Astron. Astrophys. 342, 785-798 (1999)

## Appendix A: the derivation of Eq. (23)

For fixed N and k, and an arbitrary arrangement of layers (marked by the superscript m), the outgoing intensity may be expressed in terms of as follows

wherein is the optical thickness of the atmosphere. The limits of integrations represent the optical depths that correspond to boundary planes of a given layer. These are random quantities and depend on the realized particular ordering of components in the atmosphere. It should be seen that , and .

By making use of Eq. (20), the integrals in Eq. (A1) are evaluated immediately to give

where are the optical depths which correspond to the middles of the jth layer (counting from the boundary of the atmosphere) of the first and second type, respectively. Now Eq. (A1) yields

where for brevity one of the arguments in is omitted. Taking into account the second of Eqs. (19) one may write

The coordinates and can be represented in the form

wherein and are the numbers of the components of the opposite type preceding a given layer. Thus we have

with the quantities

representing the total numbers of layers of the opposite type preceding all the layers of a given type. These numbers admit an alternative interpretation that is convenient for further discussion. This rests on the concept of `transposition' widely used in the combinatorial analysis. By transposition we mean here any exchange, or swap, of two adjacent layers of different kinds. Let us agree to call `direct' the arrangement consisting of two series such as all the layers of the first structure type precede those of the second type. The opposite arrangement, obtained by a simple inversion of two series, will be referred to as the `inverted' order. It is easily seen that, for a certain random realization, the quantity is nothing more than the total number of transpositions needed to establish the direct ordering of layers, while is the total number of transpositions performed in obtaining the inverted order. On the other hand, the number of transpositions transforming the direct order into the inverted one is so that we may write

Now utilizing Eqs. (A6), (A8) in Eq. (A4), we find

Thus, we obtained the requisite explicit expression for (cf., Eq. 22) that allows us to elucidate the characteristic features of the emerging intensity. The layers of the same sort are obviously indistinguishable, so that for fixed N and k there exist different configurations. The values of (and then of are determined completely by the discrete quantity (or ), and obey the Fermi-Dirac statistics (see, e.g., Feller 1957). It follows from Eq. (A8), that the values taken by must lie between 0 and . The configurations with (i.e., ) are symmetrical. The other values of are grouped in pairs in such a way that the quantity is of the form where are integers from the interval . These correspond to the non-symmetrical configurations which are grouped in pairs (see Sect. 4).

## Appendix B: the RelMSD N for the NLTE athmosphere

To proceed to the derivation of an explicit expression for , we employ Eqs. (24) and (A1) to write

By virtue of Eq. (A8) the internal sum in the right-hand side of Eq. (B1) takes the form

where

The bracketed term in Eq. (B3) may be rewritten as follows

wherein the coefficients have a simple probabilistic meaning: is the probability that the particular distribution of layers may be converted into the direct distribution as a result of exactly j transpositions. The rigorous evaluation of the sum Eq. (B4) is of interest from the point of view of combinatorial analysis and may become the subject of a separate treatment. Nevertheless, the direct calculation of this sum for successively large values of N allows us to establish that

Thus, in place of Eq. (B3) we find that

Substituting Eq. (B2) into Eq. (B1), we shall use the following easily checked identity

to obtain

where

© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999