Master Equation and Fokker–Planck Equation: Comparison of Entropy and of Rate Constants

A usual approximation of the master equation is provided by the Fokker–Planck equation. For chemical systems with one species, we prove generally that the prediction of the rate constant of the metastable state given by the Master equation and the Fokker–Planck approximation differ exponentially with respect to the size of the system. We show that this is related to the fact that the entropy of the metastable state is not described correctly by the Fokker–Planck equation. We prove that the rate given by the Fokker–Planck equation overestimates that rate given by the Master equation.     

Numerical algorithm for the time fractional Fokker–Planck equation

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker–Planck equation. In this paper, firstly the time fractional, the sense of Riemann–Liouville derivative, Fokker–Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann–Liouville derivative and Caputo derivative. Then combining the predictor–corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(k^min{1+2α,2})+O(h²), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for α=1.0 with the ones of directly discretizing classical Fokker–Planck equation, some numerical results for time fractional Fokker–Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for α=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.     

Direct quadrature method of moments solution of the Fokker–Planck equation

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker–Planck equation corresponding to stochastic nonlinear dynamical systems. The theoretical details of the solution procedure are first presented. The method is then used to solve Fokker–Planck equations for both 1D and 2D (noisy van der Pol oscillator) processes which possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared to analytic (1D process) and Monte Carlo (2D process) solutions.     

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

Statistical Properties of the Polarized Radiation of a Dye–Solution Laser

Within the framework of the method of polarization components we obtained the Fokker–Planck equation for the intensity–distribution function of an individual component. Solutions for the Ornstein–Uhlenbeck process show that the experimentally observed special features in the behavior of the distribution function of the intensity and degree of polarization of laser radiation in the vicinity of the threshold are well described in the approximation of statistical independence of polarization components. However, since the Ornstein–Uhlenbeck process includes states not realizable physically for the given case, an exact solution of the Fokker–Planck equation is constructed by the method of expansion in eigenstates. It is shown that this solution is totally correct physically and yields virtually the same values for the distribution functions of the intensity and degree of polarization of radiation as the dependences obtained earlier for the Ornstein–Uhlenbeck process.     

Kinetic Equation for Micromaser Generation by Correlated States of Atoms

A kinetic equation for the light generated by atoms with states prepared by a pumping mechanism producing atomic correlation is derived in a rigorous way. It is found that only two-particle correlations need to be taken into account within the framework of the Fokker–Planck approximation. The kinetic equation derived is applied to the process of micromaser operations and the light noise is discussed. A new mechanism of pumping based on correlated atomic states prepared by telecloning protocols is considered.     

1/2-Order Fractional Fokker–Planck Equation on Comblike Model

From the generalized scheme of random walks on the comblike structure, it is shown how a 1/2-order fractional Fokker–Planck equation can be derived. The operator method for the moments associated with the distribution function p(x,t) is used to solve the resulting equation. Also the anomalous diffusion along the backbone of the structure has been considered.     

A Non-Maxwellian Steady Distribution for One-Dimensional Granular Media

We consider a nonlinear Fokker–Planck equation for a one-dimensional granular medium. This is a kinetic approximation of a system of nearly elastic particles in a thermal bath. We prove that homogeneous solutions tend asymptotically in time toward a unique non-Maxwellian stationary distribution.     

A Remark on the Kramers Problem

We present new point of view on the old problem, the Kramers problem. The passages from the Fokker–Planck equation to the Smoluchowski equation, including corrections to the Smoluchowski current, is treated through an asymptotic expansion of the solution of the stochastic dynamical equations. The case of an extremely weak force of friction is also discussed.     

Statistical Mechanics in Collective Coordinates

We study the transformation of the statistical mechanics of N particles to the statistical mechanics of fields, that are the collective coordinates, describing the system. We give an explicit expression for the functional Fourier transform of the Jacobian of the transformation from particle to collective coordinate and derive the Fokker–Planck equation in terms of the collective coordinates. Simple approximations, leading to Debye–Huckel theory and to the hard sphere Percus–Yevick equation are discussed.     

Diffusion of Finite-Mass Particles in a Turbulent Viscous Medium

Based on a Fokker–Planck equation for the Jacobian of mapping of Lagrangian into Eulerian coordinates, derived in this paper, we analyze the process of diffusion of passive-tracer particles in a turbulent viscous medium. Solving this equation allows for studying the effect of finite masses (inertia) of the tracer particles on the appearance of multi-flow motion.     

Radiation Polarization Distribution Function of a Dye Laser

The Fokker–Planck equation for the distribution function of the intensity of an individual component has been solved in the approximation of the Ornstein–Uhlenbeck process within the framework of the formalism of the method of polarization components. Based on this solution, we have constructed the distribution functions of the degree of lasing radiation polarization, analyzed experimental data for a certain geometry of laser pumping, and determined the values of the distribution parameters, including the loss coefficients for the polarization component.     

Ultra-short laser ablation of dielectrics: Theoretical analysis of threshold damage fluence and ablation depth

A coupled theoretical model based on Fokker–Planck equation for ultra-short laser ablation of dielectrics is proposed. Multiphoton ionization and avalanche ionization are considered as the sources during the generation of free electrons. The impact of the electron distribution in thermodynamic nonequilibrium on relaxation time is taken into account. The calculation formula of ablation depth is deduced based on the law of energy conservation. Numerical calculations are performed for the femtosecond laser ablation of fused silica at 526 and 1053 nm. It shows that the threshold damage fluences and ablation depths resulted from the coupled model are in good agreement with the experimental results; while the damage thresholds resulted from the approximate model significantly differ from the experimental results for lasers of long pulse width. It is concluded that the coupled model can better describe the micro-process of ultra-short laser ablation of dielectrics.     

Quantum Tomography, Wave Packets, and Solitons

The wave packets, both linear and nonlinear (like solitons) signals described by a complex time-dependent function, are mapped onto positive probability distributions (tomograms). The quasidistributions, wavelets, and tomograms are shown to have an intrinsic connection. The analysis is extended to signals obeying to the von Neumann-like equation. For solitons (nonlinear signals) obeying the nonlinear Schrödinger equation, the tomographic probability representation is introduced. It is shown that in the probability representation the soliton satisfies a nonlinear generalization of the Fokker–Planck equation. Solutions to the Gross–Pitaevskii equation corresponding to solitons in a Bose–Einstein condensate are considered.     

Moment Equations for a Granular Material in a Thermal Bath

We compute the moment equations for a granular material under the simplifying assumption of pseudo-Maxwellian particles approximating dissipative hard spheres. We obtain the general moment equations of second and third order and the isotropic moment equations of any order. Our equations describe, in the space homogeneous case, the granular system described by a Boltzmann-like collision term and subject to a Brownian motion due to the interaction with a bath, described by a Fokker–Planck term. The trend to equilibrium is studied in detail.     

Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.     

On a Kinetic Model for a Simple Market Economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker–Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis.