Dynamics of particles in cold electrons plasma: fractional actionlike variational approach versus fractal spaces approach

Abstract

We study the dynamics of particles in cold electron plasma medium based on two dissimilar approaches: the fractional actionlike variational and fractal calculus approaches. In each case, the corresponding Boltzmann and Vlasov–Boltzmann equations were derived. Although the mathematical relationships between fractional calculus and fractal calculus were established in the literature, it was revealed throughout this study that the corresponding physics for its approach is quite different. Each model is characterized by its corresponding Boltzmann and Vlasov–Boltzmann equations which describe dissimilar dynamics and gives rise to unrelated Bohm–Gross formulas for electron plasma waves (dispersion relations) and different group velocities connected to the numerical ranges of the matching fractional parameter. Several consequences were obtained and discussed accordingly.