Analytical solution for the Doppler broadening function using the Kaniadakis distribution

Several works have been done for the development of models that generalize the Maxwell-Boltzmann distribution, aimed at encompassing physical phenomena that lie outside the thermal equilibrium. Amongst these, there are distributions that result from the non-extensive statistics of Tsallis and Kaniadakis. Starting from these generalized distributions, a Doppler broadening function was proposed in recent papers, using the deformed Kaniadakis distribution, which was numerical, evaluated with the use of a Gauss-Legendre quadrature. From this perspective, this paper presents an analytical solution for the generalized Doppler broadening function through obtaining a partial differential equation, considering the Kaniadakis distribution. This equation is solved analytically using the methods of Frobenius and variation of parameters, in order to obtain a generalized solution for the Doppler broadening function, containing a deformation parameter κ , that measures the deviation in relation to the Maxwell-Boltzmann distribution. Finally, the results were produced considering several values for κ , with the intent of making a comparison with the reference values. For the validation of the deformed Doppler broadening function’s analytical solution, a numerical solution of the partial differential equation was generated. It was possible to use this numerical solution as a benchmark for the analytical solution that was derived. It was demonstrated that the analytical solution obtained is consistent, because when κ tends to zero, the solution falls in the conventional form, when the Maxwell-Boltzmann distribution is considered. Apart from this, the results were shown to be good, especially when we consider the temperature and power ranges for practical applications, as the maximum error obtained was smaller than 1%.