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# Approaching a large deviation theory for complex systems

Publikationstyp

Journal Article

Publikationsdatum

2021-10-02

Sprache

English

Institut

Enthalten in

Volume

106

Issue

3

Start Page

2537

End Page

2546

Citation

Nonlinear Dynamics 106 (3): 2537-2546 (2021-11-01)

Publisher DOI

Scopus ID

Publisher

Springer Science + Business Media B.V

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→ ∞ 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 eqx≡[1+(1-q)x]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≃ 3 × 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≃ 2 × 10 5.

More Funding Information

We acknowledge partial financial support by the Santa Fe Institute, New Mexico, and the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, for support of the SFI Micro Working Group “Large Deviations in Complex Systems” meeting. CT acknowledges as well CNPq and Faperj (Brazilian agencies). U.T. is a member of the Science Academy, Bilim Akademisi, Turkey and acknowledges partial support from TUBITAK (Turkish Agency) under the Research Project number 121F269.