  This article reports on a joint theoretical and experimental study of the
Pauli quantum-mechanical stress tensor $T_{\alpha \beta}(x,y)$ for open
two-dimensional chaotic billiards. In the case of a finite current flow through
the system the interior wave function is expressed as $\psi = u+iv$. With the
assumption that $u$ and $v$ are Gaussian random fields we derive analytic
expressions for the statistical distributions for the quantum stress tensor
components $T_{\alpha \beta}$. The Gaussian random field model is tested for a
Sinai billiard with two opposite leads by analyzing the scattering wave
functions obtained numerically from the corresponding Schroedinger equation.
Two-dimensional quantum billiards may be emulated from planar microwave
analogues. Hence we report on microwave measurements for an open 2D cavity and
how the quantum stress tensor analogue is extracted from the recorded electric
field. The agreement with the theoretical predictions for the distributions for
$T_{\alpha \beta}(x,y)$ is quite satisfactory for small net currents. However,
a distinct difference between experiments and theory is observed at higher net
flow, which could be explained using a Gaussian random field, where the net
current was taken into account by an additional plane wave with a preferential
direction and amplitude.
