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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.16400 |
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| _version_ | 1866911690316054528 |
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| author | Marassi, Lucio |
| author_facet | Marassi, Lucio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16400 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Irreversibility from Self-Reference: Gradient Flow and an H-Theorem for a Self-Referential Statistical Operator Framework Marassi, Lucio Statistical Mechanics 82B30 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. |
| title | Irreversibility from Self-Reference: Gradient Flow and an H-Theorem for a Self-Referential Statistical Operator Framework |
| topic | Statistical Mechanics 82B30 |
| url | https://arxiv.org/abs/2605.16400 |