Kramers-Moyal Expansion Research Articles

Kinetic equations for physisorption; With some remarks on the compensation effect

Starting from a set of rate equations for the bound state occupation function for gas-solid systems in which the surface potential has many physisorbed bound states, we derive a master equation; its kernel is explicitly calculated for phonon-mediated adsorption and desorption in a Morse potential. We give the equivalent Smoluchowski-Chapman-Kolomogorov equation for which we find the Kramers-Moyal expansion. Identifying Van Kampen's large parameter Ω for such gas-solid systems, we establish explicit criteria for the validity of a Fokker-Planck equation. The various kinetic equations are then used to calculate desorption times. The exact time evolution of the adsorbate as calculated from the set of rate equations shows that quasi-equilibrium is only maintained at low temperatures where perturbation theory of the master equation yields a simple analytic expression for the desorption time in weakly coupled gas-solid systems. At intermediate temperatures we derive another simple expression from the Fokker-Planck equation. Lower limits for the pre-exponential factor in the desorption time of the order 10 -16 s, proportional to the inverse of the heat of adsorption, are derived. We conclude with some remarks on the compensation effect.

Kinetic equations for desorption

Starting from a set of rate equations for the bound-state occupation functions for gas-solid systems in which the surface potential has many physisorbed bound states, we derive a master equation; its kernel is explicitly calculated for phonon-mediated adsorption and desorption in a Morse potential. We give the equivalent Smoluchowski-Chapman-Kolmogorov equation for which we find the Kramers-Moyal expansion. Identifying van Kampen's large parameter $\ensuremath{\Omega}$ for such gas-solid systems, we establish explicit criteria for the validity of a Fokker-Planck equation.

Moment expansion representation of probability density functions

A discussion is given of a formal way of representing probability density functions in terms of their central moments and the derivatives of the Dirac delta function. This generalized function representation is then used (a) to obtain an expression giving the value of density function at any point and (b) to exhibit an alternative way of deriving the Kramers–Moyal expansion from the Smoluchowski equation.

Non-gaussian, non-markov processes

A non-Gaussian treatment of the Mori theory is presented with the intention of explaining the results of recent computer simulations. The Mori equation is replaced by a multi-dimensional Markov process represented by a set of interrelated matrix equations. These are related straight-forwardly to a linear master equation and its Kramers–Moyal expansion. The non-Gaussian nature of the molecular dynamics results may then be represented by truncating the latter at third order, involving an extra operator Γ(1)L. This may be expanded in a matrix over the basis set of Hermite polynomials in terms of which may be represented eigenstates of the usual Fokker–Planck operator Γ(0)L.

Comparison of a semiclassical stochastic-master-equation approach to laser fluctuations with the Scully-Lamb theory

The analogy of a laser with an autocatalytic chemical reaction is used to write down a macroscopic stochastic master equation of the birth-and-death type for the photon probability function in a semiclassical laser formalism. This equation may be solved exactly in the steady state using a generating function. This leads to a prescription for calculating all the higher-order moments of the photon number as well as providing a unique truncation of the hierarchy of moment equations at any given order. Further, the variance is given by a Poisson distribution, in agreement with the strong-signal Scully-Lamb theory. This master equation is also shown to lead to an exact Fokker-Planck equation by using a modified Kramers-Moyal expansion. A comparison between our results and the (microscopic) Scully-Lamb theory shows that the two approaches give the same results at large photon number.

On the path integral solution of the master equation

In a letter by Etim and Basili a thermodynamic lagrangian was constructed starting from the differential operator in the Kramers-Moyal expansion of the master equation. We clarify the relation between this approach and the definition of the path integral. This allows us to reinterpret some of the results.

On the equivalence of time-convolutionless master equations and generalized Langevin equations

Abstract For Langevin equations with colored random noise in either a retarded (Mori) form or in a time-instantaneous form we derive an exact closed time-convolutionless masterequation. We show the equivalence of an extension of the usual Markovian nonlinear Langevin equation with both, white Gaussian noise and white generalized Poisson noise to the Kramers-Moyal expansion and derive the fluctuation induced drift.

Approximation of the Linear Boltzmann Equation by the Fokker-Planck Equation

In general, transformation of the linear Boltzmann integral operator to a differential operator leads to a differential operator of infinite order. For purposes of mathematical tractability this operator is usually truncated at a finite order and thus questions arise as to the validity of the resulting approximation. In this paper we show that the linear Boltzmann equation can be properly approximated only by the first two terms of the Kramers-Moyal expansion; i.e., the Fokker-Planck equation, with the retention of a finite number of higher-order terms leading to a logical inconsistency.

Small-parameter expansions of linear Boltzmann collision operators

An important problem in the study of irreversible processes is the approximation of the linear Boltzmann operator by the Fokker-Planck operator or, when this is insufficient, by still further terms. The first attempt at a solution of this problem was the Kramers-Moyal expansion. Van Kampen and Siegel have reformulated this expansion with the aim of systematizing terms according to order of magnitude. Very few studies have been made of this problem in the multi-dimensional case, particularly collision processes. The present paper derives a new and greatly simplified formula for the coefficients of the Kramers-Moyal series in collision processes. This is applied to Siegel's order-of-magnitude rearrangement (“CD expansion”), which is found to have the required property at least in the case of power-law molecular interactions. Finally, we present a new stochastic model, related to the theory of collision processes, for the sake of its mathematical interest and simplicity.