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The paper continues a series of papers devoted to treatment of thecrystalline state on the basis of the approach in equilibrium statisticalmechanics proposed earlier by the author. This paper is concerned withelaboration of a mathematical apparatus in the approach for studyingsecond-order phase transitions, both commensurate and incommensurate, andproperties of emerging phases. It is shown that the preliminary symmetryanalysis for a concrete crystal can be performed analogously with the one inthe Landau theory of phase transitions. After the analysis one is able todeduce a set of equations that describe the emerging phases and correspondingphase transitions. The treatment of an incommensurate phase is substantiallycomplicated because the symmetry of the phase cannot be described in terms ofcustomary space groups. For this reason, a strategy of representing theincommensurate phase as the limit of a sequence of long-period commensuratephases whose period tends to infinity is worked out. The strategy enables oneto obviate difficulties due to the devil's staircase that occurs in thissituation.

Incommensurate phases in statistical theory of the crystalline state