Ivancevic_Applied-Diff-Geom

1074 Applied Differential Geometry: A Modern Introductionbounded below, with no vanishing eigenvalues of its Hessian matrix). Thusthe existence of critical points of the potential V makes possible a conceptuallink between dynamics and configuration space topology, which, on thebasis of both direct and indirect evidence for a few particular models, hasbeen formulated as a topological hypothesis about the relevance of topologyfor PTs phenomena (see [Franzosi et al. (2000); Franzosi and Pettini (2004);Grinza and Mossa (2004)]).Here we give two simple examples of standard Hamiltonian systemsof the form (6.112), namely Peyrard–Bishop system and mean–field XYmodel.Peyrard–Bishop Hamiltonian SystemThe Peyrard–Bishop system [Peyrard and Bishop (1989)] 17 exhibits asecond–order phase transition. It is defined by the following potential energyN∑[ ]KV (q) =2 (qi+1 − q i ) 2 + D(e −aqi − 1) 2 + Dhaq i , (6.119)i=1which represents the energy of a string of N base pairs of reduced mass m.Each hydrogen bond is characterized by the stretching q i and its conjugatemomentum p i = m ˙q i . The elastic transverse force between neighboringpairs is tuned by the constant K, while the energy D and the inverselength a determine, respectively, the plateau and the narrowness of the on–site potential well that mimics the interaction between bases in each pair. Itis understood that K, D, and a are all positive parameters. The transverse,external stress h ≥ 0 is a computational tool useful in the evaluation of thesusceptibility. Our interest in it lies in the fact that a phase transition canoccur only when h = 0. We assume periodic boundary conditions.The transfer operator technique [Dauxois et al. (2002)] maps the problemof computing the classical partition function into the easier task ofevaluating the lowest energy eigenvalues of a ‘quantum’ mechanical Morseoscillator (no real quantum mechanics is involved, since the temperatureplays the role of ). One can then observe that, as the temperature increases,the number of levels belonging to the discrete spectrum decreases,until for some critical temperature T c = 2 √ 2KD/(ak B ) only the continuousspectrum survives. This passage from a localized ground state to an17 The Peyrard–Bishop system has been proposed as a simple model for describing theDNA thermally induced denaturation [Grinza and Mossa (2004)].