Europhys. Lett.
Volume 62, Number 6, June 2003
Page(s) 775 - 781
Section General
Published online 01 June 2003
DOI: 10.1209/epl/i2003-00439-9
Europhys. Lett., 62 (6) , pp. 775-781 (2003)

Topological signature of first-order phase transitions in a mean-field model

L. Angelani1, 2, L. Casetti3, M. Pettini3, 4, G. Ruocco1 and F. Zamponi1

1  Dipartimento di Fisica and INFM, Università di Roma "La Sapienza" P. A. Moro 2, 00185 Roma, Italy
2  INFM, Center for Statistical Mechanics and Complexity Università di Roma "La Sapienza" - P. A. Moro 2, 00185 Roma, Italy
3  INFM, UdR Firenze - via G. Sansone 1, 50019 Sesto Fiorentino, Italy
4  Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Arcetri Largo Enrico Fermi 5, I-50125 Firenze, Italy

(Received 13 February 2003; accepted in final form 29 April 2003)

We study a mean-field Hamiltonian system whose potential energy $V(\{q_i\}_{i=1\ldots N})$ is expressed as a sum of k-body interactions and we show that in the thermodynamic limit the presence and the energy position of first-order phase transitions can be inferred by the study of the topology of configuration space induced by V, without resorting to any statistical measure. The thermodynamics of our model is analytically solvable and -depending on the value of k- displays no transition ( k=1), second-order ( k=2) or first-order ( k>2) phase transition. This rich behaviour is quantitatively retrieved by the investigation of one of the topological invariants (the Euler characteristic $\chi(v)$) of the subsets Mv defined by $M_v = \{(q_1,\ldots,q_N)\mid
V(\{q_i\})/N \leq v\}$ .

02.40.-k - Geometry, differential geometry, and topology.
05.20.-y - Classical statistical mechanics.
75.10.Hk - Classical spin models.

© EDP Sciences 2003