Geometry of dynamics, Lyapunov exponents and phase transitions

Details Geometry of dynamics, Lyapunov exponents and phase transitions Geometry of dynamics, Lyapunov exponents and phase transitions

1997-02-17

arxiv.org/abs/chao-dyn/9702011v2

Phys.Rev.Lett. 79 (1997) 4361-4364

Nonlinear Sciences

Chaotic Dynamics

REVTeX, 10 pages, 5 PostScript figures, published version

Scientific paper

10.1103/PhysRevLett.79.4361

The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically estimate the largest Lyapunov exponent in terms of some curvature fluctuations. The agreement between numerical and analytical values for Lyapunov exponents is very good in a wide range of temperatures. Moreover, in the three dimensional case, in correspondence with the second order phase transition, the curvature fluctuations exibit a singular behaviour which is reproduced in an abstract geometric model suggesting that the phase transition might correspond to a change in the topology of the manifold whose geodesics are the motions of the system.

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