Classical billiards and quantum fluids

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 -> infinity) = r(infinity) and Shannon entropy s, are calculated in the vicinity of a particular line in the a x t parameter space, namely t(c) = t(0)(a) = root a(2) - 1, with a is an element of (1, root 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 psi(t) = root 1 - r(infinity)(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 tc (t < t(c)), a chi-square tolerance of 1.0 x 10(-9) is reached when the numerically calculated functions psi(t) and c(t) are fitted with renormalization group formulas with fixed parameters alpha = -0.0127, beta = 0.34, and Delta = 0.5. The results bear a remarkable resemblance to the famous lambda transition in liquid He-4, 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 lambda line in the pressure x 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 is an element of (1, root 4/3).