Towards a large deviation theory for statistical-mechanical complex systems

2011/10/28 by Guiomar Ruiz, Constantino Tsallis, Ruiz, Guiomar +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #37A60 (Primary) 37A35 #37H05 (Secondary) #Advanced Thermodynamics and Statistical Mechanics #Complex Systems and Time Series Analysis #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Statistical Mechanics and Entropy #cond-mat.stat-mech #math-ph #math.MP #math.PR #msc:37A35 #msc:37A60 #msc:37H05

paper · pdf · doi:10.48550/arxiv.1110.6303

6 pages, 4 figures

arxiv created 2011/10/28 · openalex publication_date 2011/10/28 · arxiv updated 2011/10/31 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

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_B∑_i=1^W p_i ln p_i. Its optimization under appropriate constraints yields the celebrated BG weight e^-βE_i. An elementary large-deviation connection is provided by N independent binary variables, which, in the N→∞ limit yields a Gaussian distribution. The probability of having n ≠ 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_B\frac1- ∑_i=1^W p_i^qq-1 (q ∈ \cal R; S₁=S_BG). Its optimization yields the generalized weight e_q^{-β_q E_i} (e_q^z ≡ [1+(1-q)z]^1/(1-q); e₁^z=e^z). We numerically study large deviations for a strongly correlated model which depends on the indices Q ∈ [1,2) and γ∈ (0,1). This model provides, in the N→∞ limit (∀ γ), Q-Gaussian distributions, ubiquitously observed in nature (Q→ 1 recovers the independent binary model). We show that its corresponding large deviations are governed by e_q^{-N r_q} (∝ 1/N^1/(q-1) if q>1) where q= (Q-1)/(γ(3-Q))+1 ≥ 1. This q-generalized illustration opens wide the door towards a desirable large-deviation foundation of nonextensive statistical mechanics.

Related

• The large deviation approach to statistical mechanics
• A basic introduction to large deviations: Theory, applications, simulations
• Entropic extensivity and large deviations in the presence of strong correlations
• Large deviations and the Boltzmann entropy formula
• The Entropy Method in Large Deviation Theory