Breakdown of Perturbative Expansions and Exact Algebraic Absorption of Finite-Size Fluctuations in Statistical Mechanics

Abstract

In statistical mechanics, evaluating finite-size macroscopic fluctuations typically relies on Edgeworth expansions. However, these perturbative methods append additive polynomial corrections that inevitably break down in the large deviation regime, yielding unphysical negative probabilities. We propose a structural resolution: rather than relying on additive polynomials, we absorb finite-size skewness using a globally stable $q$-algebraic framework. By introducing a dynamic scaling law $1-q_n = O(n^{-1})$ for the nonextensivity parameter, we prove this $q$-deformed framework exactly captures macroscopic higher-order fluctuations in independent and identically distributed (i.i.d.) systems. Specifically, our exact algebraic tuning completely absorbs third-order skewness while structurally guaranteeing probability density non-negativity across the entire domain. Furthermore, the $k$-th degree term of this $q$-logarithmic expansion universally corresponds to the $O(n^{1-k/2})$ asymptotic order of classical $(k+1)$-th moment Edgeworth corrections. This exact correspondence functions as a stable resummation of divergent asymptotic expansions, establishing a fundamental mathematical bridge between finite-size i.i.d. fluctuations and the Tsallis statistics governing complex systems.