Construction and application of orthogonal polynomials in kinetics of quasi-particles

To determine macroscopic hydrodynamic quantities and kinetic coefficients in different media on the basis of the kinetic theory of particles (quasiparticles), one must know the solution of the kinetic equation for the distribution function. In the kinetic theory of monatomic gases, a set of polynomials is presented by the classical Sonin-Laguerre polynomials. Here, we apply a similar method of calculating kinetic coefficients in a gas of quasiparticles, the energy of which may depend on different external parameters (electrical or magnetic fields, for instance). In this case, the use of classical polynomials appears not to be efficient. Therefore, it is advantageous to construct a special set of orthogonal polynomials on the basis of the weight function characteristic of Bose-Einstein or Fermi-Dirac statistics, a proper choice of which allows us to restrict ourselves to a finite number of first polynomials when calculating kinetic coefficients.