Thermodynamics of Fluid Elements in the Context of Turbulent Isothermal Self-Gravitating Molecular Clouds in Virial Equilibrium

In this paper, we continue the study of the thermodynamics of fluid elements in isothermal turbulent self-gravitating systems, presented by molecular clouds. We build the model again on the hypothesis that, locally, the turbulent kinetic energy per fluid element can be substituted for the macro-temperature of a gas of fluid elements. Also, we presume that the cloud has a fractal nature. The virial theorem is applicable to our system too (hence it is in a dynamical equilibrium). But, in contrast to the previous work, where the turbulent kinetic energy clearly dominates over the gravity, in the present paper, we assume that the virial relation 2Ekin+Egrav=0 holds for the entire cloud. Hence, the cloud is a dense and strongly self-gravitating object. On that basis, we calculate the internal and the total energy per fluid element. Writing down the first principle of thermodynamics, we obtain the explicit form of the entropy increment. It demonstrates untypical behavior. In the range 0≤β<0.4, for the turbulent scaling exponent, the entropy increment is positive, but in the interval 0.4<β≤1, it is negative, and for βcr=0.4, it is zero. The latter two regimes (negative and zero) cannot be explained from the classical point of view. However, we give some arguments for the reasons for these irregularities, and the main is that our cloud is an open self-organizing system driven by the gravity. Moreover, we study the system for critical points under the conditions of three thermodynamic ensembles: micro-canonical, canonical, and grand canonical. Only the canonical ensemble exhibits a critical point, which is a maximum of the free energy and corresponds to an unstable equilibrium of the system. Analysis of the equilibrium potentials also shows that the system resides in unstable states under all the conditions. We explain these results by prompting the hypothesis that the virialized cloud is in the final unstable state before its contraction and subsequent fragmentation or collapse.