  Turbulence simulations play a key role in advancing the general understanding
of the physical properties turbulence and in interpreting astrophysical
observations of turbulent plasmas. For the sake of simplicity, however,
turbulence simulations are often conducted in the isothermal limit. Given that
the majority of astrophysical systems are not governed by isothermal dynamics,
we aim to quantify the impact of thermodynamics on the physics of turbulence,
through varying adiabatic index, $\gamma$, combined with a range of optically
thin cooling functions. In this paper, we present a suite of ideal
magnetohydrodynamics simulations of thermally balanced stationary turbulence in
the subsonic, super-Alfv\'enic, high beta (ratio of thermal to magnetic
pressure) regime, where turbulent dissipation is balanced by two idealized
cooling functions (approximating linear cooling and free-free emission) and
examine the impact of the equation of state by considering cases that
correspond to isothermal, monatomic and diatomic gases. We find a strong
anticorrelation between thermal and magnetic pressure independent of
thermodynamics, whereas the strong anticorrelation between density and magnetic
field found in the isothermal case weakens with increasing $\gamma$. Similarly,
with the linear relation between variations in density and thermal pressure
with sonic Mach number becomes steeper with increasing $\gamma$. This suggests
that there exists a degeneracy in these relations with respect to
thermodynamics and Mach number in this regime, which is dominated by slow
magnetosonic modes. These results have implications for attempts to infer
(e.g.) Mach numbers from (e.g.) Faraday rotation measurements, without
additional information regarding the thermodynamics of the plasma. However, our
results suggest that this degeneracy can be broken by utilizing higher-order
moments of observable distribution functions.
