https://biblio.ugent.be/publication/685594/file/1130605.pdf

Review of Particle Physics

/pdf/review-of-particle-physics-1996-1997-jeg703df6k.pdf

Review of Particle Physics, 1996-1997

On the solution of population balance equations by discretization—II. A moving pivot technique

A mixed algorithm for Monte Carlo simulation of relativistic electron and positron transport in matter is described. Cross sections for the different interaction mechanisms are approximated by expressions that permit the generation of random tracks by using purely analytical methods. Hard elastic collisions, with scattering angle greater than a preselected cutoff value, and hard inelastic collisions and radiative events, with energy loss larger than given cutoff values, are simulated in detail. Soft interactions, with scattering angle or energy loss less than the corresponding cutoffs, are simulated by means of multiple scattering approaches. This algorithm handles lateral displacements correctly and completely avoids difficulties related with interface crossing. The simulation is shown to be stable under variations of the adopted cutoffs; these can be made quite large, thus speeding up the simulation considerably, without altering the results. The reliability of the algorithm is demonstrated through a comparison of simulation results with experimental data. Good agreement is found for electrons and positrons with kinetic energies down to a few keV.

/pdf/penelope-an-algorithm-for-monte-carlo-simulation-of-the-36f2nqzk4i.pdf

PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter

Author(s): Aldous, DJ | Abstract: Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x, y)/N, were AT is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x, y) = 1 and K(x, y) = xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear. © 1999 ISI/BS.

/pdf/deterministic-and-stochastic-models-for-coalescence-1vtarx7rxw.pdf

Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists

A systematic unified review is given of the basic statistical theory of the multiple scattering of fast charged particles in the small-angle range. The approximation considered is that of the Snyder-Scott-Moliere theory, and only slight attention is given to the less accurate Gaussian approximation. The single-scattering formulas of Moliere are derived, along with the modifications of them given by Nigam, Sundaresan, and Wu. Moliere's multiple-scattering calculation is presented by an improvement of Bethe's method, and the work of Nigam is given by the same method with newly computed tables. Snyder's calculations are outlined, and previously unpublished work on spatial-angle scattering is reported with tables. Calculations by Keil, Zeitler, and Zinn for very thin films are given, as well as a detailed discussion following Lenz on scattering at very small angles. The work of Muhlschlegel and Koppe on the multiple scattering of polarized electrons is included, with the important correction that no depolarization appears in the approximation to which they worked. The distributions of lateral deflections and other characteristics are considered, but the details of applications to emulsions, cloud and bubble chambers, etc, are not entered into, nor are the electron-penetration and pathlength problems handled. Asymptotic formulas for relatively large anglesmore » are treated, as are various types of mean values. (auth)« less

The Theory of Small-Angle Multiple Scattering of Fast Charged Particles

The kinetic equation for the pure growth-by-coalescence process is solved exactly for three types of overall collection probability: proportional to the sum of droplet volumes, proportional to the product of droplet volumes, and constant. Series and asymptotic expressions are given for a variety of initial conditions, and methods indicated for use of arbitrary initial functions. Calculations are presented for initial volume distributions equivalent to Gaussian distributions in radius with σ/ṙ = 0.37, O.25, 0.15 and O.123, and several stages of real time from 69 to 1600 sec. We assume 1 gm m−8 water content, a mean volume radius of 10 μ, and normalization of the collection efficiency formulas to fit those of Shafrir and Neiburger for a 30–10 μ collision.

Analytic Studies of Cloud Droplet Coalescence I

Two approximate interpolating formulas suitable for rapid computation are given that fit a combined version of the Shafrir-Neiburger and Davis-Sartor collision efficiencies for unchanged spherical cloud droplets. The method of fitting is indicated for use in approximating future improvements in the theory.

Approximate Formulas Fitted to the Davis-Sartor-Schafrir-Neiburger Droplet Collision Efficiency Calculations

The Telford statistical approach to cloud droplet growth by coalescence and the kinetic-equation approach are shown to give identical results when the Telford assumptions are used to linearize the latter. It is concluded that in this case the kinetic equation contains all relevant statistical information. A brief demonstration is given that the nonlinear kinetic equation also is statistically complete.

On the Connection Between the Telford and Kinetic-Equation Approaches to Droplet Coalescence Theory

A new growth rate formula (NGRF) is developed for the rate of growth of cloud droplets by condensation. The theory used is a modification of the Lees-Shankar theory in which the two-stream Maxwellian distribution function of Lees is used in Maxwell's method of moments to determine the transport of water vapor to and heat away from the droplet. Boundary conditions at the droplet are the usual conditions set in terms of accommodation coefficients, and the solution passes smoothly into diffusion flow in the far region. Comparisons are given between NGRF and the conventional formula showing close agreement (approximately 0.1%) for large radii with significant difference (approximately 5%) for small radii (not greater than 1 micron). Growth times for haze droplets in a Laktionov chamber are computed.

William T. Scott

Papers

The Theory of Small-Angle Multiple Scattering of Fast Charged Particles

Analytic Studies of Cloud Droplet Coalescence I

Approximate Formulas Fitted to the Davis-Sartor-Schafrir-Neiburger Droplet Collision Efficiency Calculations

On the Connection Between the Telford and Kinetic-Equation Approaches to Droplet Coalescence Theory

TWO-Stream Maxwellian Kinetic Theory of Cloud Droplet Growth by Condensation