On Interactions for Large Scale Interacting Systems

[2410.06778] On Interactions for Large Scale Interacting Systems Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2410.06778 (math) [Submitted on 9 Oct 2024] Title: On Interactions for Large Scale Interacting Systems Authors: Kenichi Bannai , Jun Koriki , Makiko Sasada , Hidetada Wachi , Shuji Yamamoto View a PDF of the paper titled On Interactions for Large Scale Interacting Systems, by Kenichi Bannai and 3 other authors View PDF HTML (experimental) Abstract: Statistical mechanics explains the properties of macroscopic phenomena based on the movements of microscopic particles such as atoms and molecules. Movements of microscopic particles can be represented by large-scale interacting systems. In this article, we study a combinatorial object which we call interactions, given as a symmetric directed graph representing the possible transition of states on adjacent sites of large-scale interacting systems. Such interactions underlie various standard processes such as the exclusion processes, generalized exclusion processes, multi-species exclusion processes, lattice-gas processes with energy, and the multi-lane particle processes. We introduce the notion of equivalences of interactions using their space of conserved quantities. This allows for the classification of interactions reflecting corresponding macroscopic properties. In particular, we prove that when the set of local states consists of two, three or four elements, then the number of equivalence classes of separable interactions are respectively one, two and five. We also define the wedge sums and box products of interactions, which give systematic methods for constructing new interactions from existing ones. Furthermore, we prove that the irreducibly quantified condition for interactions, which has implicitly played an important role in the theory of hydrodynamic limits, is preserved by wedge sums and box products. Our results provide a systematic method to construct and classify interactions, offering abundant examples suitable for considering hydrodynamic limits. Comments: 30 pages, 17 figures Subjects: Probability (math.PR) ; Combinatorics (math.CO) MSC classes: Primary: 82C22, Secondary: 05C63 Cite as: arXiv:2410.06778 [math.PR] (or arXiv:2410.06778v1 [math.PR] for this version) https://doi.org/10.48550/arXiv.2410.06778 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J. Stat. Phys. 193, 4 (2026) Related DOI : https://doi.org/10.1007/s10955-025-03550-6 Focus to learn more DOI(s) linking to related resources Submission history From: Kenichi Bannai [ view email ] [v1] Wed, 9 Oct 2024 11:14:08 UTC (7,035 KB) Full-text links: Access Paper: View a PDF of the paper titled On Interactions for Large Scale Interacting Systems, by Kenichi Bannai and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2024-10 Change to browse by: math math.CO References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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