We consider the application of the supersymmetric quantum-mechanical formalism to the Schrödinger equation describing a particle characterized by a position-dependent effective mass $m\left(x\right).$ We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential $V_2\mathrm{}\left(x\right)\mathrm{}$ depends on both the form of the original potential $V_1\mathrm{}\left(x\right)\mathrm{}$ and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form $m\left(x\right)\mathrm{}$ of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.

The kinetic foundations of Tsallis' nonextensive thermostatistics are investigated through Boltzmann's transport equation approach. Our analysis follows from a nonextensive generalization of the “molecular chaos hypothesis.” For $q>\mathrm{}0\mathrm{}$, the $q$-transport equation satisfies an $H$ theorem based on Tsallis entropy. It is also proved that the collisional equilibrium is given by Tsallis' $q$-nonextensive velocity distribution.