Towards a large deviation theory for strongly correlated systems

A large-deviation connection of statistical mechanics is provided by N   independent binary variables, the (N→∞N→∞) limit yielding Gaussian distributions. The probability of n≠N/2n≠N/2 out of N   throws is governed by e−Nre−Nr, r   related to the entropy. Large deviations for a strong correlated model characterized by indices (Q,γ)(Q,γ) are studied, the (N→∞N→∞) limit yielding Q  -Gaussians (Q→1Q→1 recovers a Gaussian). Its large deviations are governed by eq−Nrq (∝1/N1/(q−1)∝1/N1/(q−1), q>1q>1), q=(Q−1)/(γ[3−Q])+1q=(Q−1)/(γ[3−Q])+1. This illustration opens the door towards a large-deviation foundation of nonextensive statistical mechanics.

► We introduce the formalism of relative entropy for a single random binary variable and its q-generalization. ► We study a model of N strongly correlated binary random variables and their large-deviation probabilities. ► Large-deviation probability of strongly correlated model exhibits a q-exponential decay whose argument is proportional to N, as extensivity requires. ► Our results point to a q-generalized large deviation theory and suggest a large-deviation foundation of nonextensive statistical mechanics.