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→∞)=r_∞ and Shannon entropy s, are calculated in the vicinity of a particular line in the a×t parameter space, namely tc=t₀(a)=√a²-1, with a∈(1,√4/3). If t<tc, the billiard is known to exhibit a mixed phase space (regular and chaotic regions). As the line tc is crossed upwards by increasing t with fixed a, we observe that the function ψ(t)=√1-r_∞(t) critically vanishes at t=tc. In addition, we show that the function $c(t)=t(ds/dt)$ displays a pronounced peak at t=tc. In the vicinity of tc (t<tc), 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 ⁴He, 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 t₀(a) is a strong candidate for the bound for chaos in the elliptical stadium billiard if a∈(1,√4/3).