Mean field bipartite spin models treated with mechanical techniques

• Published in: The European physical journal. B, Condensed matter physics Vol. 87; no. 3
• Main Authors: Barra, Adriano, Galluzzi, Andrea, Guerra, Francesco, Pizzoferrato, Andrea, Tantari, Daniele
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-03-2014; Springer
• Subjects: Analysis; Complex Systems; Condensed Matter Physics; Ferromagnetism; Fluid- and Aerodynamics; Methods; Physics; Physics and Astronomy; Regular Article; Solid State Physics; Thermodynamics; Statistical and Nonlinear Physics
• ISSN: 1434-6028; 1434-6036
• DOI: 10.1140/epjb/e2014-40952-4

Inspired by a continuously increasing interest in modeling and framing complex systems in a thermodynamic rationale, in this paper we continue our investigation in adapting well-known techniques (originally stemmed in fields of physics and mathematics far from the present) for solving for the free energy of mean field spin models in a statistical mechanics scenario. Focusing on the test cases of bipartite spin systems embedded with all the possible interactions (self and reciprocal), we show that both the fully interacting bipartite ferromagnet, as well as the spin glass counterpart, at least at the replica symmetric level, can be solved via the fundamental theorem of calculus, trough an analogy with the Hamilton-Jacobi theory and lastly with a mapping to a Fourier diffusion problem. All these technologies are shown symmetrically for ferromagnets and spin-glasses in full details and contribute as powerful tools in the investigation of complex systems.