Thermodynamics of fluid elements in the context of saturated isothermal turbulence in molecular clouds

The presented paper is an attempt to investigate the dynamical states of an hydro-dynamical isothermal turbulent self-gravitating system using the powerful tools of classical thermodynamics. Our main assumption, inspired by the work of Keto et al. (2020), is that turbulent kinetic energy can be substituted for the macro-temperature of chaotic motion of fluid elements. As a proper sample for our system we use a model of turbulent self-gravitating isothermal molecular cloud which is at the final stages of its life-cycle, when the dynamics is nearly in steady state. Starting from this point, we write down the internal energy for a physically small cloud's volume, and then using the first principle of thermodynamics obtain in explicit form the entropy, free energy, and Gibbs potential for this volume. Setting fiducial boundary conditions for the latter system (small volume) we explore its stability as a grand canonical ensemble. Searching for extrema of the Gibbs potential we obtain conditions for its minimum, which corresponds to a stable dynamical state of the hydrodynamical system. This result demonstrates the ability of our novel approach.