Generalized Onsager-Machlup function as solution of a multi-dimensional Fokker-Planck equation

The multi-dimensional Fokker-Planck equation is solved by Feynman path integrals. The solution may be interpreted as generalized Onsager-Machlup function.     

Solution of the Fokker-Planck equation with spatial coordinate-dependent moments

The solution of the Fokker-Planck equation with spatial coordinate-dependent moments is given in the form of the path integral, involving the conditional probability expressed in terms of the moments at the “postpoint”. It is easily seen that it satisfies the Fokker-Planck equation.     

On the path integral solutions of the master equation

The lagrangian in the path integral solution of the master equation of a stationary Markov process is derived by application of the Ehrenfest-type theorem of quantum mechanics and the Cauchy method of finding inverse functions. Applied to the non-linear Fokker-Planck equation we reproduce the result obtained by integrating over Fourier series coefficients and by other methods.     

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.     

Stochastic quantization and path integral formulation of Fokker-Planck equation

The operator formalism (Fokker-Planck dynamics) associated to a general n-dimensional, non-linear drift, non-constant diffusion Fokker-Planck equation, is derived by a stochastic quantization from the corresponding path integral formulation in phase space.     

The Fokker-Planck equation of a laser with many modes and multi-level atoms

Using the Glauber-SudarshanP-representation for the field modes and a quasi-distribution function recently presented for arbitrary quantum systems we derive an exact generalized Fokker-Planck equation for a multi-mode laser containing a set of multi-level atoms with homogeneous and inhomogeneous level broadening. By introduction of suitable collective atomic coordinates this generalized Fokker-Planck equation is reduced to an ordinary one which may serve as a basis for the adequate treatment of laser light statistics.     

Solution of Fokker-Planck equation using Trotter's formula

Solution of Fokker-Planck equation using Trotter's formula is discussed. The method is illustrated on the linear Fokker-Planck equation and the Ornstein-Uhlenbeck solution is obtained. For the case of a general nonlinear Fokker-Planck equation the method yields an integral representation amenable to approximations. In the lowest order approximation Suzuki's scaling result emerges. Physical interpretation and limitations of the approximations are also discussed.     

Fractional generalized Langevin equation approach to single-file diffusion

Fractional generalized Langevin equation with external force is used to model single-file diffusion. It is found that for external force that varies with power law the solution for such a fractional Langevin equation gives the correct short and long time behavior for the mean square displacement of single-file diffusion when appropriate choice of parameters associated with fractional generalized Langevin equation are used. By considering some special cases of the fractional generalized Langevin equation, a new class of closed analytic expressions for the mean square displacement of single-file diffusion can be obtained. The effective Fokker-Planck equation associated with single-file diffusion is briefly considered.     

On the application of truncated generalized Fokker-Planck equations

The Pawula theorem states that the generalized Fokker-Planck equation with finite derivatives greater than two leads to a contradiction to the positivity of the distribution function. Though negative values are inconsistent from a logical point of view, we show that such distribution functions with negative values can be very useful from a practical point of view. For a Poisson-process, where the exact solution is known, we compare the solution of the second order Fokker-Planck equation to the solutions of Fokker-Planck equations of finite order. It turns out that for certain parameters the approximations of the distribution function and the moments are much better for some higher order and that the magnitude of negative values may be very small in the relevant region of variables.     

Stochastic quantization and detailed balance in Fokker-Planck dynamics

The path integral and operator formulations of the Fokker-Planck equation are considered as stochastic quantizations of underlying Euler-Lagrange equations. The operator formalism is derived from the path integral formalism. It is proved that the Euler-Lagrange equations are invariant under time reversal if detailed balance holds and it is shown that the irreversible behavior is introduced through the stochastic quantization. To obtain these results for the nonconstant diffusion Fokker-Planck equation, a transformation is introduced to reduce it to a constant diffusion Fokker-Planck equation. Critical comments are made on the stochastic formulation of quantum mechanics.     

Masterequation and Fokker-Planck equation for coupled modes of a laser with axial magnetic field

The distribution function for two circularly polarized modes in a laser with axial magnetic field is calculated. If the resonator has different damping constants for waves polarized in different directions, we find a coupling between the modes. To describe this coupling it is necessary to couple both modes to the same heatbath. Starting from a masterequation describing the damping of the atoms and the light field, we may transform the masterequation to a Fokker-Planck equation by introducing a quasi distribution function. By eliminating the atomic variables adiabatically we find a Fokker-Planck equation for the light field variables only. The solution of the Fokker-Planck equation is given in the stationary case above threshold and in the non stationary case below threshold.     

Generalized Fokker-Planck equation: Derivation and exact solutions

We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and Lévy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.     

Chapman-Kolmogorov equation and path integrals for discrete chaos in presence of noise

We derive an equation of the Chapman-Kolmogorov type for discrete multi-dimensional mappings under the action of additive and multiplicative noise with arbitrary distribution function. The resulting equation is reduced to a Fredholm integral equation. By iteration of the Chapman-Kolmogorov equation as usual, a path integral solution is found. Specializing the distribution function of the noise to a Gaussian distribution and taking the Fourier transform contant can be made with the path integral formulation used by Shraiman, Wayne and Martin.     

A solution of radiative transfer in isotropic plane-parallel atmosphere by integrating Milne equation

In this paper we solve the inversion problem of the radiative transfer process in the isotropic plane-parallel atmosphere by iterative integrations of the Milne integral equation. As a result, we obtain the scattering function in the form of a cubic polynomial in optical thickness. The author has already solved the same problem by iterative integrations of Chandrasekhar's integral equation. In the Milne integral equation, both the cosines of the viewing angles and the optical thickness are integral variables, while in Chandrasekhar's integral equation the cosines of the viewing angles are variables but the optical thickness is not. We derive several series of exponential-like functions as intermediate derivations. Their convergences are evaluated by the author's previous work in the solution of Chandrasekhar's integral equation. The truncated scattering function up to the third order in optical thickness thus obtained is identical to that obtained from Chandrasekhar's integral equation, though their apparent forms are different. Chandrasekhar pointed out that the solution of Chandrasekhar's integral equation does not have a uniqueness of solution. The Milne equation, in contrast, has been proven to have a unique solution. We discuss the uniqueness of the solution by these two methods.     

The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems. In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form. Received 26 February 1999     

The Fokker-Planck equation in the second-order pitch angle approximation and its exact solution

The diffusive particle propagation and its pitch angle scattering is studied using kinetic equation of the Fokker-Planck form. The case is considered when charged particles preferable propagate along the strong mean magnetic field direction and undergo the pitch angle scattering with respect to it. The paper deals with solution of the equation for particle distribution function in the second-order approximation in the pitch angle. The exact analytical solution is obtained in an integral form. The well-known solution in the first-order pitch angle approximation can be restored performing the small time limit in the result. Unlike the first-order solution the obtained solution in the second approximation rightly shows that the pitch angle diffusion is closely connected with the particle transport along the mean magnetic field. The expression for particle density for the point instantaneous unidirectional source also has been obtained.