Theory of the $β$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment
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| Format: | Preprint |
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2025
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| author | Laudicina, Corentin C. L. Janssen, Liesbeth M. C. Szamel, Grzegorz |
| author_facet | Laudicina, Corentin C. L. Janssen, Liesbeth M. C. Szamel, Grzegorz |
| contents | We develop a systematic extension of mode-coupling theory (MCT) that incorporates critical dynamical fluctuations. Starting from a microscopic diagrammatic theory, we identify dominant classes of divergent diagrams near the mode-coupling transition and show that the corresponding asymptotic series dominates the mean-field below an upper critical dimension $d_c=8$. To resum these divergences, we construct a mapping to a stochastic dynamical process in which the order parameter evolves under random spatiotemporal fields. This reformulation provides a controlled, fully dynamical derivation of an effective theory for the $β$-relaxation which remarkably coincides with stochastic beta-relaxation theory [T. Rizzo, EPL 106, 56003 (2014)]. All coupling constants of the latter theory are expressed microscopically in terms of the liquid static structure factor and are computed for the paradigmatic hard-sphere system. The analysis demonstrates that fluctuations alone restore ergodicity and replace the putative mean-field transition by a smooth crossover. Our results establish a predictive framework for structural relaxation beyond mean-field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_13092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Theory of the $β$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment Laudicina, Corentin C. L. Janssen, Liesbeth M. C. Szamel, Grzegorz Statistical Mechanics Disordered Systems and Neural Networks Soft Condensed Matter Chemical Physics We develop a systematic extension of mode-coupling theory (MCT) that incorporates critical dynamical fluctuations. Starting from a microscopic diagrammatic theory, we identify dominant classes of divergent diagrams near the mode-coupling transition and show that the corresponding asymptotic series dominates the mean-field below an upper critical dimension $d_c=8$. To resum these divergences, we construct a mapping to a stochastic dynamical process in which the order parameter evolves under random spatiotemporal fields. This reformulation provides a controlled, fully dynamical derivation of an effective theory for the $β$-relaxation which remarkably coincides with stochastic beta-relaxation theory [T. Rizzo, EPL 106, 56003 (2014)]. All coupling constants of the latter theory are expressed microscopically in terms of the liquid static structure factor and are computed for the paradigmatic hard-sphere system. The analysis demonstrates that fluctuations alone restore ergodicity and replace the putative mean-field transition by a smooth crossover. Our results establish a predictive framework for structural relaxation beyond mean-field. |
| title | Theory of the $β$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment |
| topic | Statistical Mechanics Disordered Systems and Neural Networks Soft Condensed Matter Chemical Physics |
| url | https://arxiv.org/abs/2512.13092 |