2025/02/28 by Lima, J. A. S., Benetti, M. H. · 1 citation
#FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2502.21061
In 1988, Constantino Tsallis proposed an extension of the Boltzmann statistical mechanics by postulating a new entropy formula, Sq = kBlnq W, where W is the number of microstates accessible to the system, and lnq defines a deformation of the logarithmic function. This ``top-down" , approach recovers the celebrated Boltzmann entropy in the limit q → 1 since S1 = kBln W. However, for q≠ 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.