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The standard Large Deviation Theory (LDT) mirrors the Boltzmann-Gibbs (BG)factor which describes the thermal equilibrium of short-range Hamiltoniansystems, the velocity distribution of which is Maxwellian. It is genericallyapplicable to systems satisfying the Central Limit Theorem (CLT), among others.When we focus instead on stationary states of typical complex systems (e.g.,classical long-range Hamiltonian systems), both the CLT and LDT need to begeneralized. We focus here on a scale-invariant stochastic process involvingstrongly-correlated exchangeable random variables which, through the Laplace-deFinetti theorem, is known to yield a long-tailed $Q$-Gaussian $N\to\infty$attractor in the space of distributions ($1 < Q<3)$. We present strongnumerical indications that the corresponding LDT probability distribution isgiven by $P(N,z)=P_0\,e_q^{-r_q(z)N}=P_0[1-(1-q)r_q(z)N]^{1/(1-q)}$ with$q=2-1/Q \in (1,5/3)$. The rate function $r_q(z)$ seemingly equals the relativenonadditive $q_r$-entropy per particle, with $q_r \simeq \frac{7}{10} +\frac{6}{10}\frac{1}{Q-1}$, thus exhibiting a singularity at $Q=1$ andrecovering the BG value $q_r=1$ in the $Q \to 3$ limit. Let us emphasize thatthe extensivity of $r_q(z)N$ appears to be verified, consistently with what isexpected, from the Legendre structure of thermodynamics, for a total entropy.The present analysis of a relatively simple model somewhat mirroring spin-1/2long-range-interacting ferromagnets (e.g., with strongly anisotropic XYcoupling) might be helpful for a deeper understanding of nonequilibrium systemswith global correlations and other complex systems.

Entropic extensivity and large deviations in the presence of strong correlations