  We examine the effect of non-equilibrium processes modeled by the
introduction of a generalized Boltzmann factor on the thermal and magnetic
properties of an array of two-dimensional GaAs quantum dots in the presence of
an external uniform and constant magnetic field. The model consists of a
single-electron subject to a confining Gaussian potential with a spin-orbit
interaction in the Rashba approach. We compute the specific heat and the
magnetic susceptibility within the formalism of $\chi^2$-superstatistics from
the exact solution of the Schr\"odinger equation. Furthermore, an analytic
solution for the partition function allows a study of the impact of the number
of subsystems on the superstatistical corrections and confirms that the
ordinary thermo-magnetic properties are recovered whenever the thermal
distribution can be approximated by a Dirac delta. Also, we found a progressive
disappearance of the Schottky anomaly with decreasing number of subsystems,
while the specific heat ceases to be a monotonically increasing function with
respect to the average temperature when the $\chi^2$-distribution is spread
over a large range of temperatures. Remarkably, the introduction of
fluctuations in the temperature is found to suppress the paramagnetic phase
transition that would otherwise appear at low temperatures. Finally, we
emphasize that an appropriate construction of the definition of physical
observables is crucial for obtaining a correct description of the physics
derived from a non-extensive construction of the entropy.
