Statistics of molecular excitons and magnons at high concentrations

Exact commutation relations are obtained for the operators of quasiparticles in a periodic lattice (Frenkel' excitons, magnons). The commutation relations are trilinear in the operators of creation and annihilation of quasipartieles and contain a conservation law of the quasimomentum. The statistical properties of the quasiparticles are described by modified parastatistics of rank $\mathfrak N$ ($\mathfrak N$ is the number of sites of the lattice in the delocalization region of the excitation). It is shown that a gas of quasiparticles satisfying parastatistics is always nonideal since the corrections for the non-Bose nature of the quasiparticle operators are of the same order as the corrections for the nonideal behavior. The number of quasiparticles does not exceed the rank of the parastatistics describing them, which shows that there are no fundamental prohibitions of the phenomenon of Bose condensation. The commutation relations can be used to take into account exactly the kinematic interaction in any order in the dynamical interaction.