Irreversibility from Self-Reference: Gradient Flow and an H-Theorem for a Self-Referential Statistical Operator Framework

TL;DR

This paper extends previous work on a self-referential operator in statistical mechanics by analyzing stability, convergence, and an H-theorem, supported by numerical evidence and exploring phase transitions.

Contribution

It provides the first-order perturbative expansion beyond the local kernel approximation, establishes an H-theorem within this framework, and characterizes phase transitions related to the self-coupling parameter.

Findings

Structural stability of q = alpha + beta at leading order.

Monotone decrease of free energy F observed numerically.

Identification of a re-entrant disordered phase at kappa > 0.50.

Abstract

This paper is a direct companion to arXiv:2605.06705, where the self-referential operator Omega was introduced and the Tsallis index q = alpha + beta was derived as a fixed-point condition within the local kernel approximation (LKA). Here we address four aspects deferred from the previous work. First, we carry out the first-order perturbative expansion of Omega beyond the LKA and demonstrate structural stability of q = alpha + beta at leading order in (xi/L)^2. Second, we define the iterative dynamical scheme Psi_(n+1) = Omega[Psi_n] and analyze convergence via Frechet spectral radius. Third, and centrally, we establish an H-theorem rigorously within the LKA for both the discrete iteration and the continuous gradient flow: we compute dF/dtau explicitly along the flow, identify the negative semi-definite dissipation term, establish the result rigorously in the LKA using strict convexity of F proved in the companion paper, and provide numerical evidence showing monotone decrease of F[Psi_n] across 53 iterations on an N = 80 discrete system. Fourth, we characterize the non-perturbative role of the self-coupling parameter kappa, identifying a re-entrant disordered phase at kappa > kappa* approximately 0.50 +/- 0.05. The paper is explicit about what is proved, what is established numerically, and what open problems remain for a complete analytical proof beyond the LKA.