title: 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).
authors: Departamento de Física, Universidade Federal de Pernambuco, PE 50670-901 Recife, Brazil.
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authors: Araújo Lima
authors: Departamento de Física, Universidade Federal de Pernambuco, PE 50670-901 Recife, Brazil.
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