Approaching a large deviation theory for complex systems

Abstract

The standard large deviation theory (LDT) is mathematically illustrated by the Boltzmann–Gibbs factor which describes the thermal equilibrium of short-range-interacting many-body Hamiltonian systems, the velocity distribution of which is Maxwellian. It is generically applicable to systems satisfying the central limit theorem (CLT). When we focus instead on stationary states of typical complex systems (e.g., classical long-range-interacting many-body Hamiltonian systems, such as self-gravitating ones), the CLT, and possibly also the LDT, need to be generalized. Specifically, when the $N\rightarrow \infty$ attractor (N being the number of degrees of freedom) in the space of distributions is a Q-Gaussian (a nonadditive q-entropy-based generalization of the standard Gaussian case, which is recovered for $Q=1$) related to a Q-generalized CLT, we expect the LDT probability distribution to asymptotically approach a power law. Consistently with available strong numerical indications for probabilistic models, this behavior possibly is that associated with a q-exponential (defined as $e_q^x\equiv \left[ 1+(1-q)x\right] ^{1/(1-q)}$, which is the generalization of the standard exponential form, straightforwardly recovered for $q=1$); q and Q are expected to be simply connected, including the particular case $q=Q=1$. The argument of such q-exponential would be expected to be proportional to N, analogously to the thermodynamic entropy of many-body Hamiltonian systems. We provide here numerical evidence supporting the asymptotic power law by analyzing the standard map, the coherent noise model for biological extinctions and earthquakes, the Ehrenfest dog-flea model, and the random walk avalanches. For the particular case of the strongly chaotic standard map, we numerically verify (below $5\%$ error bar) the validity of the asymptotic exponential behavior predicted by the usual LDT once the initial transient elapses typically beyond $N \simeq 3 \times 10^6$. Analogously, for the standard map with vanishing Lyapunov exponent, we provide numerical evidence (below the same error bar) for the asymptotic validity of the q-exponential behavior once the initial transient elapses typically beyond $N \simeq 2 \times 10^5$.