Emergence of Tsallis statistics in Thomas-Fermi theory.

This work establishes a direct connection between semiclassical Thomas-Fermi theory and Tsallis statistics. The kinetic energy term of the Thomas-Fermi framework is shown to be associated with the Tsallis entropy S_{q}, while the particle interactions play the role of internal energy U, allowing the Thomas-Fermi-Lenz functional to be interpreted as a free-energy functional within a generalized thermostatistical setting. Accordingly, the minimization of the energy functional is equivalent to a maximum-entropy principle. The entropic index q is found to depend on spatial dimension: q=(d+2)/d in the nonrelativistic case, q=(d+1)/d in the ultrarelativistic limit, and q=3 under strong magnetic fields. A time-dependent hydrodynamic formulation leads to a nonlinear diffusion equation consistent with a nonlinear Fokker-Planck description. An anomalous diffusion characterized by a universal subdiffusive regime, independent of the spatial dimension, was identified. The time-dependent and time-independent results reveal a correspondence between Thomas-Fermi theory and Tsallis statistics, indicating that the former can be viewed as a generalized thermostatistics; Tsallis statistics is not assumed, it emerges from Thomas-Fermi theory.