Average-ion level-population correlations in off-equilibrium plasmas

Abstract

We present techniques to calculate statistical distributions in time-dependent off-equilibrium atomic physics. Starting from the master equation, a first method assumes a Gaussian distribution and deduces the time evolution of the means and correlations. Alternatively, the discrete probability distribution is written as the product of the known statistical factor with the exponential of an unknown function. This function of the electronic populations can then be fitted using a second-order polynomial. Another method sets up a continuous version of the master equation, then expands the probability around the most probable configuration. It is remarkable that the obtained equation set is the same as in the Gaussian approximation of the first method. A major property of all these models is that they recover the probability distribution of thermodynamical equilibrium, when external conditions make it possible. Numerical tests on a two-level system are presented.

A Coherent Derivation of an Average Ion Model Including the Evolution of Correlations Between Different Shells

“…• the evolution equation for correlations (3.1) has new terms depending on correlations (second line) in addition to those (first line) already present in [2] and well-known references in statistics (see [4], Sec.5, especially eq.5.17).…”

“…The ion charge is given by the formula 2) and to each level j is associated an energy E j ( f ) which is also assumed to be a function of the average populations f .…”

“…[3,2]). Those models give a simplified macroscopic statistical description of a large set of ions, by calculating the populations of N "levels".…”

“…If more detail about the distribution of states (for example the N (N + 1)/2 correlations between level populations) is needed in an average-ion model, this will be calculated afterwards (cf. [2]). In a previous paper (cf.…”

“…We recall the setting and notations used in [1] and [2]: we consider a set of ions which belong to the same species of atoms in a bath of particles (electrons) at Maxwellian equilibrium at a given temperature T . We denote by Z the charge of the nucleus of the considered atomic species.…”