Generalized algebra grounded on nonadditive entropies
Leandro Lyra Braga Dognini, Constantino Tsallis · Journal of Mathematical Physics · 2025
The class of N-body complex systems with total number of microscopic states given by W(N)∼νNγ(ν>1,γ>0) can be thermostatistically handled with the nonadditive entropic functional Sδ({pi})=k∑i=1Wpiln1piδ(δ>0,S1=SBG), SBG=k∑i=1Wpiln1pi being the Boltzmann-Gibbs functional. Indeed, Sδ=1/γ({1/W(N)})=k[lnW(N)]1γ∝N, as mandated by thermodynamics. Another wide class is that with W(N) ∼ Nρ (ρ > 0) and a generalized statistical mechanics grounded on the nonadditive entropic functional Sq({pi})=k∑i=1Wpilnq1pi(q∈R,S1=SBG), with lnqz=z1−q−11−q(z≥0,q∈R,ln1z=lnz), satisfactorily handles such systems with q = 1 − 1/ρ. Furthermore, for this class, the size of the corresponding admissible phase space is characterized by lnq(x ⊗qy) = lnqx + lnqy, x, y ≥ 1, q ≤ 1, and the q-product x⊗qy=[x1−q+y1−q−1]+11−q(x⊗1y=xy) also leads to the definition of a q-algebra. The entropic functional Sq,δ({pi})=k∑i=1Wpilnq1piδ(q∈R,δ>0) unifies both cases above: Sq,1 = Sq, S1,δ = Sδ and S1,1 = SBG. In this paper, we generalize the q-algebra associated with Sq to a new one associated with Sq,δ, namely the (q, δ)-algebra.