Non-Gaussian velocity distributions Maxwell would understand

In 1988, Constantino Tsallis proposed an extension of the Boltzmann statistical mechanics by postulating a new entropy formula, $S_q = k_B\ln_q W$, where $W$ is the number of microstates accessible to the system, and $\ln_q$ defines a deformation of the logarithmic function. This ``top-down" , approach recovers the celebrated Boltzmann entropy in the limit $q \rightarrow 1$ since $S_1 = k_B\ln W$. However, for $q\neq 1$ the entropy is non-additive and has been successfully applied for a variety of phenomena ranging from plasma physics to cosmology. For a system of particles, Tsallis' formula predicts a large class of power-law velocity distributions reducing to the Maxwellian result only for a particular case. Here a more pedagogical ``bottom-up" path is adopted. We show that a large set of power-law distributions for an ideal gas in equilibrium at temperature T is derived by slightly modifying the seminal Maxwell approach put forward in 1860. The emergence of power-laws velocity distribution is not necessarily related with the presence of long-range interactions. It also shed some light on the long-standing problem concerning the validity of the zeroth law of thermodynamics in this context. Potentially, since the new method highlights the value of hypotheses in the construction of a basic knowledge, it may have an interesting pedagogical and methodological value for undergraduate and graduate students of physics and related areas.