Topological aspects of geometrical signatures of phase transitions

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 60, Issue 5, 1999, Pages

  Franzosi, Roberto ^a,b   Casetti, Lapo ^c   Spinelli, Lionel ^b,d   Pettini, Marco ^b,e  

^a UNIVERSITY OF FLORENCE   (Italy)

^b INFM   (Italy)

^c POLITECNICO DI TORINO   (Italy)

^d CNRS   (France)

^e INAF OSSERVATORIO ASTROFISICO DI ARCETRI   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE;

EID: 0001013648     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.60.R5009     Document Type: Article

References (20)

1. C.N. Yang and T.D. Lee, Phys. Rev. 87, 404 (1952).
2. L. Caiani, L. Casetti, C. Clementi, and M. Pettini, Phys. Rev. Lett. 79, 4361 (1997).
3. L. Caiani, L. Casetti, C. Clementi, G. Pettini, M. Pettini, and R. Gatto, Phys. Rev. E 57, 3886 (1998).
4. Here topology is meant in the sense of de Rham’s cohomology.
5. L. Caiani, L. Casetti, and M. Pettini, J. Phys. A 31, 3357 (1998).
6. C. Clementi, Master thesis, SISSA/ISAS, 1996 (unpublished).
7. M.C. Firpo, Phys. Rev. E 57, 6599 (1998).
8. M. Cerruti-Sola, R. Franzosi, and M. Pettini, Phys. Rev. E 56, 4872 (1997).
9. L. Casetti, R. Livi, and M. Pettini, Phys. Rev. Lett. 74, 375 (1995).
10. L. Casetti, C. Clementi, and M. Pettini, Phys. Rev. E 54, 5969 (1996), and references cited therein.
11. L. Casetti and M. Pettini, Phys. Rev. E 48, 4320 (1993).
12. J. Milnor, Morse Theory (Princeton University Press, Princeton, 1969).
13. M. W. Hirsch, Differential Topology (Springer-Verlag, New York, 1976).
14. J.A. Thorpe, Elementary Topics in Differential Geometry (Springer-Verlag, New York, 1979).
15. This classic co-area formula can be found in H. Federer, Geometric Measure Theory (Springer, Berlin, 1969).
16. M. P. do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992).
17. K. Binder, Monte Carlo Methods in Statistical Physics (Springer-Verlag, Berlin, 1979).
18. It is remarkable that (Formula presented) appears rather insensitive to the system size. This fact does not seem too surprising in light of recent arguments about finite-size scaling effects in statistical mechanics. The claim is that these effects are peculiar to the canonical ensemble and have nothing to do with the physics of the phase transition. See D.H.E. Gross, Phase transitions without thermodynamic limit, e-print cond-mat/9805391
19. D.H.E. GrossPhys. Rep. 279, 119 (1997), and references therein. Now, (Formula presented) is computed through a geometric integral that has nothing to do with canonical ensemble averages, whence a possible explanation of its near independence of the system size.
20. This hypothesis is strenghtened by recent analytical results reported in L. Casetti, E.G.D. Cohen, and M. Pettini, Phys. Rev. Lett. 82, 4160 (1999).