. We show how the solution of a many-dimensional Fokker-Planck equation may be reduced to a Fokker-Planck equation of dimensionn, wheren is the number of relevant constants of motion. We treat also a Fokker-Planck equation with continuously many variables and the time-dependent one. The usefulness of the present procedure to determine explicitly distribution functions is exhibited by several examples. If all temperatures are equal the Boltzman distribution function is obtained as a special case. Using the method of quantum-classical correspondence, the distribution function for quantum systems may be found.     

Distribution- and correlation-functions for a laser amplitude

In this paper a contribution to the nonlinear theory of laser noise is given. The lasing field is treated as a classical random variable, the noise is introduced by the concept of fluctuating dipoles. In order to obtain correlation functions the method of distribution functions is employed. The distribution functions are calculated by the Fokker-Planck equation.     

Optical bistability due to a two-photon absorption resonance with fluctuations

Using the intensity-dependent complex dielectric function for a two-photon absorption resonance we derive the Langevin equation for the fluctuating light-field in the non-linear resonator. The corresponding Fokker-Planck equation is solved by expanding the distribution function in terms of products of trigonometric functions and generalized Laguerre polynomials. The expansion coefficients are calculated using the method of matrix continued fractions. Numerical results for the stationary case are given.     

Fokker-Planck equation,distribution and correlation functions for laser noise

As an application of a preceding paper we set up a Fokker-Planck equation with quantum mechanically defined dissipation and fluctuation coefficients for a distribution function of the atomic variables (dipole moments and level occupation numbers) as well as of the lasing light amplitude in a laser with a homogeneously broadened line. Since the nonlinear coefficients can be linearized in appropriate coordinates well below and well above threshold, the equation can be solved with the Wang-Uhlenbeck method. Then it is easy to obtain correlation functions, spectral densities and expressions for linewidth.     

Correlation functions and correlation times for models with multiplicative white noise

Correlation functions and correlation times for the Stratonovich and Verhulst model are investigated. By transforming the Fourier transform of the corresponding Fokker-Planck equation into a tridiagonal vector recurrence relation, the Fourier transform of the correlation function and the correlation time are expressed in terms of matrix continued fractions or by similar iterations and are thus obtained numerically. By using the inverse Fourier transform, the correlation function itself is calculated. Furthermore an analytic expression in terms of an integral is obtained for the correlation time, which is evaluated exactly in the Verhulst model and asymptotically for large and weak noise strength in the Stratonovich model. A Padé expansion approximating the correlation time for all noise strength is also given.     

Projection operators in classical kinetic theory

Zwanzig's projected kinetic equation is rederived by a perturbation method. A choice of projection is proposed which, in conjunction with appropriate initial-value conditions, yields kinetic equations for the two time distribution functions of phase subsets for a system in equilibrium. These equations are generalizations of the Fokker-Planck equations in which the dissipative terms are non-Markoffian.

It is shown that exact equations for the van Hove self and distinct correlation functions are particular cases of these equations.     

Correlations in classical and quantum systems far from thermal equilibrium

We consider classical systems described by a Fokker-Planck equation or a generalized Fokker-Planck equation and quantum systems described by a density matrix equation or by a generalized Fokker-Planck equation using the principle of quantum classical correspondence. We split the corresponding operators of the equation of motion into a part which refers to the proper system and another one which describes the coupling of the proper system to the external world (reservoirs). We demonstrate that by use of conservation laws, referring to the proper systems, exact relations hold for certain moments, valid for all temperatures and coupling constants of the reservoirs. Using the concepts of a previous paper we describe then a perturbation theoretical approach which allows in a simple manner to determine a number of important correlation functions (moments of the total system). The time dependent case is briefly discussed. The applicability and usefulness of the present procedure is demonstrated by the example of the single-mode laser yielding e.g. expressions for the atom-field correlation.     

Dynamical properties of colloidal systems: II. Correlation- and response functions

For systems of interacting Brownian particles a Fokker-Planck equation is derived for the probability distribution function of the concentration fluctuations, using assumption of a Gaussian static distribution function. The drift- and the diffusion term are determined by static correlation functions. By this approach specific properties of different systems as e.g., suspensions of charged spherical particles or chain polymers are taken into account. Although the diffusion term is fluctuation dependent the properties of detailed balance and both fluctuation dissipation theorems are satisfied. Using the formalism of Martin, Siggia and Rose, Dyson- and vertex-equations for the two-particle correlation functions are derived. An explicit calculation of these functions, together with related quantities as the dynamic structure factor, and of the diffusion coefficients, is given in a mean-field approximation. The results are compared with several earlier theories, which were developed for specific systems.     

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.     

Charged-particle distribution in velocity,angle and time by Fokker-Planck equation

Summary The present paper describes a direct method of obtaining are expression of the distribution function of charged particles which diffuse in a plasma. The distribution function in velocity, angle and time is presented starting from the Fokker-Planck equation. Temperature, average velocity and energy of charged particles (electron or ions) have also been calculated as functions of time. The authors of this paper have agreed to not receive the proofs for correction.     

Quantum mechanical solutions of the laser masterequation

We treat a laser consisting of one mode described by a running wave and a set of atoms with two optically active levels which are homogeneously broadened. We start from the laser density matrix equations ofWeidlich andHaake and define a distribution functionf for lightfield and atomic variables, where we use for the lightfield the coherent state representation and for the atomic system a modified version of the distribution function used bySchmid andRisken in a previous paper. We derive a partial differential equation forf which is completely exact and is of the type of a generalized Fokker-Planck equation, i.e. it contains higher derivatives. Using a recently stated theorem ofHaken andWeidlich we show that this distribution function allows to calculate single-time as well as multitime quantum mechanical correlation functions. If the leading terms of the generalized Fokker-Planck equation are retained we find the semiclassical Fokker-Planck equation ofRisken,Schmid andWeidlich. Our treatment can be extended to several modes connected with standing waves and multilevel atoms.     

Wick's theorem and Fokker-Planck dynamics

The correlation and response functions are studied through a Wick's theorem for a Fokker-Planck system in a non-stationary situation. Non gaussian correlations are considered. The effect of the stochastic force is a dressing of the free correlation function.     

Functional Fokker-Planck treatment of electromagnetic field propagation in a thermal medium

The quantum statistics of continuous space time dependent electromagnetic fields is analyzed by means of functionals. The case of a field propagating in a thermal reservoir serves as a simple example to illustrate the succeeding steps: a masterequation is derived for the density operator which is a functional of the field operators. By means of the coherent state representation for continuous fields the masterequation is transformed into a functional differential equation in the function space, spanned by the coherent state amplitudes. This equation is of the Fokker-Planck type and determines a Gaussian process for a continuum of variables or a field. It is solved by determining the characteristics in function space of the associated equation of motion for the characteristic functional and subsequent functional integration. The solution is used to calculate some correlation functions and the spectral function of the field.     

On the theory of a detuned single mode laser near threshold

A theory of a detuned single mode laser near threshold is given using the Fokker-Planck equation technique. The Fokker-Planck equation is solved by an eigenfunction expansion. The eigenfunctions and the corresponding eigenvalues are determined by a nonhermitian operator and are calculated numerically in the threshold region. The dependence of the linewidth from the detuning is shown. In the intensity distribution the detuning enters only via a change of the scaling parameter. For the linewidth, however, an additional broadening is found. Finally it is shown that in certain cases the correlation function must not be approximated by a single exponential term.     

Stationary Response of Lotka-Volterra System with Real Noises

A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It o stochastic differential equations. The Ito formula is then carried out to obtain the It o stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged Ito stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

On the validity of the Onsager-Machlup postulate for nonlinear stochastic processes

It is shown that: (i) the Onsager-Machlup postulate applies to nonlinear stochastic processes over a time scale that, while being much longer than the correlation times of the random forces, is still much shorter than the time it takes for the nonlinear distortion to become visible; (ii) these are also the conditions for the validity of the generalized Fokker-Planck equation; and (iii) when the fine details of the space-time structure of the stochastic processes are unimportant, the generalized Fokker-Planck equation can be replaced by the ordinary Fokker-Planck equation.