Kramers-Moyal Expansion Research Articles

A test for two Fokker-Planck modellings of a master equation

The kind of two Fokker-Planck equation modellings of a non-equilibrium system, defined by a two-variable master equation, is studied. We compute explicitly their associated stationary potentials in the weak noise limit, and we compare them with the exact one coming from the master equation. We Find that a recently proposed Fokker-Planck equation is far better than the other coming from the commonly used Kramers-Moyal expansion.

Non-Markovian process driven by quadratic noise: Kramers-Moyal expansion and Fokker-Planck modeling.

A linear non-Markovian process driven by additive exponentially correlated Gaussian quadratic noise is considered. An exact master equation for a probability distribution of the process is presented. A Kramers-Moyal-type expansion of the master equation is studied. Some limiting cases of the expansion are investigated. It is demonstrated that any finite truncation of the expansion fails. An alternative and correct Fokker-Planck modeling is constructed.

The relationship between the transition matrix model and the diffusion model

The diffusion equation model and the Lefkovitch matrix model have been employed independently in plant population ecology in order to analyze the dynamics of growth and size structure. The two models describe the dynamics of size structure in biological populations, and thus there must be some relationship between them. In the present paper, we examine the theoretical relationship between these two models. We demonstrate, on a certain assumption, that the one-step Lefkovitch matrix model corresponds to a difference equation of the diffusion equation and that the two- and three-step Lefkovitch matrix model correspond to difference equations of the 4th- and 6th-order Kramers-Moyal expansions, respectively. It is also shown that 2n moments (the first to the 2n-th moments) of growth rate are necessary and sufficient to rewrite uniquely the n-step Lefkovitch matrix model in terms of the linear combination of the moments. We finally discuss the relationship between the species characteristics of census data and the appropriate types of the Lefkovitch matrix.

Diffusion equation analysis of energy transfer

Collisional energy transfer in unimolecular systems is often described using a master equation. The lack of detailed knowledge of energy transfer events combined with the computationally intensive nature of this approach makes approximate methods attractive, especially for parametrizing experimental results. One such approximation is based on the diffusion equation. In this paper a number of diffusion equation approximations is examined. Troe and Nikitin have derived such approximations by truncation of the Kramers–Moyal expansion of the master equation. A new approximation, using a similar approach is presented here. The diffusion equation can also be obtained as the limit of a sequence of master equations as the collision energy transfer becomes infinitely small and frequent. The derivation of such a limiting diffusion equation is presented. A further set of approximations in which the energy transfer process is imbedded in a diffusion process is also examined. The accuracy of all approximations is assessed by comparison with full solutions of the corresponding master equations for energy transfer in azulene and ethane, and for the kinetics of methane dissociation.

Boltzmann-like and Boltzmann-Fokker-Planck equations as a foundation of behavioral models

It is shown that the Boltzmann-like equations allow the formulation of a very general model for behavioral changes. This model takes into account spontaneous (or externally induced) behavioral changes and behavioral changes by pair interactions. As most important social pair interactions, imitative and avoidance processes are distinguished. The resulting model turns out to include as special cases many theoretical concepts of the social sciences. A Kramers-Moyal expansion of the Boltzmann-like equations leads to the Boltzmann- Fokker-Planck equations, which allows the introduction of “social forces” and “social fields”. A social field reflects the influence of the public opinion, social norms and trends on behavioural changes. It is not only given by external factors (the environment) but also by the interactions of the individuals. Variations of the individual behavior are taken into account by diffusion coefficients.

A Hopf-like equation and perturbation theory for differential delay equations

We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion.

Dispersion, convection, and reaction in porous media

The problem of transport of a reactive solute in a porous medium by convection and diffusion is studied for the case in which the solute particles undergo a first-order chemical reaction on the surface of the bed. Assuming that the geometry is periodic, the method of homogenization is applied, showing explicitly that the effective equation is given by a Kramers–Moyal expansion, i.e., a partial differential equation of infinite order in which the nth term is the product of the nth gradient of the mean concentration by an nth-order constant tensor. The effective values of reactivity, solute velocity, diffusivity, and of all the tensorial coefficients in the expansion are independent of the initial solute distribution and are expressed in terms of Peclet’s and Damkohler’s numbers, Pe=aV/D and Da=ak/D, respectively, where a is the cell size, V is the solvent mean velocity, D is the solute molecular diffusivity, and k is the surface reactivity, showing that they are independent of the initial solute distribution. Since the ratio between two successive terms in the effective equation equals the small ratio ε between the micro- and macrolength scales, truncating the expansion after the nth term allows us to find the effective concentration up to O(εn) terms. The impact of this fact is exemplified, showing that in the case of a solute flowing in a pipe with small Damkohler number Da, the effective concentration can be determined up to O(Da) terms only if the effective equation includes the skewness term. When Pe and Da are either small or large, after determining a two-parameter expansion of the solution, it is shown that the ratios between the diffusion, convection, and reaction macroscopic characteristic time scales cannot always be inferred through a naive dimensional analysis at the microscale. For example, when Da≫1 we find that the effective reaction rate tends to a constant value, independent of Da. When Pe≫1, Taylor-like dispersion, proportional to Pe2, is obtained when the mean flow is perpendicular to any vector of the reciprocal lattice. If this condition is not satisfied, the result strongly depends on the magnitude of the volume fraction of the bed particles Φ. If Pe−3≪Φ≪1, then the main mechanism causing dispersion is convection alone and the effective diffusivity is proportional to Pe; on the contrary, when Φ≪Pe−3, the effective diffusivity tends to a constant value independent of Pe.

Operator analysis of the langevin algorithm

By studying the Fokker-Planck equation in its operator form, various Langevin algorithms can be systematically derived and analyzed without the use of stochastic differential equations or the Kramers-Moyal expansion. New, but in a way canonical, second order Langevin algorithms are presented. The convergence behaviors of these new algorithms are numerically shown to be superior to that of the usual Runge-Kutta algorithm.

Transport coefficients and interaction potentials for lithium ions in helium and argon

A new approach to the calculation of ion swarm parameters has been developed, based upon a Kramers-Moyal expansion of the Boltzmann collision integral, and has made it possible to avoid the convergence problems associated with the so-called two- and three-temperature theories. The new approach has been used, together with new experimental data, to investigate the 7Li+-He and 7Li+-Ar systems. For the 7Li+-He system, excellent agreement is found between the experimental data and data calculated from a recent ab initio potential. For the 7Li+-Ar system, a similarly good agreement is found when the best available ab initio potential is modified in a direction indicated by theoretical considerations.

Bistability in active circuits: Application of a novel Fokker-Planck approach

The problem of metastability in electronic circuits with negative differential resistance, originally pioneered by Landauer in 1962, is reconsidered from the viewpoint of a Fokker-Planck modeling for nonlinear shot noise (master equation). A novel Fokker-Planck approximation scheme is presented that describes correctly the deterministic flow and the long-time dynamics of the master equation. It is demonstrated that the conventional scheme of a truncated Kramers-Moyal expansion at the second order overestimates the transition rates in leading exponential order. In order to obtain the correct relative stability, the novel scheme uses a diffusion coefficient which incorporates information about global nonlinear fluctuations characterized by the whole set of higher-order Kramers-Moyal transport coefficients.

Random walk in a quasicontinuum.

A continuous-time random walk on a spatial lattice described by the Kramers-Moyal expansion has a continuum limit described by a Fokker-Planck equation. It is often desirable to know corrections to quantities computed in the continuum limit, but truncation of the Kramers-Moyal expansion at any level other than the Fokker-Planck either breaks down or yields unphysical results. Here we introduce an alternative approximation to the Kramers-Moyal expansion which circumvents the problems of a naive truncation and correctly incorporates the first-order corrections due to the discrete lattice.

On solutions of truncated Kramers-Moyal expansions; continuum approximations to the Poisson process

We discuss the solutions of a Kramers-Moyal-expansion-type master equation for a discrete Poisson process, truncated at an arbitrary orderM. As was shown some time ago solutions withM=3, 7, 11 are in better agreement with the exact solution than the solution truncated atM=2. If a δ function is used as an initial condition, the solutions start to oscillate very rapidly with increasingM leading to a δ-function behaviour at the integer points for largeM. If, however, more smooth initial conditions are used, the rapid oscillations die out for increasingM and the solutions converge to interpolations of the exact solution.

Cycle slipping in an optical communication system employing subcarrier angle modulation

The output of an ideal photodetector that is modeled as a doubly stochastic Poisson process is used to excite a phase-locked loop. The rate of this Poisson process contains an angle-modulated subcarrier process that is modeled as a Gauss-Markov process. A measure transformation involving a martingale translation is applied to the suboptimal filter used to estimate the phase. Cycle slipping is then discussed with respect to the evolution of the phase error process under the new measure. The approach is in the context of a first-passage problem and uses a Kramers-Moyal expansion of the equation satisfied by the mean slip time. Various approximations to the jump rate and size are assumed, and a perturbation analysis results in asymptotic representations of the corresponding solutions. Comparisons with solutions obtained from so-called diffusion approximations are also included, as well as a specific example to illustrate the procedure.

Generalized random forcing in random-walk turbulent dispersion models

The Kramers–Moyal expansion is used to derive an infinite hierarchy of Eulerian moment conservation equations from a random-walk model with non-Gaussian random forcing, thus generalizing the Langevin-equation–Fokker–Planck analysis of van Dop et al. [H. van Dop, F. T. M. Nieuwstadt, and J. C. R. Hunt, Phys. Fluids 28, 1639 (1985)]. By imposing the condition that an initially well-mixed state should remain so, equations for random forcing moments of arbitrary order are derived in terms of the Eulerian velocity moments of the turbulence. This procedure makes explicit the equivalence of the different procedures used by van Dop et al. and Thomson [D. J. Thomson, Q. J. R. Meteorol. Soc. 110, 1107 (1984)] to derive the first few forcing moments and extends their results. It is then shown that the random forcing approximation implies an infinite hierarchy of Eulerian closure assumptions, the first few of which were derived by van Dop et al. The analysis is extended to a class of rescaled random-walk equations, and it is shown that the version developed by Wilson et al. [J. D. Wilson, B. J. Legg, and D. J. Thomson, Boundary-Layer Meteorol. 27, 163 (1983)] is unique in that it alone is realizable for inhomogeneous Gaussian turbulence and then has Gaussian random forcing.

Master equation without strict detailed balance

The potentials of master equations without strict detailed balance are obtained in leading order of the system-size Ω by means of the Kramers-Moyal expansion. The approach of the new Fokker-Planck equation by Hanggi et al. is suggested to be extended to many-step Markov processes.

Systematic derivation of a macroscopic equation and the Fokker-Planck equation by a scaling expansion method in stochastic processes

The coarse-graining method for a gain-loss type master equation which describes time evolution of a system following a Markov process, are discussed by introducing a scaling concept. The master equation expressed in new variables scaled by two characteristic sizes of a system, is expanded in Ω-1 and Ω-1/2. Thus, not only a macroscopic equation of motion and the Fokker-Planck equation, but also a continuity equation of hydrodynamics, are automatically obtained through the Kramers-Moyal expansion.

Asymptotic solution of the Kramers-Moyal equation and first-passage times for Markov jump processes

We calculate the activation rates of metastable states of general one-dimensional Markov jump processes by calculating mean first-passage times. We employ methods of singular perturbation theory to derive expressions for these rates, utilizing the full Kramers-Moyal expansions for the forward and backward operators in the master equation. We discuss various boundary conditions for the first-passage-time problem, and present some examples. We also discuss the validity of various diffusion approximations to the master equation, and their limitations.

Bistable systems: Master equation versus Fokker-Planck modeling

Relaxation and fluctuations of nonlinear macroscopic systems, which are frequently described by means of Fokker-Planck or Langevin equations, are studied on the basis of a master equation. The problem of an approximate Fokker-Planck modeling of the dynamics is investigated. A new Fokker-Planck modeling is presented which is superior to the conventional method based on the truncated Kramers-Moyal expansion. The new approach is shown to give the correct transition rates between deterministically stable states, while the conventional method overestimates these rates. An application to the Schl\ogl models for first- and second-order nonequilibrium phase transitions is given.

Fluctuations in reversible chemical reactions

The paper proposes a new approach to describe fluctuations of reversible chemical reactions in closed systems. The deterministic rate laws are cast into the form of nonlinear Onsager type closed laws. By means of nonlinear transport theory a Fokker-Planck equation describing the stochastic process of concentration fluctuations is obtained. It is shown that the stochastic formulation reduces to the correct deterministic rate laws in the thermodynamic limit V → ∞ with the concentrations kept fixed. Concrete examples of reactions in ideal mixtures are given and the results of the presented approach are compared with those of the usual approach by means of a birth and death type master equation. It is shown that both approaches lead to the same stationary probability and exhibit the same natural boundaries reflecting the fact of a restricted state space. The proposed Fokker-Planck equation is different from the Fokker-Planck equation obtained from the master equation by truncating its Kramers-Moyal expansion. However, the two equations are shown to have identical Fokker-Planck coefficients in the vicinity of the deterministic equilibrium state. Compared with the usual master equation approach the proposed stochastic modeling of chemical reactions has the advantage of allowing for a straightforward extension to reactions in non-ideal mixtures.

Derivation of Kramer's equation, friction coefficient, and macroscopic laws for physisorption

Starting from a master equation with the transition probabilities calculated quantum mechanically for one-phonon cascades, we derive via a Kramers-Moyal expansion the classical Kramers's equation and also van Kampen's linear Fokker-Planck equation for physisorption. The friction coefficient is calculated in terms of the microscopic parameters of the gas-solid system; it agrees with the result from a classical model. The macroscopic laws of friction for an adsorbing or desorbing gas particle are derived and solved establishing the time scales for the intermediate quasiequilibration and for the overall adsorption and desorption processes.