Correlation functions for the diffusive motion of particles in a periodic potential

The diffusive motion of particles in a one-dimensional periodic potential is treated using the Fokker-Planck equation method. First, a concise form of the Fokker-Planck equation and of the correlation function for this problem is set up. By expanding the distribution function into suitable eigenfunctions, a general method for calculating the correlation functions is then given. Finally, explicit calculations are presented for the velocity correlation functions, some of these are compared with those which were obtained by continued fraction methods.     

Photon statistics of the optical parametric oscillator including the threshold region

Starting from an effective Hamiltonian the derivation of a set of classical Langevin equations for the amplitudes of signal, idler, and pump is briefly reconsidered. From these equations all variables except those describing the signal mode are eliminated with the help of an adiabatic approximation and certain others, which are valid in the threshold region and somewhat above (i.e. photonumbers ? 10¹⁴). The signal mode amplitude then satisfies a van der Pol equation in the rotating wave approximation and is driven by a fluctuating force. With the exception of a slight difference due to the undamped phase diffusion of the pumping laser, the same Langevin equation has been derived earlier for the amplitude of a laser mode near threshold. We present the stochastically equivalent Fokker-Planck equation, whose solution is reduced to the known solution of the laser Fokker-Planck equation. Thus the complete photon statistics of the signal mode is revealed at once. In particular we obtain the stationary distribution and the amplitude and intensity correlation functions as well as the transient solution.     

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- 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.     

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.     

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.     

A polynomial expansion method for solving the laser Fokker-Planck equation

Distribution functions of the laser amplitude and intensity can be determined by solving the laser Fokker-Planck equation. By a suitable expansion of the distribution functions in Laguerre polynomials, a system of ordinary differential equations for the coefficients of the expansion is derived and is shown to have the form of a recurrence relation with length four. Applying it to the transient solution, the averaged amplitude and the first four cumulants of the intensity distribution are obtained even for those pump parameters where the hitherto known numerical solution is not applicable.     

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.     

Amplitude-energy fluctuations of optical pulses and statistics of the exit distance

We consider the propagation of an isolated optical pulse in the presence of additive noise. We apply an approach based on the Fokker-Planck equation to determine the probability distribution functions of the propagation distance at which the pulse energy and the pulse amplitude leave a certain region.     

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.     

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.     

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.     

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.     

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.     

A General Solution of the Fokker-Planck Equation

The Fokker-Planck equation is useful to describe stochastic processes. Depending on the force acting in the system, the solution of this equation becomes complicated and approximate or numerical solutions are needed. The relation with the Schrödinger equation allows building a method to obtain solutions of the Fokker-Planck equation. However, this approach has been limited to the study of confined potentials, restricting its applicability. In this work, we suggest a general treatment for non-confining potentials through the use of series of functions based on the solution of the Schrödinger equation, with part of discrete spectrum and part of continuum spectrum. Two examples, the Rosen-Morse potential and a limited harmonic potential, are analyzed using the suggested approach.     

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.     

The influence of nuclear spin on theD 2 optical pumping of alkali metal vapors

We investigate theoretically a new way of producing laser light by means of parametric processes. For this end we treat Bose fields (photons, phonons, magnons, excitons etc.) which interact via nonlinear coupling and which are coupled individually to reservoirs. The action of the reservoirs alone would lead to a Gaussian distribution of the field amplitude. We solve the corresponding Fokker-Planck equation by a recently developed novel method and show that under certain conditions for the thermal photon numbers and decay constants a laser-like photon distribution may be found for the signal.     

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.     

Distribution functions for the Brownian motion of particles in a periodic potential driven by an external force

The stationary distribution functions for the Brownian motion of particles driven by an external force are calculated by expanding the velocity part into Hermite functions and the space part into a Fourier series. Insertion into the Fokker-Planck equation leads to a matrix continued fraction for the lowest two coefficients of the Hermite functions. Higher order terms are found by reverse iteration. Results are shown for a cosine potential. The good convergence allows the calculation in the full range of damping constants. For small friction the distribution function is in good agreement with previous results and the maxima are given by the solutions without noise.     

用二维含时Fokker-Planck方程分析Tokamak中α粒子的输运

按分布函数的定义不同,描述高能带电粒子在等离子体中输运的-Planck方程有不同的形式。从数值计算的观点出发对两种不同形式的Fokker-Planck方程作了比较和评价,并指出Fokker-Planck碰撞项可解释为速度空间的对流扩散项。在此基础上用有限差分方法求解二维(速度一维,几何一维)含时Fokker-Planck方程,编制了计算程序CAPT,并将其应用于α粒子的输运研究。最后计算了典型的Tokamak D-T聚变堆参数下α粒子的损失,并给出了堆内α粒子的分布及损失α粒子的速度分布。