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.
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