Classical billiards and quantum fluids

Abstract

The dynamics of a particle confined in the elliptical stadium billiard with rectangular thickness 2t, major axis 2a, and minor axis 2b=2 is numerically investigated in a reduced phase space with discrete time n. Both relative measure r(n), with asymptotic value r(n→∞)=r(∞) and Shannon entropy s, are calculated in the vicinity of a particular line in the a×t parameter space, namely t(c)=t(0)(a)=√a(2)-1, with a∈(1,√4/3). If t<t(c), the billiard is known to exhibit a mixed phase space (regular and chaotic regions). As the line t(c) is crossed upwards by increasing t with fixed a, we observe that the function ψ(t)=√1-r(∞)(t) critically vanishes at t=t(c). In addition, we show that the function c(t)=t(ds/dt) displays a pronounced peak at t=t(c). In the vicinity of t(c) (t<t(c)), a chi-square tolerance of 1.0×10(-9) is reached when the numerically calculated functions ψ(t) and c(t) are fitted with renormalization group formulas with fixed parameters α=-0.0127, β=0.34, and Δ=0.5. The results bear a remarkable resemblance to the famous λ transition in liquid (4)He, where the two-component (superfluid and normal fluid) phase of He-II is critically separated from the fully entropic normal-fluid phase of He-I by the so-called λ line in the pressure × temperature parameter space. The analogy adds support to a set of previous results by Markarian and coworkers, which indicate that the line t(0)(a) is a strong candidate for the bound for chaos in the elliptical stadium billiard if a∈(1,√4/3).

Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets

“…The ergodic property (ρ c = 1) is numerically guaranteed in black regions. This diagram also supports previous works [28,29], where a critical transition from a mixed phase space to a fully ergodic was found to cross a critical line t(a) = √ a 2 − 1. The E-C 3 B's classical dynamical properties will be studied in the same way but are characterized through the collisions of the orbits with the horizontal side of its FD shown in Fig.…”

“…[28] showed that in the region a ∈ (1, √ 2) and t ∈ (0, ∞) are possible to find chaotic dynamics or a mixed phase space depending on the parameters. In [29] is presented a critical behavior of the billiard dynamics near a transition curve, t(a) = √ a 2 − 1 for the interval a ∈ (1, 4/3). Based on these previous works, we focus our analysis on this last interval and t ∈ (0, 1/ √ 3).…”

“…Quantitative characterization of these mixed-phase spaces can be made through the chaotic (regular) fraction ρ c (ρ r ) of each phase portrait with ρ c + ρ r = 1 and 0 ρ c 1. The phase plane is partitioned into N c small disjoint cells to measure these quantities [27,29,[35][36][37]. For a given orbit, let N (n) be the number of different cells in the phase space, which are visited up to n impacts in the cross-section.…”

“…This paper presents numerical results on classical dynamics and quantization in two bi-parametric billiard families. The ESB comprises two ellipses of minor semiaxe unitary, major semi-axe a, and a rectangular region of length 2t [28,29]. The other family, introduced here as E-C 3 B, presents the C 3 symmetry [18,25,27] and is formed by an equilateral triangle with rounded corners by two ellipses with semi-axis A e = 2a e and B e = 2b e .…”

“…For this, we perform numerical calculations on the energy spectra of two bi-parametric families of billiards with elliptical sectors on their boundaries and analyze the short correlations. The first one is the Elliptical Stadium Billiard (ESB), a perturbation of the Bunimovich Stadium, whose mixed classical phase space was studied in [28,29]. In sequence, we introduce a perturbation of the C-C 3 B, replacing the circumferences with ellipses.…”