Path-integral approach to scale anomaly at finite temperature

We derive the relativistic thermodynamic scale equation using imaginary-time path integrals, with complex scalar field theory taken as a concrete example. We use Fujikawa's method to derive the scaling anomaly for this system using a matrix regulator. We make a general scaling argument to show how for anomalous systems, the $\beta$ function of the vacuum theory can be derived from measurement of macroscopic thermodynamic parameters.