Probability distributions extremizing the nonadditive entropySδand stationary states of the corresponding nonlinear Fokker-Planck equation

Abstract

Under the assumption that the physically appropriate entropy of generic complex systems satisfies thermodynamic extensivity, we investigate the recently introduced entropy S(δ) (which recovers the usual Boltzmann-Gibbs form for δ=1) and establish the microcanonical and canonical extremizing distributions. Using a generalized version of the H theorem, we find the nonlinear Fokker-Planck equation associated with that entropic functional and calculate the stationary-state probability distributions. We demonstrate that both approaches yield one and the same equation, which in turn uniquely determines the probability distribution. We show that the equilibrium distributions asymptotically behave like stretched exponentials, and that, in appropriate probability-energy variables, an interesting return occurs at δ=4/3. As a mathematically simple illustration, we consider the one-dimensional harmonic oscillator and calculate the generalized chemical potential for different values of δ.

Hypergeometric foundations of Fokker–Planck like equations

“…For the time being, we restrict ourselves to the case K = constant. A similar hypergeometric derivation applies to a non linear generalization of equation (1.1), in the spirit if the one advanced 20 years ago by Plastino and Plastino [2], that has arisen interest till today [4,5,6,7]. This papers continues a line of research initiated by uncovering hypergeometric connotations of quantum equations [3].…”

Nonextensive statistical mechanics and high energy physics

“…where g(x) is a well behaved function which, in the x → 0 limit, asymptotically satisfies g(x) ∝ |x| λ with λ ∈ R. Obviously the case g(x) = constant (hence λ = 0) recovers definition (13). Examples with g(x) constant are very frequent in the literature.…”

“…4. We may extend the q-exponential class quite naturally by introducing, in definition (13), the analog of a density of states g(x), as usually done in condensed matter physics and elsewhere (see in [70] one such example in economics, for the distribution of stock-market volumes). The more general …”

Nonlinear inhomogeneous Fokker-Planck equations: Entropy and free-energy time evolution

“…In particular, the Boltzmann-Gibbs entropy will be recovered for linear FPEs, while other commonly used generalized entropies are naturally associated to a wide class of nonlinear FPEs [25,37,38]. Differentiating Eq.…”