The computational study commented by Touchette opens the door to a desirable generalization of standard large deviation theory for special, though ubiquitous, correlations. We focus on three interrelated aspects: (i) numerical results strongly suggest that the standard exponential probability law is asymptotically replaced by a power-law dominant term; (ii) a subdominant term appears to reinforce the thermodynamically extensive entropic nature of q-generalized rate function; (iii) the correlations we Keywordsdiscussed, correspond to Q.-Gaussian distributions, differing from Levy's, except in the case of CauchyProbability theory Lorentz distributions. Touchette has agreeably discussed point (i), but, unfortunately, points (ii) and (iii) Statistical mechanics escaped to his analysis. Claiming the absence of connection with q-exponentials is unjustified. Entropy Before addressing in detail the Comment by Touchette [1] on our paper [2], let us describe the physical scenario within which we have undertaken a possible generalization of the standard large deviation theory (LDT). A standard many-body Hamiltonian system in thermal equilibrium with a thermostat at temperature T is described by the Boltzmann-Gibbs (BG) weight, proportional to e -0H N = e -fi[H N /N]N^ wnere ii N i s tne jv-particle Hamiltonian, and /) = l/kfiT. For standard Hamiltonian systems (typically involving short-range interactions and an ergodic behavior), the total energy is extensive. Consequently, the quantity [Tijv/JV] scales with JV, analogously to a (thermodynamically) intensive variable. This is to be compared with the LDT probability P(N) ~e~r iN , where the rate function r\ (the meaning of the subindex 1 will soon become clear) is related to a BG entropic quantity per particle, and plays a role analogous to /3[H.N/N] (we remind that, for such standard systems, /) is an intensive variable).If now we focus on say a d-dimensional classical system involving two-body interactions whose potential asymptotically decays at long distance r like -A/r a (A > 0; a > 0), the canonical BG partition function converges whenever the potential is integrable, i.e. for a/d > 1 (short-range interactions), and diverges whenever it is nonintegrable, i.e. for 0 < a/d < 1 (long-range interactions). The use of the BG weight becomes unjustified ("illusory" in Gibbs ) has already emerged, in a considerable amount of nonextensive and similar systems (see among others), as the appropriate generalization of the exponential one (and its associated Gaussian). Therefore, it appears as rather natural to conjecture that, in some sense that remains to be precisely defined, the LDT expression e~r iN becomes generalized into something close to e q q (q e 1Z), where the generalized function rate r q should be some 0375-9601/$ -see front matter
