We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure (named κ-algebra), motivated by the Kappa-statistics. From this structure, we obtain deformed versions of the position and momentum operators, which allow us to define a point canonical transformation that maps a particle with a constant mass in a deformed space into a particle with a position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews–Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.

Fractional diffusion equations imply non-Gaussian distributions that allow the characterisation of a wide variety of diffusive processes. Recent advances in fractional calculus lead to a class of new fractional operators defined by non-singular memory kernels. In this work we propose a generalisation of the Fokker–Planck equation in terms of a non-singular fractional temporal operator and considering a non-constant diffusion coefficient. We obtain analytical solutions for the Caputo–Fabrizio and the Atangana–Baleanu fractional kernel operators, from which non-Gaussian distributions emerge having unimodal or bimodal behavior according to whether the diffusion index ν is positive or negative respectively, where the diffusion coefficient of the power law type is considered. Thereby, we show that several types of diffusion can be characterised in the context of fractional kernel operators.

We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the $H$-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.