The evolution of white dwarfs (WDs) at their lowest luminosities is investigated by computing a reference track with solar metal and helium abundances down to the beginning of WD evolution. The main characteristics of the cooling tracks are described, including the onset of crystallization and its completion, and the differentiation in the relation T(c) - T(eff) is shown for the tracks. It is shown why the evolutionary times do not shorten abruptly at a given luminosity as a result of Debye cooling. The structure of the coolest models is shown to consist of dense atmospheres, with photospheres lying at the boundary of pressure ionization. A study of the resulting luminosity functions (LFs) shows that fast cooling never occurs, and that the LF in the crucial region log L/L(solar) between -4 and -6 is either flat or slowly decreasing. Comparisons with the observed LFs explains well the peak or flattening of the LF at log L/L(solar) = -3 or less but fails to reproduce the drop at log L/L(solar) = -4.5.
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D’Antona et al. (1989) studied this question.