Study of a curvature-based criterion for chaos in Hamiltonian systems with two degrees of freedom

Abstract

A non-Euclidean geometric criterion for chaos (Horwitz et al 2007 Phys. Rev. Lett. 98 234301) is investigated in bound systems with two degrees of freedom. It is shown that the criterion is partly equivalent to a simpler indicator of chaos based on the shape of equipotential curves. The method is numerically tested in a restricted Bohr model and in an extended Creagh-Whelan model. Although in some cases the geometric criterion approximately predicts the onset of chaos at low energies, manifest counterexamples disproving its general applicability are demonstrated.

Classification of excited-state quantum phase transitions for arbitrary number of degrees of freedom

“…For increasing numbers of degrees of freedom f , the ESQPTs affect higher and higher derivatives of the level density and flow of the spectrum. Their effects in systems with f = 2 have been thoroughly studied in our recent works [14,15]. These analyses contain the prerequisites for an ESQPT theory in an arbitrary number of degrees of freedom, but only for systems whose Hamiltonian is of the form…”

Heat capacity for systems with excited-state quantum phase transitions

“…The esqpt represents an N → ∞ singularity in the level density of a finite-f quantum system caused by a stationary point of the corresponding classical Hamiltonian. The singularity affects also many other observables, e.g., the "flow" of energy levels with a running control parameter, the form of eigenstates and evolution of some expectation values in the critical energy region, response of the system to external probes and so on [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Here we will investigate some consequences on thermal behavior.…”

“…The level density of the (f − 1)-dimensional oscillator is given by ρ osc ∝ E f −2 osc , while that of the 1-dimensional potential well system reads as ρ well ∝ m k=1 L k Θ(E well − E k )/ √ E well −E k (we are using the semiclassical approximation). The total level density of the whole system is given by a convolution of the two partial level densities with E = E well +E osc [13], yielding:…”

“…An interesting phenomenon that occurs in the infinite-size limit of finite-dof systems is the excited-state quantum phase transition (esqpt). It is a singularity of the density and flow of the quantum spectrum as a function of excitation energy and suitable control parameters [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The esqpts extend the groundstate quantum phase transitions at zero temperature (qpts) [20] and their possible links to thermal phase transitions were naturally considered.…”

Free motion around black holes with discs or rings: between integrability and chaos -- V

“…To say it straight away, a firm reply was given by Vieira & Letelier (1996b): while admitting that there surely exist links between global dynamics and local (curvature) properties of the corresponding configuration manifold of the system, they pointed out that "any local analysis, in effective or even physical spaces, is far from being sufficient to predict a global phenomenon like chaotic motion" (more recently, the same statement has e.g. been voiced by Stránský & Cejnar (2015)). One can easily imagine, for example, a system living in an Euclidean space and subject to an interaction only acting at discrete locations (and/or times), like e.g.…”