Eur. Phys. J. B (2014) 87: 74
https://doi.org/10.1140/epjb/e2014-40952-4

Regular Article

Mean field bipartite spin models treated with mechanical techniques

¹ Sapienza Università di Roma, Dipartimento di Fisica and GNFM Gruppo di Roma, 00185 Rome, Italy
² Sapienza Università di Roma, Dipartimento di Matematica and GNFM Gruppo di Roma, 00185 Rome, Italy
³ Sapienza Università di Roma, Dipartimento di Fisica and INFN Sezione di Roma, 00185 Rome, Italy
⁴ The University of Warwick, Mathematics Institute, CV47 AL Coventry, United Kingdom

^a e-mail: galluzzi@mat.uniroma1.it

Received: 22 October 2013
Published online: 24 March 2014

Abstract

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.