A closer look at coupled logistic maps at the edge of chaos

We focus on a linear chain of $N$ first-neighbor-coupled logistic maps at their edge of chaos in the presence of a common noise. This model, characterised by the coupling strength $epsilon$ and the noise width $sigma_{max}$, was recently introduced by Pluchino et al [Phys. Rev. E {bf 87}, 022910 (2013)]. They detected, for the time averaged returns with characteristic return time $tau$, possible connections with $q$-Gaussians, the distributions which optimise, under appropriate constraints, the nonadditive entropy $S_q$, basis of nonextensive statistics mechanics. We have here a closer look on this model, and numerically obtain probability distributions which exhibit a slight asymmetry for some parameter values, in variance with simple $q$-Gaussians. Nevertheless, along many decades, the fitting with $q$-Gaussians turns out to be numerically very satisfactory for wide regions of the parameter values, and we illustrate how the index $q$ evolves with $(N, tau, epsilon, sigma_{max})$. It is nevertheless instructive on how careful one must be in such numerical analysis. The overall work shows that physical and/or biological systems that are correctly mimicked by the Pluchino et al model are thermostatistically related to nonextensive statistical mechanics when time-averaged relevant quantities are studied.

Emergence of $q$-statistical functions in a generalized binomial distribution with strong correlations

We study a symmetric generalization $mathfrak{p}^{(N)}_k(eta, alpha)$ of the binomial distribution recently introduced by Bergeron et al, where $eta in [0,1]$ denotes the win probability, and $alpha$ is a positive parameter. This generalization is based on $q$-exponential generating functions ($e_{q^{gen}}^z equiv [1+(1-q^{gen})z]^{1/(1-q^{gen})};,e_{1}^z=e^z)$ where $q^{gen}=1+1/alpha$. The numerical calculation of the probability distribution function of the number of wins $k$, related to the number of realizations $N$, strongly approaches a discrete $q^{disc}$-Gaussian distribution, for win-loss equiprobability (i.e., $eta=1/2$) and all values of $alpha$. Asymptotic $Nto infty$ distribution is in fact a $q^{att}$-Gaussian $e_{q^{att}}^{-beta z^2}$, where $q^{att}=1-2/(alpha-2)$ and $beta=(2alpha-4)$. The behavior of the scaled quantity $k/N^gamma$ is discussed as well. For $gamma<1$, a large-deviation-like property showing a $q^{ldl}$-exponential decay is found, where $q^{ldl}=1+1/(etaalpha)$. For $eta=1/2$, $q^{ldl}$ and $q^{att}$ are related through $1/(q^{ldl}-1)+1/(q^{att}-1)=1$, $forall alpha$. For $gamma=1$, the law of large numbers is violated, and we consistently study the large-deviations with respect to the probability of the $Ntoinfty$ limit distribution, yielding a power law, although not exactly a $q^{LD}$-exponential decay. All $q$-statistical parameters which emerge are univocally defined by $(eta, alpha)$. Finally we discuss the analytical connection with the P{o}lya urn problem.

Reply to Comment on Towards a large deviation theory for strongly correlated systems

The paper that is commented by Touchette contains a computational study which opens the door to a desirable generalization of the standard large deviation theory (applicable to a set of $N$ nearly independent random variables) to systems belonging to a special, though ubiquitous, class of strong correlations. It focuses on three inter-related aspects, namely (i) we exhibit strong numerical indications which suggest that the standard exponential probability law is asymptotically replaced by a power-law as its dominant term for large $N$; (ii) the subdominant term appears to be consistent with the $q$-exponential behavior typical of systems following $q$-statistics, thus reinforcing the thermodynamically extensive entropic nature of the exponent of the $q$-exponential, basically $N$ times the $q$-generalized rate function; (iii) the class of strong correlations that we have focused on corresponds to attractors in the sense of the Central Limit Theorem which are $Q$-Gaussian distributions (in principle $1 < Q < 3$), which relevantly differ from (symmetric) Levy distributions, with the unique exception of Cauchy-Lorentz distributions (which correspond to $Q = 2$), where they coincide, as well known. In his Comment, Touchette has agreeably discussed point (i), but, unfortunately, points (ii) and (iii) have, as we detail here, visibly escaped to his analysis. Consequently, his conclusion claiming the absence of special connection with $q$-exponentials is unjustified.

Towards a large deviation theory for statistical-mechanical complex systems

The theory of large deviations constitutes a mathematical cornerstone in the foundations of Boltzmann-Gibbs statistical mechanics, based on the additive entropy $S_{BG}=- k_Bsum_{i=1}^W p_i ln p_i$. Its optimization under appropriate constraints yields the celebrated BG weight $e^{-beta E_i}$. An elementary large-deviation connection is provided by $N$ independent binary variables, which, in the $Ntoinfty$ limit yields a Gaussian distribution. The probability of having $n e N/2$ out of $N$ throws is governed by the exponential decay $e^{-N r}$, where the rate function $r$ is directly related to the relative BG entropy. To deal with a wide class of complex systems, nonextensive statistical mechanics has been proposed, based on the nonadditive entropy $S_q=k_Bfrac{1- sum_{i=1}^W p_i^q}{q-1}$ ($q in {cal R}; ,S_1=S_{BG}$). Its optimization yields the generalized weight $e_q^{-beta_q E_i}$ ($e_q^z equiv [1+(1-q)z]^{1/(1-q)};,e_1^z=e^z)$. We numerically study large deviations for a strongly correlated model which depends on the indices $Q in [1,2)$ and $gamma in (0,1)$. This model provides, in the $Ntoinfty$ limit ($forall gamma$), $Q$-Gaussian distributions, ubiquitously observed in nature ($Qto 1$ recovers the independent binary model). We show that its corresponding large deviations are governed by $e_q^{-N r_q}$ ($propto 1/N^{1/(q-1)}$ if $q>1$) where $q= frac{Q-1}{gamma (3-Q)}+1 ge 1$. This $q$-generalized illustration opens wide the door towards a desirable large-deviation foundation of nonextensive statistical mechanics.

A closer look at time averages of the logistic map at the edge of chaos

The probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown [see U. Tirnakli, C. Beck and C. Tsallis, Phys. Rev. E 75, 040106(R) (2007)] to be numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, maximizes the nonadditive entropy S_q, the basis of nonextensive statistical mechanics. This analysis was based on a study of the tails of the distribution. We now check the entire distribution, in particular its central part. This is important in view of a recent q-generalization of the Central Limit Theorem, which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q-Gaussian limit distribution. We numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q-Gaussian both in the central region and in the tail region, and find a scaling law involving the Feigenbaum constant delta. Our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy S_q and its associated concepts.