qExponential

q-exponential, often written exp_q(x) or e_q^x, is a one-parameter generalization of the exponential function used in Tsallis nonextensive statistics. It is defined by exp_q(x) = [1 + (1 − q)x]^{1/(1 − q)} for 1 + (1 − q)x > 0, with exp_1(x) = exp(x) in the limit q → 1. The function is monotone increasing on its domain and reduces to the ordinary exponential in the limit q = 1.

Key properties include a simple derivative: d/dx exp_q(x) = exp_q(x)^q. This makes exp_q solutions to certain nonlinear differential equations and endows the function with a close link to q-deformed algebra.

Domain and boundary behavior depend on q. If q < 1, the domain is x > −1/(1 − q); as x approaches that boundary from above, exp_q(x) → 0. If q > 1, the domain is x < 1/(q − 1); as x approaches the boundary from below, exp_q(x) tends to infinity. In many statistical applications, the argument is x = −βE, so the physically relevant regime is q > 1, which yields power-law tails in the resulting distributions.

Applications and typical use include modeling in nonextensive thermodynamics and statistical mechanics. The maximum-entropy derivation with Tsallis entropy S_q yields probability distributions p_i ∝ exp_q(−βE_i). The q-exponential also underpins q-Gaussian distributions, defined as exp_q(−αx^2), which generalize Gaussians as q → 1.

Relation to broader frameworks: exp_q is part of the family of q-exponential functions tied to Tsallis entropy, nonextensive statistics, and q-deformed calculus. It is a widely used tool for describing systems with heavy tails or finite-support behaviors in a single-parameter family.