Approaching a large deviation theory for complex systems

TL;DR

This paper explores a generalized large deviation theory for complex systems with long-range interactions, proposing that their probability distributions asymptotically follow a power law described by q-exponentials, supported by numerical evidence.

Contribution

It introduces a generalized large deviation framework for complex systems where distributions tend to a Q-Gaussian, extending classical LDT to nonextensive cases with numerical validation.

Findings

Numerical evidence supports power-law asymptotics in complex systems.

Q-exponential distributions describe the asymptotic behavior.

The framework generalizes classical large deviation theory to complex, long-range 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 generalised. Specifically, when the $N\to\infty$ attractor ($N$ being the number of degrees of freedom) in the space of distributions is a $Q$-Gaussian (a nonadditive $q$-entropy-based generalisation of the standard Gaussian case, which is recovered for $Q=1$) related to a $Q$-generalised CLT, we expect the LDT probability distribution to asymptotically approach a power law. Consistently with available strong numerical indications for probabilistic models, this behaviour possibly is that associated to a $q$-exponential (defined as $e_q^x\equiv\left[1+(1-q)x\right]^{1/(1-q)}$, which is the generalisation 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 thermodynamical entropy of many-body Hamiltonian systems. We provide here numerical evidence supporting the asymptotic power-law by analysing the standard map, the coherent noise model for biological extinctions and earthquakes, the Ehrenfest dog-flea model, and the random-walk avalanches.