Hamiltonian dynamics and geometry of phase transitions in classicalXYmodels

Abstract

The Hamiltonian dynamics associated to classical, planar, Heisenberg XY models is investigated for two-and three-dimensional lattices. Besides the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively new information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests to consider other observables of geometric meaning tightly related with the largest Lyapunov exponent. The numerical computation of these observables -unusual in the study of phase transitions -sheds a new light on the microscopic dynamical counterpart of thermodynamics also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces ΣE of phase space can be naturally established. In this framework, an approximate formula is worked out, determining a highly non-trivial relationship between temperature and topology of the ΣE. Whence it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of the ΣE. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

“…Several results strongly support this Topological Hypothesis and suggest that a phase transition might well be the consequence of an abrupt transition between different rates of change in the topology above and below the critical point. More details can be found in the review paper [36] and in the subsequent papers: [58] where the topology of the M v is analytically studied for the mean-field XY model; [59,60] where the topology of the M v is analytically studied for a trigonometric model undergoing also a first-order phase transition; [60,61] where an analytic relationship between topology and thermodynamic entropy is given among other results; [62,63] where a preliminary account of a general theorem on topology and phase transitions is given.…”

Topology and phase transitions: From an exactly solvable model to a relation between topology and thermodynamics

“…This can be achieved by improving some preliminary results on the subject reported in [22,23]. Consider a generic classical system described by a Hamiltonian…”

Exact result on topology and phase transitions at any finiteN

“…We have thus shown that as v changes from its minimum −h (corresponding to n π = 0) to 1 2 (corresponding to n π = N 2 ) the manifolds M v undergo a sequence of topology changes at the N critical values v(n π ) given by Eq. (9). There might be a TC also at the last (maximum) critical value v c = 1 2 + h 2 2 .…”

“…This is the more surprising since in many cases, knowing a priori that a system undergoes a PT, several relevant properties of the PT can be predicted just in terms of very general features of V (ϕ) (e.g., by means of renormalization-group techniques). In the light of some recent results [5][6][7][8][9][10][11] an alternative new approach approach seems actually possible by resorting to topological concepts: PTs would then be related to topology changes (TCs) of suitable submanifolds of configuration space, defined by the potential energy V (ϕ). The connection between topology of configuration space and PTs can be heuristically understood as follows.…”