Transition Probability Rates Research Articles

Markov Models of Steady Crystal Growth

We consider solid-on-solid models of two- and three-dimensional crystal growth described by Markov rate processes whose states are representations of the entire crystal surface and whose transitions are captures of single atoms by this surface or escapes of atoms. Under natural conditions on the transition probability rates, we prove ergodicity of the Markov process. This implies the existence of states of steady growth and statistically stationary surface structure. Microscopic surface instabilities do not therefore occur under such conditions. For two dimensions, our conditions are very general; for three dimensions, they are less general.

Hard-Sphere Model of Activation Energy Originated in Ion Size: Gate Model

The gate model is proposed such that a mobile ion happens to hop between adjacent sites only when immobile (gate) ions probabilistically move away to keep an enough space: By taking account of hard-sphere ions vibrating around their stable sites, the transition probability rate Γ is statistical-mechanically evaluated according to the Vineyard method. The obtained Γ is of activation-type where the activation energy increases with the overlap length defined with radii of mobile and immobile ions, and also increases with the gate frequency.

Calculations of oscillator strengths for n = 3 to 2 and 3 transitions in F-like selenium

The oscillator strengths (and corresponding wavelengths) for stronger n = 3 to 2 and 3 transitions in F-like Selenium have been computed in both Coulomb gauge and length gauge using the Multi-Configurational Dirac-Fock (MCDF) code [Grant, I. P. et al., Comput. Phys. Commun. 21, 207 (1980)] and the successive subroutines developed by ourselves for calculating electric dipole transition probability rates. For most n = 3 to 2 transitions, our results from both gauges are in good agreement with those of Hagelstein's [Phys. Rev. A34, 924 (1986)], but for n = 3 to 3 transitions, only in length gauge, our results are close to his. It is shown that the discrepancies between our results from different gauges are large for n = 3 to 3 transitions.

Realistic evaluation of impact ionisation and Auger recombination rates for the ccch transition in InSb and InGaAsP

Accurate Bloch function overlap integrals and anisotropic energy band structures derived from the same band structure calculations have been used for the first time in the computation of impact ionisation and Auger recombination rates for the ccch transition in InSb and In0.72Ga0.28As0.6P0.4. The small-signal Auger lifetimes in InSb are shown to be more affected by the anisotropy of the energy bands than by the anisotropy of the overlap integrals. The large-signal lifetimes in InGaAsP are found to be greater by between one and two orders of magnitude than those obtained in previous theoretical calculations which had used approximate estimates for the overlap integrals. In impact ionisation the use of accurate overlap integrals introduces an additional degree of 'softness' into the threshold so that for a given direction the transition probability rate varies as (E-ET)3 for an electron of energy E near the threshold energy ET, and the direction-averaged probability rate varies as (E-ET(minimum))4.

Lasers without inversion: Single-atom transient response.

We discuss the effect of the transient response on the dynamics of lifetime broadened lasers that operate without the need for population inversion. A relationship between the steady-state absorptive transition probability rate and the transient gain and loss is given.

Microcomputer programs for markov and modified markov chain disease models

Two simulation models were created using an electronic spreadsheet. The models, IV and V, were developed from a multi-agent mastisis example and assumed Markov and modified Markov chain processes, respectively. Model IV simulated disease occurrence in which the probability of transmission remained constant. Model V simulated disease occurrence in a population which had constant state transition probability rates, except for those of a single agent, which were assumed to be dynamic and follow the assumptions of the classic Reed-Frost model. The methodology developed for these models should be easily adopted to other disease problems in either human or veterinary medicine.

Impact ionisation rate and soft energy thresholds for anisotropic parabolic band structures

The effect of anisotropic parabolic energy bands and of indirect parabolic energy bands (conduction and valence band extrema at different points in wavevector space) on the transition probability per unit time for impact ionisation is investigated. The resulting direction-dependent threshold energy is described by introducing a threshold ellipsoid. It is found that for the 'lucky-drift' model for impact ionisation in an electric field the transition probability rate which is applicable increases from zero above the energy threshold with a cubic power for anisotropic or indirect parabolic bands, or, in special circumstances a 5/2 power compared with the more usual square law of the Keldysh result for direct isotropic parabolic band structures.

Effects of Autoionizing Levels in Highly Ionized Atoms

A review is given of the contribution of autoionizing levels to photoionization, collisional ionization, dielectronic recombination, and collisional excitation, all based on a simple approach involving only perturbation-theory calculation of autoionization and radiative transition probability rates. Numerical examples are treated for dielectronic-recombination contributions to observed 2p-3d spectra in Mo29+ to Mo32+, and for resonance contributions to 2s-3s and 2p-3s collision strengths in O5+.

A function useful for analyzing single-step one-dimensional Markov processes

We introduce, for a single-step, one-dimensional Markov process, the quantity T+∞(x), defined as the fully relaxed mean waiting time for the occurrence of the transition x→x+1. We show that T+∞(x) is just as easy to calculate from the given transition probability rates as is the fully relaxed state probability function P∞(x). We further show that many interesting first-passage quantities, particularly those describing transitions and fluctuations in bimodal systems, can be expressed simply and compactly in terms of the functions T+∞(x) and P∞(x).

The convergence of a branching Brownian motion used as a model describing the spread of an epidemic

A spatial epidemic process where the individuals are located at positions in the Euclidean space R 2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R 2 according to a Brownian motion with a diffusion coefficient σ 2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.

The convergence of a branching Brownian motion used as a model describing the spread of an epidemic

A spatial epidemic process where the individuals are located at positions in the Euclidean space R2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R2 according to a Brownian motion with a diffusion coefficient σ2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.

Approximating the master equation by Fokker–Planck-type equations for single-variable chemical systems

The stochastic evolution of a well stirred chemically reacting system containing a single time-varying species X is accurately described by the master equation, in which the total number x of X molecules is an integer variable. We investigate here the legitimacy of approximating the master equation by a Fokker–Planck type partial differential equation in which x is treated as a real variable. Taking the position that any partial differential equation may be regarded as a legitimate approximation to the master equation if and only if it reduces to the master equation when subjected to a proper discretization procedure, we deduce the following: For the special case in which the various chemical reactions can alter the X molecule population by no more than one molecule at a time, a second order (two term) Fokker–Planck equation suffices. However, for the more general case in which reactions are allowed that alter the X molecule population by two molecules at a time, it is necessary to use at least a fourth order (four term) Fokker–Planck equation. In all cases it is assumed that x is large compared to unity; however, no assumptions are made about the way in which the transition probability rates and the molecular population fluctuations scale with the size of the system.

Monte Carlo simulation of random walks with residence time dependent transition probability rates

An exact, computer-oriented, Monte Carlo procedure is derived for numerically simulating continuous-time/discrete-state random walks in which the transition probability per unit time from state S m to state S n may depend upon the residence time τ in the state S m . Conditions for applicational feasibility of the simulation procedure are briefly indicated, and explicit stepping algorithms for some simple τ-dependencies are obtained.

Exact long-term total transition probability rates and Fermi's 'golden rules'

Fermi's 'golden rule' expressions refer to first- and second-order time-dependent perturbation theory results for transition probability rates involving designated final states which are members of a continuum. By use of a Laplace-average formalism, it is shown that the long-term time-averaged total transition probability rate involving designated initial and final states which are all members of a continuum is exactly the sum of the pertinent 'golden rule' expressions. Generalization to higher-order perturbation theory results is suggested.

Asymptotic behavior of Markoffian kinetics

The ergodic state with the least mean recurrence time in an irreducible Markoffian process having countable states, between any pair of which transition probability rates are definable, is the state with the least irreversible decay rate if neither the microscopic reversibility nor the doubly stochastic property holds.