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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.
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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.
The standard Large Deviation Theory (LDT) represents the mathematical counterpart of the Boltzmann-Gibbs factor which describes the thermal equilibrium of short-range 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 Hamiltonian systems), both the CLT and LDT need to be generalized. Specifically, when the N->infinity attractor in the space of distributions is a Q-Gaussian related to a Q-generalized CLT (Q=1 recovers Gaussian attractors), we expect the LDT probability distribution to approach a q-exponential (where q=f(Q) with f(1)=1, thus recovering the standard LDT exponential distribution) with an argument proportional to N, consistently with thermodynamics. We numerically verify this conjectural scenario for the standard map, the coherent noise model for biological extinctions and earthquakes, the Ehrenfest dog-flea model, and the random-walk avalanches.
In two recent articles [PRL 90, 026802 (2003); PRB 69, 085307 (2004)], we developed a transport theory for an extended tunnel junction between two interacting fractional-quantum-Hall edge channels, obtaining analytical results for the conductance. Ponomarenko and Averin (PA) have expressed disagreement with our theoretical approach and question the validity of our results (cond-mat/0602532). Here we show why PAs critique is unwarranted.
The theory of large deviations has been applied successfully in the last 30 years or so to study the properties of equilibrium systems and to put the foundations of equilibrium statistical mechanics on a clearer and more rigorous footing. A similar approach has been followed more recently for nonequilibrium systems, especially in the context of interacting particle systems. We review here the basis of this approach, emphasizing the similarities and differences that exist between the application of large deviation theory for studying equilibrium systems on the one hand and nonequilibrium systems on the other. Of particular importance are the notions of macroscopic, hydrodynamic, and long-time limits, which are analogues of the equilibrium thermodynamic limit, and the notion of statistical ensembles which can be generalized to nonequilibrium systems. For the purpose of illustrating our discussion, we focus on applications to Markov processes, in particular to simple random walks.
We reply to Comment by J. Gemmer, L. Knipschild, R. Steinigeweg (arXiv:1712.02128) on our paper Phys. Rev. Lett. 119, 100601 (2017).
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