Elliptical Stadium Billiard: Classical Dynamics and Quantization

Billiards are prototype models in the ergodic theory of Hamiltonian systems. They describe the classical dynamics of particle (unit mass
and speed) free to move between specular reflections in a closed 2D domain; The dynamics can be regular, chaotic or may display a mixed phase
space, depending only on the geometry of the border. The Elipitical Stadium Billiard (ESB) is composed by two half-ellipses (major
axis $2a$ and minor axis $2b$) that bracket a rectangular sector of thickness $2t$ and height $2b$, as usual, we set $b=1$ [1]. Here,
we study numerically the vicinity of a particular line in the paramater space $a \times t$, namely $t_c=t_0(a)=\sqrt{a^2-1}$.
If $t\geqslant t_c$, there is chaos almost everywhere[1]. If $t<t_c$, the billiard exhibits a mixed phase space. From the relative measure and
Shannon entropy we define a order parameter and a billiard capacity respectively. By fixing $a$ and variyng $t$, a phase transition
is observed at $t_0(a)$, which is characterized by exponents $\beta=0.34$ and $\alpha=-0.0127$. The results bear a remarkable resemblance
to the famous $\lambda$ 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 $\lambda$ line in the pressure $\times$ 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 ESB[2].\\

The quantized version of a given billiard corresponds to a 2D infinite quantum well with the same geometry, one hes to solve the Helmholtz
equation $(\nabla^2+k^2)\phi=0$, where $\phi$ is the energy eigenfunction and $k^2=2mE/\hslash^2$, where $E$ is the energy eigenvalue.
For the quantization of the ESB, we are using a numerical method[3] to obtain the first 150,000 energy eigenvalues with great efficiency.
Spectra are being statiscally characterized through the nearest neighbor spacing distribution, $p(s)$ and the Dyson-Mehta spectral
rigidity, $\Delta_3$.\\

[1] E. Canale, R. Markarian, S. O. Kamphorst, and S. P. de Carvalho, A lower bound for chaos on the elliptical stadium,
Physica D 115, 189 (1998);\\

[2] T. Ara?jo Lima and F. M. de Aguiar, Classical billiards and quantum fluids, Phys. Rev. E 91, 012923 (2015);\\

[3] E. Vergini and M. Saraceno, Calculation by scaling of highly excited states of billiards, Phys. Rev. E 52, 2204 (1995).