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

Abstract

In the present work, we suggest a new approach for studying the equilibrium states of an hydrodynamic isothermal turbulent self-gravitating system as a statistical model for a molecular cloud. The main hypothesis is that the local turbulent motion of the fluid elements is purely chaotic and can be regarded as a perfect gas. Then, the turbulent kinetic energy per fluid element can be substituted for the temperature of the chaotic motion of the fluid elements. Using this, we write down effective formulae for the internal and total the energy and for the first principal of thermodynamics. Then, we obtain expressions for the entropy, the free energy, and the Gibbs potential. Searching for equilibrium states, we explore two possible systems: the canonical ensemble and the grand canonical ensemble. Studying the former, we conclude that there is no extrema for the free energy. Through the latter system, we obtain a minimum of the Gibbs potential when the macro-temperature and pressure of the cloud are equal to those of the surrounding medium. This minimum corresponds to a possible stable local equilibrium state of our system.

1. Introduction

This paper is dedicated to considering the local turbulent motion of fluid elements, in the case of purely saturated isothermal turbulence which demonstrates an inertial range of scales, as a perfect gas and making use of the thermodynamic laws to obtain conclusions for the dynamical state of a turbulent self-gravitating fluid. Molecular clouds (MCs) are entities in the cold neutral interstellar medium that perfectly match this aim. They are (supersonically) turbulent, self-gravitating, and isothermal [1] fluids consisting mostly of molecular hydrogen [2,3,4]. Their turbulence is driven at scales several times larger than the cloud size. All the cloud’s scales, for several orders of magnitude, belong to an inertial range [2,3,4], which is well defined in both cases—subsonic isothermal turbulence [5,6] and supersonic isothermal turbulence [7,8,9]. Magnetic fields and newborn stars also play a significant role in their physics, but we neglect them in our considerations in order to simplify the first step to a fiducial model. In the inertial range, which includes the scales from the cloud’s size (∼100 pc) down to the pre/proto-stellar core’s size (∼0.05 pc), the dominant energies are the turbulent kinetic and the gravitational. The magnetic energy is subdominant, so it can be neglected in our analysis (see, e.g., [3,10]).

It is easy for one to see the advantage of using thermodynamic laws to study the dynamical states of a MC, regarded as an ideal gas of fluid elements. The idea to apply thermodynamic models for studying the dynamical stability of physical systems consisting of macro-particles (for example: stars, galaxies, etc.) comes from the works of [11,12,13,14,15] and, more recently, from the works by [16,17,18,19,20,21]. Although this approach is fairly developed, there are many uncertainties regarding the one-to-one correspondence between the thermodynamic equilibrium and the dynamical stability. What makes the situation even more intricate is the fact that the thermodynamic ensembles of self-gravitating systems are not equivalent (for discussion on these topics, see [15,16,22], and references therein). In spite of the latter, we find that the simplicity of thermodynamics makes it a very attractive alternative for studying the hydrodynamical turbulent systems; of course, this should be conducted with caution. In this work, we focus on searching for possible equilibrium states (stable and/or unstable) of the system. For example, the problem with stability of a molecular cloud against collapse or dispersing is a key issue concerning star formation [2,3,4,23]. This issue can be explored observationally, through numerical experiments, or analytically. Here, we study it analytically by using thermodynamic potentials like energy, entropy, free energy, and Gibbs potential as tools to draw conclusions about the stability of a cloud.

As the basis of our model, we adopt the idea that the turbulent kinetic energy can be substituted for the temperature (hereafter referred to as the macro-temperature or the temperature of the macro-gas). This idea was first suggested by [24] in order to investigate the instability and fragmentation of star-forming clouds by the use of turbulent–entropic instability as a thermodynamic tool. In the present work, starting from the above assumption, we write down the internal and total energy, per one fluid element, of a turbulent self-gravitating and isothermal MC. We also presume that there exists an inertial range for the turbulence and its averaged velocity dispersion obeys a scaling law in this range. The latter determines, also, a scaling relation for the macro-temperature. The natural variables for the thermodynamic system studied here are the macro-temperature and the number density. That is why starting from the first principle of thermodynamics, assuming virial balance between gravity and turbulence, and putting a limitation on the turbulent velocity scaling exponent, we succeeded in obtaining the explicit equation for the entropy at an arbitrary scale in the inertial range (see Section 4.1).

In these variables (macro-temperature and number density), the natural thermodynamic potential is the free energy, and hence we deal with a canonical ensemble. We obtain the free energy in an explicit form in Equation (12). Then, we search for extrema of the latter potential and conclude that it may have a minimum when the macro-temperature of the cloud is equal to the macro-temperature of the surrounding medium, playing a role of a large “thermal” reservoir. Obtaining the derivatives with respect to the number density, we see that the free energy is a monotonically increasing function, and hence it does not have extrema. All possible states of the canonical ensemble are unstable, because if one perturbs the number density then the cloud will disperse or collapse, depending on its Jeans’ mass.

Then, extending our considerations, we study the case of a grand canonical ensemble, which corresponds to Gibbs potential (see Equation (14)) in variables: macro-temperature and pressure. Seeking for extrema, we conclude that the Gibbs potential has a local minimum when the macro-temperature and pressure of the macro-gas in our cloud are equal to the corresponding quantities of the surrounding medium. In this case, the cloud may reside in a stable dynamical equilibrium.

The paper structure is as follows: in the next Section 2, we make a short retrospection (from our point of view) of the main works in the field, and try to find the place of our study; in Section 3, we present the model and give the equations for the internal and the total energy, and also make the assumptions for our mean field consideration; in Section 4, we write down the first principle for the model and obtain the explicit form of the entropy (Section 4.1), then we obtain the formulae for the free energy (Section 4.2) and the Gibbs potential (Section 4.3); after that, in Section 4.4, we consider the cloud stability in two cases: canonical ensemble (Section 4.4.1), and grand canonical ensemble (Section 4.4.2); in Section 5, we comment on the basis of the model and the main results (Section 5.1), on the statistical interpretation of the entropy, the free energy, and the Gibbs potential (Section 5.2), and on the possible caveats (Section 5.3); finally, in Section 6, we present our conclusions.

2. The General Frame

The thermodynamics of self-gravitating systems started with Antonov’s [11] discovery that, when a self-gravitating system, with a mass M, is confined within a box of radius R, no maximum entropy state can exist below a certain critical energy $E=-0.335G{M}^2/R$. Hence, the system, regarded as a microcanonical ensemble, does not have an equilibrium state. This very interesting result was further developed by Lynden-Bell and Wood [13], who guessed that for $E<-0.335G{M}^2/R$, the system would collapse and overheat. This is called “gravo-thermal catastrophe” or “Antonov’s instability”. In [13], the authors have related this phenomenon to the property of self-gravitating systems to obtain negative specific heats. The gravo-thermal catastrophe picture is expected to play a crucial role in the evolution of globular clusters. It is found that the collapse proceeds self-similarly and that the central density becomes infinite in a finite time. This instability has been known as “core collapse” and many globular clusters have probably endured core collapse [25]. In the case of dense clusters of compact stars (neutron stars or stellar mass black holes), the gravo-thermal catastrophe should probably lead to the formation of supermassive black holes of the size to explain quasars and AGN.

Later, Katz [12] investigated, from a theoretical point of view, the stability of isothermal spheres with a very powerful method extending Poincare’s theory of linear series of equilibrium. He found that instability sets in at the point of minimum energy. Padmanabhan [14] reconsidered this stability analysis. He studied the sign of the second variations of entropy and reduced the problem of stability to an eigenvalue equation. His considerations lead to the same stability limit as Katz’s, but in addition, Padmanabhan’s method provides the form of perturbation that induces instability in the critical point. This perturbation presents a “core–halo” structure. Padmanabhan has performed his analysis in a microcanonical ensemble in which the energy and mass are fixed. The microcanonical ensemble is perhaps the most relevant for studying stellar systems like elliptical galaxies or globular clusters [25]. In addition, from a statistical mechanics point of view, only the microcanonical ensemble is rigorously justified for non-extensive systems, as discussed in the review of Padmanabhan [15]. However, it is possible to formally consider a canonical or grand canonical ensemble, where the temperature is fixed instead of the energy. Here, we should mention the works of de Vega, Sanchez and Combes [19,20], who have developed a theory of the cold neutral interstellar medium, and especially of the MCs, using the formalism of the renormalization field theory group. They considered the medium (MC) as a gas, composed of micro-particles (atoms or molecules), which are in thermal equilibrium. These particles interact with each other through Newtonian gravity. Turbulence is not accounted for. They showed that this non-relativistic self-gravitating gas in thermal equilibrium with variable number of particles (they regarded the system as a grand canonical ensemble) is exactly equivalent to a field theory of a single scalar field with exponential self-interaction. They analysed this field theory perturbatively and non-perturbatively through the renormalization group approach. As a main result, they successfully obtained the scaling relations for the mass of the cloud: $M\left(R\right)$∼$R^d$, and for the velocity dispersion: $\mathsf{\Delta }v\left(R\right)$∼$R^q$, where R is the characteristic size of the system. Moreover, assuming virialization of the gas, they obtained a relation for the scaling exponents: $q=(d-1)/2$. The mean field theory in their approach yields for the scaling exponents $d=2$ and $q=1/2$, accordingly. So they theoretically derived the fractal structure of MCs and fiducial values for scaling exponents, assuming the self-gravitating medium to be in thermal equilibrium and steady state, without turbulence. Also, in the paper by de Vega and Sanchez [21], the authors built the statistical mechanics of non-relativistic self-gravitating gas in thermal equilibrium (again without accounting for the turbulence), using Monte Carlo simulations, analytic mean field methods, and low-density expansions. The system is shown to possess an infinite volume limit, both in the canonical and in the microcanonical ensemble when $N,V\to \infty$, keeping $N/V^{1/3}$ fixed. They computed the equation of state, the entropy, the free energy, the chemical potential, the specific heats, the compressibility, and the speed of sound, and analysed their properties, signs, and singularities. They showed, as well, that the system suggests a fractal structure with fractal dimension $1\lesssim d⩽3$, with d slowly decreasing with increasing density.

The canonical ensemble was addressed theoretically by Chavanis [16], who found an analytical solution for the marginally stable perturbations using the method developed by Padmanabhan [14]. The constant $T,P-$ case, which is the grand canonical ensemble, was addressed previously with synthetic arguments by Bonnor [26] and Ebert [27], and also by Yabushita [28], who studied the stability of the system numerically; by Lombardi and Bertin [29] who extended the analysis by Bonnor [26] to the non-spherically symmetric case; and by Chavanis [17], who provided an analytical solution for the case of marginally stable perturbations using the same method as for the canonical ensemble.

It is well known, for self-gravitating systems, that the thermodynamic ensembles do not coincide in the whole range of parameters [15]. Using toy models, Padmanabhan [15] demonstrated that the region of negative specific heats possible in the microcanonical ensemble is replaced by a phase transition in the canonical ensemble. This phase transition separates a dilute “gaseous” phase from a dense “collapsed” phase. Since these toy models are not very realistic, the self-gravitating gas was also studied in a mean field approximation. In this consideration, an isothermal sphere is stable if and only if it is a local maximum/minimum of an appropriate thermodynamic potential (the entropy in the microcanonical ensemble and the free energy in the canonical ensemble). As expected from physical grounds, phase transition occurs when the gaseous sphere stops being a local maximum/minimum of this potential and becomes a saddle point.

Other problems arise when one uses the equation of Boltzmann entropy to calculate the critical points and, eventually, the equilibrium states and the saddle points of a self-gravitating thermodynamic system. For example, this is the approach used in the works of Antonov [11], Lynden-Bell and Wood [13], and Chavanis [16]. As it is commented, in detail, in Sormani and Bertin [22], the Boltzmann entropy is equivalent to the classical thermodynamic entropy ($\mathrm{d}S=\delta Q/T$), if and only if the system has local equation of state of ideal gas, and if the system is in hydrostatic equilibrium. However, due to the attractive gravitational force, the pressure should be lower than that of an ideal gas. Hence, strictly speaking, the equation of state cannot be defined in terms of local quantities. A different way to justify the use of Boltzmann entropy was suggested by Padmanabhan [15], by means of the so-called mean field approximation. However, the range of applicability of this approximation is still not clear.

In the frame, sketched above, our model can be characterized as a thermodynamic picture of a hydrodynamical isothermal turbulent self-gravitating system (molecular cloud in the cold neutral media), which presumes an ideal gas equation of state that is valid locally. Also, the system is in steady state, assuming hydrostatic equilibrium, and gravity is presented by mean field approximation. The novel contribution is that the ideal gas is composed of fluid elements, the motion of which is due to turbulence. The latter was initially suggested by Keto et al. [24], but their approach and study are different. Accounting for the inertial range of turbulence and fractal structure of the cloud, we obtain scaling laws for the temperature and the self-gravity potential, accordingly. Then, we explicitly write down the total energy and derive formulae for the entropy, the free energy, and the Gibbs potential. Analysing the canonical and the grand canonical ensembles, we conclude that for the former, there is no equilibrium state for the system, but for the latter there is a critical point, which is a local minimum from a thermodynamic point of view, and may be a possible hydrodynamical stable equilibrium. We believe these considerations to be of use in relation to our suggested thermodynamic picture of turbulent self-gravitating MCs.

3. Set Up of the Model

We consider a cloud of molecular gas. This gas is isothermal with Kelvin temperature T. We model the gas as a turbulent self-gravitating fluid. The turbulence is fully developed and saturated. We suppose that there exists an inertial range for turbulence in the interval: $l_{\mathrm{up}}\ge l\ge l_{\mathrm{d}}$1, where l is the scale of consideration. $l_{\mathrm{up}}$ is the upper scale of the inertial range and $l_{\mathrm{d}}$ is the scale at which the turbulence starts to dissipate. We presume that this inner part with characteristic size $l_{\mathrm{d}}$ has very small volume and mass in regard to the whole cloud with size l, hence: $l\gg l_{\mathrm{d}}$. We suppose, as well, the cloud to be a homogeneous entity with an averaged number density $n=n\left(l\right)$. The latter, of course, is a very rough approximation (in regard to the very fragmented structure of molecular clouds), but our model is based on the theory of simple thermodynamic systems and this simplification is needed at this first step.

We also assume that for the turbulence, in the inertial range, the following scaling law holds:

$$\sigma \left(l\right)=u_0{l}^{\beta },$$

(1)

where $\sigma \left(l\right)$ is the 3D turbulence velocity dispersion, $u_o$∼1 is a normalizing coefficient, and $0\le \beta \le 1$ is a scaling exponent [8,30,31,32].

We presume that the cloud is submerged in a very large, but not infinite 2, medium which acts as a reservoir of thermal and turbulent energy. This medium has some averaged number density $n_0$, Kelvin temperature $T_0$, and 3D turbulence velocity dispersion ${\sigma }_0$. Also, in the volume of the molecular cloud, this medium creates gravitational potential ${\phi }_{\mathrm{m}}=\mathrm{const}$.

Our main hypothesis is that the macroscopic motion of the fluid elements caused by the turbulence in the inertial range is locally purely chaotic and isotropic—in other words, the turbulence is of the Kolmogorov type, and can be regarded as a motion in an ideal gas containing of fluid elements treated as particles without internal degrees of freedom. In this sense we can introduce the notion of macro-temperature $\theta$ (the temperature of the chaotic motion of fluid elements) which is related to the velocity dispersion $\sigma$ through the following expression:

$${\displaystyle \frac{1}{2}}m\sigma {\left(l\right)}^2\equiv {\displaystyle \frac{3}{2}}\kappa \theta \left(l\right),$$

(2)

where m is the mass of the fluid elements and $\kappa$ is the Boltzmann constant. The Equation (2) means that the turbulent kinetic energy, per fluid element, at scale l is equivalent to the kinetic energy of the chaotic motion of the gas of fluid elements, which we term as “macro-gas”. This equation must be considered as an averaged relation (at regarded scale l), because, according to our assumption, the motion of the fluid elements is only locally purely chaotic.

Make use of Equations (1) and (2) which one obtains for the macro-temperature $\theta$ the following scaling law:

$$\theta \left(l\right)={\displaystyle \frac{m}{3\kappa }}\sigma {\left(l\right)}^2={\displaystyle \frac{m{u}_0^2}{3\kappa }}{l}^{2\beta }.$$

(3)

Therefore, if one accounts for the turbulent kinetic energy, the thermal energy (energy of the molecule motion) 3, and the gravitational energy, one can write down the following expressions for the internal and the total energy of the macro-gas, per fluid element, accordingly:

$$u={\epsilon }_{\mathrm{turb}}+{\epsilon }_{\mathrm{th}}={\displaystyle \frac{3}{2}}\kappa \theta +{\displaystyle \frac{3}{2}}{\displaystyle \frac{m}{m_0}}\kappa T,$$

(4)

and

$$\epsilon ={\epsilon }_{\mathrm{turb}}+{\epsilon }_{\mathrm{th}}+{\epsilon }_{\mathrm{grav}}=u+{\epsilon }_{\mathrm{grav}}={\displaystyle \frac{3}{2}}\kappa \theta +{\displaystyle \frac{3}{2}}{\displaystyle \frac{m}{m_0}}\kappa T+m\phi ,$$

(5)

where $m_0$ is the mean molecular mass of the gas and $\phi ={\phi }_{\mathrm{m}}+{\phi }_{\mathrm{s}}$ is the total gravitational potential in the cloud’s volume caused by both the surrounding medium (${\phi }_{\mathrm{m}}$) and the self-gravity of the cloud (${\phi }_{\mathrm{s}}$).

In principle, in the expression of the internal energy of one thermodynamic system taking part in the kinetic energy of the particles, their internal energy (already neglected in our considerations, as was mentioned above), if they are not simple material points, we should consider the potential energy of their interaction. In our model, the interaction between the fluid elements is only due to gravitational attraction. This interaction, in a small volume, is negligible in comparison to their kinetic energy. But one cannot neglect the gravitational energy of these elements caused by the self-gravity of the whole cloud. Thus, we account for it, including in the Equation (5) (for the total energy, per fluid element) the total potential $\phi$. The latter means that we consider every fluid element as it is submerged in an averaged external gravitational field caused by the surrounding medium and the self-gravity of the cloud. For the self-gravity, we assume that ${\phi }_{\mathrm{s}}\propto n\left(l\right)l^2$, which is natural from dimensional considerations. Accounting for the fractal nature of molecular clouds, we use the scaling law for the mass of our cloud: $M\left(l\right)\propto l^{\gamma }$, to obtain that $n\left(l\right)\propto l^{\gamma -3}$, and hence one can write for the self-gravity the following simple expression:

$${\phi }_{\mathrm{s}}=B{n}^{\delta },\delta ={\displaystyle \frac{\gamma -1}{\gamma -3}},$$

(6)

where $B<0$ is a negative constant, and the mass scaling exponent varies in the range $1\le \gamma <3$. The latter is in agreement with the literature (see the review by Hennebelle and Falgarone [3]).

One more point has to be clarified in the above context. The self-gravity of large systems is one of the major issues, which arise, if they are regarded as thermodynamic systems. This is due to two reasons. The first one is that the gravitation causes a long acting force, which can violate the thermodynamic limit hypothesis 4. We avoid this problem assuming that the cloud and the surrounding medium (which is very large compared to the cloud) are nearly homogeneous, and if the cloud’s size changes with a small amount $\mathrm{d}l$ (hence the change in the density will also be small, $\mathrm{d}n\propto l^{\gamma -4}\mathrm{d}l$), then the thermodynamic limit will hold. The second assumption is that the formula for the self-gravitational energy depends on $N^2$ (where N is the number of particles in the system) and therefore the total energy (and hence the other thermodynamic potentials) of the whole system are not additive. That is why we make sure to consider the thermodynamic potentials not for the whole system, but rather per fluid element (see, for example, the book of Shapiro and Teukolsky [33]). The latter allows us to consider the gravitational energy as caused by some external averaged field, where a fluid element is submerged. This energy is already additive.

4. Results

4.1. First Principle and Entropy of Macro-Gas

In the context of the previous Section, the first principle of thermodynamics of the fluid elements, written per fluid element, reads:

$$\mathrm{d}\epsilon =\theta \mathrm{d}s-P\mathrm{d}(1/n),$$

(7)

where s is the entropy per fluid element, $P=n\kappa \theta$ is the pressure of macro-gas expressed through the equation of state, and n is the number density ($1/n$ has a meaning of volume per fluid element).

If one accounts that from Equation (5) it stems that $\mathrm{d}\epsilon =(3/2)\kappa \mathrm{d}\theta +mB\delta n^{\delta -1}\mathrm{d}n$, then using (7) it is easy to obtain that

$$\mathrm{d}s={\displaystyle \frac{3}{2}}{\displaystyle \frac{\kappa }{\theta }}\mathrm{d}\theta +\left[{\displaystyle \frac{mB}{\theta }}\delta n^{\delta -1}-{\displaystyle \frac{\kappa }{n}}\right]\mathrm{d}n.$$

(8)

Here, we must note that the expression for $\mathrm{d}\epsilon$ is a full differential, because the second mixed derivatives of the energy are both equal, they are zero, and hence the energy is a function of the state. But the entropy does not satisfy this condition; the second mixed derivatives are not equal due to the gravity term $(mB\delta n^{\delta -1}/\theta )\mathrm{d}n$. The latter means that the entropy is not a function of state. This problem can be seen alternatively if one integrates the Equation (8) to obtain the function $s(\theta ,n)$ along two different paths. First, in respect to $\theta$,

$$s_1(\theta ,n)=\int {\displaystyle \frac{3}{2}}{\displaystyle \frac{\kappa }{\theta }}\mathrm{d}\theta ={\displaystyle \frac{3}{2}}\kappa ln(\theta /{\theta }_{\mathrm{d}})+q\left(n\right).$$

Then, the unknown function $q\left(n\right)$ is obtained using the partial derivative with respect to n. One arrives at the following expression:

$$s_1(\theta ,n)={\displaystyle \frac{3}{2}}\kappa ln(\theta /{\theta }_{\mathrm{d}})-\kappa ln(n/n_{\mathrm{d}})+{\displaystyle \frac{mB}{\theta }}\left(n^{\delta }-n_{\mathrm{d}}^{\delta }\right),$$

(9)

integrating from the scale of dissipation $l_{\mathrm{d}}$ to the considered scale l to obtain the free constant. Here, ${\theta }_{\mathrm{d}}$ and $n_{\mathrm{d}}$ are, accordingly, the macro-temperature and number density at dissipation scale. Also, we assume that $s_{\mathrm{d}}=s({\theta }_{\mathrm{d}},n_{\mathrm{d}})=0$, which plays the role of third principle of the thermodynamics of fluid elements.

Along the alternative path, one integrates first with respect to n and then uses the partial derivative with respect to $\theta$ to obtain the unknown function of $\theta$. One gets the following:

$$s_2(\theta ,n)={\displaystyle \frac{3}{2}}\kappa ln(\theta /{\theta }_{\mathrm{d}})-\kappa ln(n/n_{\mathrm{d}}),$$

(10)

in which the gravity term is missing.

To investigate this problem, we assess the absolute value of gravity term in the Equation (9) for $s_1(\theta ,n)$, according to the assumptions of the model. First we rewrite this term using the scaling laws for temperature $\theta ={\theta }_{\mathrm{d}}{(l/l_{\mathrm{d}})}^{2\beta }$ and density $n\left(l\right)=n_{\mathrm{d}}{(l/l_{\mathrm{d}})}^{\gamma -3}$:

$${\displaystyle \frac{m\left|B\right|}{\theta }}\left(n^{\delta }-n_{\mathrm{d}}^{\delta }\right)=m\left|B\right|{\displaystyle \frac{n_{\mathrm{d}}^{\delta }}{{\theta }_{\mathrm{d}}}}\left({\displaystyle \frac{{\theta }_{\mathrm{d}}}{\theta }}\right)\left[{\left({\displaystyle \frac{n}{n_{\mathrm{d}}}}\right)}^{\delta }-1\right]=m\left|B\right|{\displaystyle \frac{n_{\mathrm{d}}^{\delta }}{{\theta }_{\mathrm{d}}}}{\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{-2\beta }\left[{\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{\gamma -1}-1\right]=$$

$$=m\left|B\right|{\displaystyle \frac{n_{\mathrm{d}}^{\delta }}{{\theta }_{\mathrm{d}}}}{\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{\gamma -2\beta -1}\left[1-{\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{1-\gamma }\right].$$

Here, we recall that the main goal of this paper is to study the hydrodynamic stability of the system using the tools of equilibrium thermodynamics. It is natural, then, to suppose the system is in a hydrodynamic equilibrium, or at least near to this state. In the case of MCs, where turbulence and gravity are the main physical factors, we assume virial equilibrium (e.g., see [3,4,10]). In other words, $\left|W\left(l\right)\right|\approx 2E_{\mathrm{k}}$, where $\left|W\left(l\right)\right|\approx GM{\left(l\right)}^2/l\sim l^{2\gamma -1}$ is the gravitational energy at scale l and $E_{\mathrm{k}}\approx M\left(l\right)\sigma {\left(l\right)}^2/2\sim l^{\gamma +2\beta }$ is the turbulent kinetic energy at the same scale. Therefore, in order for the system to be virialized in the inertial range, the relation $\gamma =2\beta +1$ for the scaling indexes $\beta$ and $\gamma$ must hold. Then, the gravity term in regard gets

$${\displaystyle \frac{m\left|B\right|}{\theta }}\left(n^{\delta }-n_{\mathrm{d}}^{\delta }\right)=m\left|B\right|{\displaystyle \frac{n_{\mathrm{d}}^{\delta }}{{\theta }_{\mathrm{d}}}}\left[1-{\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{1-\gamma }\right].$$

Moreover, the ranges of the two indexes are $0\le \beta \le 1$ and $1\le \gamma <3$, accordingly. These values for $\beta$ and $\gamma$ are in agreement with the virial relation $\gamma =2\beta +1$ ($\gamma =3$ for $\beta =1$ and $\gamma =1$ for $\beta =0$, where the latter values nullify the gravitational term). It is useful to see that the exponent $\delta =\beta /(\beta -1)$ can be expressed through the turbulent velocity scaling parameter $\beta$. We note, also, that the expression in the parenthesis is less than 1 and $\ge 0$ for all scales $l\ge l_{\mathrm{d}}$. To complete the analysis, let us evaluate the coefficient $m\left|B\right|(n_{\mathrm{d}}^{\delta }/{\theta }_{\mathrm{d}})$. Considering the gravitational potential for 3D object ${\phi }_{\mathrm{s}}\approx -GM\left(l\right)/l$ and the adopted formula ${\phi }_{\mathrm{s}}=B{n}^{\delta }$, one can easily obtain that $B\approx -mG$. Also, from Equation (3), it is clear that ${\theta }_{\mathrm{d}}=\theta \left(l_{\mathrm{d}}\right)=m{u}_0^2{(l_{\mathrm{d}}/1\mathrm{pc})}^{2\beta }/3\kappa$. Then, accounting for $u_0\sim 1 \mathrm{km}/\mathrm{s}\sim {10}^3 \mathrm{m}/\mathrm{s}$, for the order of the considered coefficient, one gets the following:

$$m\left|B\right|(n_{\mathrm{d}}^{\delta }/{\theta }_{\mathrm{d}})\sim \kappa \times {10}^{-6}\times mG{n}_{\mathrm{d}}^{\delta }/{(l_{\mathrm{d}}/1\mathrm{pc})}^{2\beta }\equiv \kappa \times j\left(\beta \right),$$

where $j\left(\beta \right)={10}^{-6}\times mG{n}_{\mathrm{d}}^{\delta }/{(l_{\mathrm{d}}/1\mathrm{pc})}^{2\beta }$ is a function of $\beta$, the order of magnitude of which we wish to estimate. To assess the order of the fluid element mass m, we remember that it must be constant through the scales and also the set of fluid elements can be regarded as a thermodynamic system (ideal gas) at all scales. Therefore, at the smallest scale $l_{\mathrm{d}}$ containing the smallest mass $M_{\mathrm{d}}=M\left(l_{\mathrm{d}}\right)\approx m_0\begin{array}{c}\hfill n_{0,\mathrm{d}}\end{array}{l}_{\mathrm{d}}^3$. The ratio $M_{\mathrm{d}}/m$∼$N_{\mathrm{A}}\approx 6\times {10}^{23}{\mathrm{mol}}^{-1}$ must be of the order of the Avogadro constant. Then, adopting fiducial values [3,4] for the molecule number density $n_{0,\mathrm{d}}$∼${10}^5{\mathrm{cm}}^{-3}={10}^{11}{\mathrm{m}}^{-3}$ and for the dissipation scale $l_{\mathrm{d}}$∼$0.05\mathrm{pc}\approx 3\times {10}^{13}\mathrm{m}$, we obtain for the dissipation scale mass $M_{\mathrm{d}}\approx {10}^{25}\mathrm{kg}$ (where $m_0\approx 3.4\times {10}^{-27}\mathrm{kg}$ is the mass of the hydrogen molecule). Hence, the mass of the fluid element, according to our assumptions, will be of the order m∼$10\mathrm{kg}$. To assess the density $n_{\mathrm{d}}$ of the fluid elements at the dissipation scale, we take into account that $m{n}_{\mathrm{d}}=m_0{n}_{0,\mathrm{d}}$; hence, $n_{\mathrm{d}}$∼$3\times {10}^{-17}{\mathrm{m}}^{-3}$. Finally, if we have the value for the gravitational constant $G=6.67\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$ and accounting for the exponents $\delta =\beta /(\beta -1)<0$ and $2\beta \ge 0$, one gets for the coefficient $j\left(\beta \right)$ of the gravitational term:

$$\begin{array}{c}\hfill j\left(\beta \right)={10}^{-6}\times 10\times 6.67\times {10}^{-11}\times {(3\times {10}^{-17})}^{{\displaystyle \frac{\beta }{\beta -1}}}\times {(5\times {10}^{-2})}^{-2\beta }\end{array}.$$

We take the logarithm of $j\left(\beta \right)$ and search for roots of the equation $log\left(j\right(\beta \left)\right)=0$, which corresponds to the relation $j\left(\beta \right)\sim 1$. The function $log\left(j\right(\beta \left)\right)$ is plotted on Figure 1. It is strictly monotonically increasing in the range $0<\beta <1$ and the only root of the equation $log\left(j\right(\beta \left)\right)=0$ is $\beta \approx 0.45$. We should note also that $log\left(j\right(\beta \left)\right)$ is very sensitive. For example, the root of equation $log\left(j\right(\beta \left)\right)=-1$ is $\beta \approx 0.43$. Hence, we can neglect the gravitational term in the Equation (9) for $\beta <0.45$, and hence the two expressions for the entropy approximately coincide $s_1(\theta ,n)\approx s_2(\theta ,n)$. Hereafter, we denote entropy with $s(\theta ,n)$ assuming Equation (10). Therefore, for $\beta <0.45$ the entropy is a function of state, and of course the same is valid for other potentials derived from the entropy, like the free energy and the Gibbs potential.

As a final note, we have to say that the obtained limit ${\beta }_{\mathrm{cr}}\approx 0.45$ depends on the parameters of the medium like $l_{\mathrm{d}}$ and $n_{\mathrm{d}}$. If the former decreases, then ${\beta }_{\mathrm{cr}}$ increases. If the latter decreases, then ${\beta }_{\mathrm{cr}}$ increases, as well. An extended study of the parametric freedom of the limit ${\beta }_{\mathrm{cr}}$ is left for future work.

4.2. Free Energy

Free energy, per fluid element, can be defined through the following expression 5:

$$f\equiv \epsilon -\theta s\Rightarrow \mathrm{d}f=-s\mathrm{d}\theta -P\mathrm{d}(1/n).$$

(11)

To obtain the explicit form of free energy, we make use of Equations (5) and (10). The formula for $f(\theta ,n)$ is as follows:

$$\begin{array}{c}\hfill f(\theta ,n)={\displaystyle \frac{3}{2}}\kappa \theta [1-ln(\theta /{\theta }_{\mathrm{d}})]+\kappa \theta ln(n/n_{\mathrm{d}})+\\ \hfill m\left[B{n}_{\mathrm{d}}^{\delta }+{\phi }_{\mathrm{m}}\right].\end{array}$$

(12)

Here, we have added and subtracted the constant term $mB{n}_{\mathrm{d}}^{\delta }$ in order to obtain the gravitational term $mB[n^{\delta }-n_{\mathrm{d}}^{\delta }]$, which by absolute value is much smaller than $\kappa \theta$ for $\beta <0.45$ (according to the above considerations) and can be neglected in the expression for $f(\theta ,n)$.

4.3. Gibbs Potential

The Gibbs potential, per fluid element, reads as follows:

$$g\equiv \epsilon -\theta s+P/n\Rightarrow \mathrm{d}g=-s\mathrm{d}\theta +(1/n)\mathrm{d}P.$$

(13)

To obtain the explicit form of Gibbs potential, we make use of Equations (5) and (10), and the equation of state: $P=n\kappa \theta$. The formula for $g(\theta ,n)$ is as follows:

$$\begin{array}{c}\hfill g(\theta ,n)={\displaystyle \frac{3}{2}}\kappa \theta [1-ln(\theta /{\theta }_{\mathrm{d}})]+\kappa \theta [1+ln(n/n_{\mathrm{d}})]+\\ \hfill m\left[B{n}_{\mathrm{d}}^{\delta }+{\phi }_{\mathrm{m}}\right].\end{array}$$

(14)

Here, again, we have added and subtracted the constant term $mB{n}_{\mathrm{d}}^{\delta }$ in order to obtain the gravitational term $mB[n^{\delta }-n_{\mathrm{d}}^{\delta }]$, which by absolute value is much smaller than $\kappa \theta$ for $\beta <0.45$ (according to the above considerations) and can be neglected in the expression for $g(\theta ,n)$.

4.4. Stability Analysis

Using the model presented in Section 3 and the equations of thermodynamics of fluid elements obtained above in this Section 4, one can perform a stability analysis of the considered molecular cloud in the context of the thermodynamics of macro-gas. In this Section, we perform the analysis in two different cases. In the first case, we regard the cloud as a macro-gas which comes into contact with a huge reservoir (the surrounding medium) at fixed temperature ${\theta }_0$ and fixed number density $n_0$—this realises a case of canonical ensemble. Hence, the relevant potential is the free energy. The second case is a grand canonical ensemble: the cloud comes into contact with a huge reservoir at fixed temperature ${\theta }_0$ and fixed pressure $P_0$; the relevant potential is the Gibbs potential, accordingly. To perform the analysis, in both cases, we consider the equation for energy in the following form (according to the above approximations):

$$\epsilon ={\displaystyle \frac{3}{2}}\kappa \theta +mB\left[n^{\delta }-n_{\mathrm{d}}^{\delta }\right]+m\left[B{n}_{\mathrm{d}}^{\delta }+{\phi }_{\mathrm{m}}\right]\approx {\displaystyle \frac{3}{2}}\kappa \theta +m\left[B{n}_{\mathrm{d}}^{\delta }+{\phi }_{\mathrm{m}}\right].$$

(15)

Regarding the following considerations, we refer the reader to a book by Reif [34], specifically to Chapter 8.

4.4.1. Canonical Ensemble

In this case, the cloud is regarded as a macro-gas which comes into contact with a large surrounding medium at fixed macro-temperature ${\theta }_0$ and fixed number density $n_0$. The free energy in the off-equilibrium form 6 reads:

$$f_0(\theta ,n)=\epsilon (\theta ,n)-{\theta }_{0}s(\theta ,n),$$

and it will be varied as a function of $\theta$, which plays the role of a parameter determining the state of the system (our cloud). Therefore, we take the first derivative of $f_0(\theta ,n)$ in regard to $\theta$ (see Equations (7) and (8)) and seeking extrema. This is as follows:

$${\left({\displaystyle \frac{\partial f_0}{\partial \theta }}\right)}_n={\left({\displaystyle \frac{\partial \epsilon }{\partial \theta }}\right)}_n-{\theta }_0{\left({\displaystyle \frac{\partial s}{\partial \theta }}\right)}_n={\displaystyle \frac{3}{2}}\kappa -{\displaystyle \frac{3}{2}}\kappa {\displaystyle \frac{{\theta }_0}{\theta }}=0.$$

Hence, the free energy might have an extremum for macro-temperature $\theta ={\theta }_0$. One needs to calculate the second derivative of $f_0(\theta ,n)$ in regard to $\theta$ to say what kind of extremum this may be. It is obvious that the second derivative at temperature $\theta ={\theta }_0$ is positive:

$${\left({\displaystyle \frac{{\partial }^2{f}_0}{\partial {\theta }^2}}\right)}_n={\displaystyle \frac{3}{2}}{\displaystyle \frac{\kappa }{{\theta }_0}}>0.$$

This corresponds to a possible minimum of the free energy. The conclusion is that if the system (our cloud) resides at macro-temperature $\theta ={\theta }_0$, then it can be in a stable dynamical equilibrium. To complete this study, one has to calculate the derivatives of $f_0(\theta ,n)$ in regard to n, at temperature $\theta ={\theta }_0$. They are as follows:

$${\left({\displaystyle \frac{\partial f_0}{\partial n}}\right)}_{\theta }={\left({\displaystyle \frac{\partial \epsilon }{\partial n}}\right)}_{\theta }-{\theta }_0{\left({\displaystyle \frac{\partial s}{\partial n}}\right)}_{\theta }={\displaystyle \frac{\kappa {\theta }_0}{n}},$$

and

$${\left({\displaystyle \frac{{\partial }^2{f}_0}{\partial n^2}}\right)}_{\theta }=-{\displaystyle \frac{\kappa {\theta }_0}{n^2}}.$$

From the above formulae, it stems that there are no extrema for the free energy in regard to n. $f_0(\theta ,n)$ is an increasing function of variable n and the first derivative decreases and tends to zero.

Hence, the free energy does not have extrema in two variable spaces $(\theta ,n)$. The critical point for the temperature $\theta ={\theta }_0$ exists, but with regard to the number density n, the free energy is a monotonically increasing function. Therefore, there is no local equilibrium state for the system (our cloud) regarded as a canonical ensemble.

4.4.2. Grand Canonical Ensemble

In the second case, the cloud is considered as a macro-gas which comes into contact with a large surrounding medium at fixed macro-temperature ${\theta }_0$ and fixed pressure $P_0$. The Gibbs potential, written in the off-equilibrium form 7 reads

$$g_0(\theta ,n)=\epsilon (\theta ,n)-{\theta }_{0}s(\theta ,n)+P_0(1/n),$$

and it will be varied, separately, as a function of $\theta$ and n, which are the variables determining the state of the cloud.

Thus, we take the first derivative of $g_0(\theta ,n)$ with regard to $\theta$ (see Equations (7) and (8)) and search for extrema, as follows:

$${\left({\displaystyle \frac{\partial g_0}{\partial \theta }}\right)}_n={\left({\displaystyle \frac{\partial \epsilon }{\partial \theta }}\right)}_n-{\theta }_0{\left({\displaystyle \frac{\partial s}{\partial \theta }}\right)}_n={\displaystyle \frac{3}{2}}\kappa -{\displaystyle \frac{3}{2}}\kappa {\displaystyle \frac{{\theta }_0}{\theta }}=0.$$

Hence, the Gibbs potential might have an extremum for macro-temperature $\theta ={\theta }_0$. We need the second derivative of $g_0(\theta ,n)$ in regard to $\theta$ to say what kind of extremum this is. For one, it is easy to see that the second derivative at temperature $\theta ={\theta }_0$ is positive:

$${\left({\displaystyle \frac{{\partial }^2{g}_0}{\partial {\theta }^2}}\right)}_n={\displaystyle \frac{3}{2}}{\displaystyle \frac{\kappa }{{\theta }_0}}>0.$$

This means that if the cloud resides at macro-temperature $\theta ={\theta }_0$, then it might be in a stable dynamical equilibrium, because the Gibbs potential may have a minimum.

Suppose that the macro-temperature of the cloud is considered fixed at $\theta ={\theta }_0$. Then, the variation in the Gibbs potential is only due to n. The first derivative in regard to n, accordingly reads

$${\left({\displaystyle \frac{\partial g_0}{\partial n}}\right)}_{\theta }={\left({\displaystyle \frac{\partial \epsilon }{\partial n}}\right)}_{\theta }-{\theta }_0{\left({\displaystyle \frac{\partial s}{\partial n}}\right)}_{\theta }-{\displaystyle \frac{P_0}{n^2}}={\displaystyle \frac{\kappa {\theta }_0}{n}}-{\displaystyle \frac{P_0}{n^2}}=0.$$

But the pressure of macro-gas in the cloud, at considered conditions, is $P=n\kappa {\theta }_0$. Therefore, the condition for an extremum, in regard to variable n, is $P=P_0$. The second partial derivative of $g_0(\theta ,n)$ about n, under the conditions $\theta ={\theta }_0$ and $P=P_0$, reads

$${\left({\displaystyle \frac{{\partial }^2{g}_0}{\partial n^2}}\right)}_{\theta }=-{\displaystyle \frac{\kappa {\theta }_0}{n^2}}+2{\displaystyle \frac{P_0}{n^3}}={\displaystyle \frac{P_0}{n^3}}>0.$$

The mixed second derivatives of Gibbs potential are obviously zero, and therefore the functional determinant built up from the second partial derivatives of $g_0(\theta ,n)$, calculated at $\theta ={\theta }_0$ and $P=P_0$, will be

$$D={\left({\displaystyle \frac{{\partial }^2{g}_0}{\partial {\theta }^2}}\right)}_n{\left({\displaystyle \frac{{\partial }^2{g}_0}{\partial n^2}}\right)}_{\theta }-\left({\displaystyle \frac{{\partial }^2{g}_0}{\partial \theta \partial n}}\right)\left({\displaystyle \frac{{\partial }^2{g}_0}{\partial n\partial \theta }}\right)={\displaystyle \frac{3}{2}}{\displaystyle \frac{\kappa }{{\theta }_0}}{\displaystyle \frac{P_0}{n^3}}>0.$$

Hence, if the parameters of our system are set at $\theta ={\theta }_0$ and $P=P_0$, then it resides in a local stable dynamical equilibrium.

To illustrate our results for the grand canonical ensemble, in Figure 2 and Figure 3, we present the extrema of the off-equilibrium Gibbs potential $g_0(\theta ,n)$ for different boundary conditions and the equilibrium Gibbs potential $g(\theta ,n)$, which crosses $g_0(\theta ,n)$ through its minima. In Figure 2, the Gibbs potential is plotted as a function of the macro-temperature $\theta$. The equilibrium Gibbs potential $g\left(\theta \right)$ is represented by a thick line. The off-equilibrium Gibbs potential $g_0\left(\theta \right)$ is represented by dotted, dashed and dot-dashed lines for three different values of the ratio ${\theta }_0/{\theta }_{\mathrm{d}}=2,3,4$8, accordingly, which correspond to scales $l>l_{\mathrm{d}}$. The density ratios are chosen to be $n/n_{\mathrm{d}}=n_0/n_{\mathrm{d}}=0.4$ (note that if the temperature is varied, then the density must be equal to its equilibrium value, which, as chosen here, is arbitrary, but corresponds to a scale larger than the dissipation one). The pressure of the thermal reservoir (the surrounding medium) is $P_0=n\kappa {\theta }_0$, respectively. At Figure 3, in turn, the Gibbs potential is plotted as a function of the density n. The equilibrium Gibbs potential $g\left(n\right)$ is represented again by a thick line. The off-equilibrium Gibbs potential $g_0\left(n\right)$ is represented by dotted, dashed, and dot-dashed lines for three different values of the ratio $n_0/n_{\mathrm{d}}=0.1,0.25,0.4$, accordingly, which again correspond to scales $l>l_{\mathrm{d}}$. The macro-temperature ratios are chosen to be $\theta /{\theta }_{\mathrm{d}}={\theta }_0/{\theta }_{\mathrm{d}}=2$ (the temperature is fixed at its arbitrary chosen equilibrium value). The pressure of the thermal reservoir (the surrounding medium) is $P_0=n_0\kappa \theta$, respectively. One can conclude that the behaviour of the Gibbs potential (both equilibrium and off-equilibrium) is expected for ideal gas.

Finally, it is worth noting that the considered cases, namely of canonical and grand canonical ensembles, are appropriate to real molecular clouds, because at their boundaries, phase transition occurs between atomic warm neutral media and molecular cold neutral media of the ISM [2,3,4]. This transition is a non-linear process which corresponds with a jump in number density of the gas 9 at constant pressure [2,3,4].

5. Discussion

5.1. Basic Assumptions and Main Results

In this paper, we have made an attempt at applying the powerful tools of thermodynamic equilibrium analysis to study the dynamical state of turbulent isothermal self-gravitating hydrodynamical systems presented by MCs. We base our model on the assumption that the turbulent kinetic energy can be locally substituted for the macro-temperature of purely chaotic turbulent motion of fluid elements, an idea initially proposed by [24]. Setting up the model, we also presume that there is an inertial range of spatial scales for this turbulence, which is substantiated for the subsonic [5,6] as well as for the supersonic [7,8,9] case. The latter presumption seems to be unnecessary for our considerations, but it provides a continuity for $\theta$ and n in regard to the scales of consideration. This is very important for a cloud’s stability in the inertial range (see Section 4.4.2). The thermodynamic analysis is valid for twice differentiable thermodynamic potentials and coordinates, so intermittency cannot be considered. Additionally, energies (our approach is energetic) are related to the second order structure functions. Intermittency does not have a significant effect on such low-order structure functions. Also, we regard our cloud to be submerged in a very large, but not infinite, medium which serves as a reservoir of turbulent (and also thermal) energy and supplies confining pressure for the system. MCs are a distinct phase in the multi-phase ISM (e.g., [3,4]). According to classical thermodynamics, there is a local equilibrium at the phase boundary. The thermodynamic interaction between the cloud and the ISM, i.e., the thermostat, is defined by the boundary conditions, which are specified by the choice of an ensemble. We give, also, an account of the gravity of the surrounding medium and for the self-gravity of the cloud, as well, assuming they cause potentials in the cloud’s volume. The latter is the so-called “averaged field approach”.

Starting with the model described previously, we have succeeded in writing down the internal and total energy and the first principle per fluid element. Making use of them, we have obtained the explicit equations for entropy, and for free energy, and Gibbs potential, as well. The latter two potentials are tools for studying the cloud’s stability in the cases of canonical and grand canonical ensembles, accordingly.

We must note here that due to gravity, a problem arises with the thermodynamic potentials (entropy, free energy, and Gibbs potential) except for the energy. Their mixed second-order partial derivatives are not equal. So, these quantities are not functions of the system’s state. Hence, we cannot develop equilibrium thermodynamics for our model. This problem we overcome by assessing the absolute value of the gravitational term in comparison to the kinetic terms in the equations. We have achieved the latter, obtaining a fiducial assessment of the mass of the fluid element and, also, assuming that the cloud is virialized, which is in agreement with observations and theory (e.g., see [3,4,10]). In this way, we have succeeded in neglecting the gravitation (except for a constant term) in the equations for the turbulence scaling exponent $0\le \beta <0.45$, and obtained potentials, which are simple functions of the state. Therefore, the hydrodynamics of the MCs modelled through the tools of equilibrium thermodynamics are equivalent to an ideal non-gravitating macro-gas.

Considering the canonical ensemble, we draw arrived the conclusion that the system does not have a stable equilibrium state, although the free energy has a minimum when the macro-temperature is equal to the temperature of the reservoir, because, with respect to the number density, $f(\theta ,n)$ is a strictly increasing function. Hence, though one can set the system at macro-temperature $\theta ={\theta }_0$, if there happens to be a small perturbation in n, then the cloud will tend to become less or more dense (depending on its Jeans’ mass, in regard to its density and Kelvin temperature), and finally it will disperse or collapse.

Contrary to the above case, if one considers a grand canonical ensemble, the system has a stable dynamical equilibrium state. The Gibbs potential has a local minimum for $\theta ={\theta }_0$ and $P=P_0$. This means that, although small perturbations in macro-temperature and/or density might occur, the cloud will come always back to the equilibrium state.

How can one understand these ensembles and corresponding conditions for $\theta$, n and P, in the context of the cloud’s hydrodynamics? Since we deal with equilibrium thermodynamics, one natural way to interpret our model is through the virial theorem. This theorem has two forms: Eulerian and Lagrangian, depending on the chosen coordinate system. We decided to use the Eulerian form because it is more appropriate for stationary systems. It reads

$${\displaystyle \frac{1}{2}}\ddot{I}=2\left(\tau -{\tau }_S\right)+w-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\underset{S}{\int }\rho r^2\overrightarrow{u}·\overrightarrow{\mathrm{d}S},$$

(16)

where $\overrightarrow{r}$ is the position vector (with absolute value $r=\left|\overrightarrow{r}\right|$) from the origin of the Eulerian coordinate system, $\overrightarrow{u}(\overrightarrow{r},t)$ is the velocity field (with absolute value $u=\left|\overrightarrow{u}\right|$) of the fluid, and $\rho (\overrightarrow{r},t)$ is the density field. The terms in Equation (16) have the following explicit form and physical meaning:

$$I=\underset{V}{\int }\rho r^2\mathrm{d}V,$$

(17)

is the moment of inertia of the fluid in the volume V, calculated about the origin, and $\ddot{I}$ is its second time derivative;

$$\tau =\underset{V}{\int }\left({\displaystyle \frac{1}{2}}\rho u^2+{\displaystyle \frac{3}{2}}{P}_{\mathrm{th}}\right)\mathrm{d}V,$$

(18)

is the total kinetic plus thermal energy of the fluid ($P_{\mathrm{th}}$ is the thermal pressure);

$${\tau }_S=\underset{S}{\int }\overrightarrow{r}·\mathsf{\Pi }·\overrightarrow{\mathrm{d}S},$$

(19)

is the confining pressure on the volume surface, including both the thermal pressure and the ram pressure of any gas flowing across the surface; where ${\mathsf{\Pi }}_{ij}=\rho u_i{u}_j+P_{\mathrm{th}}{\delta }_{ij}$ is the general form of the fluid pressure tensor:

$$w=-\underset{V}{\int }\rho \overrightarrow{r}·\overrightarrow{\nabla }\phi \mathrm{d}V,$$

(20)

is the gravitational energy of the cloud, where the potential $\phi$ accounts for both self-gravity of the cloud and gravity of the external matter. The last term on the r.h.s. in Equation (16) represents the rate of change of the momentum flux across the cloud surface.

We have to point out that the terms concerning the thermal energy and the pressure (in Equations (18) and (19)), due to the microscopic motion of the fluid molecules, are not in question. They have their standard interpretation. The terms concerning the kinematics and the dynamics of the fluid elements need to be clarified. Let us begin with gravitational term (Equation (20)). This is the volume energy term which is accounted for the total energy, in our model (through the mean field potential $\phi$ in Equation (5)). Then it appears in the Equations (12) and (14) for free energy and Gibbs potential, accordingly (please note that we neglect in these formulae the term that includes the variable $n^{\delta }$ and leave only the constant part). The terms containing macro-temperature in Equations (5), (12) and (14) correspond to the volume kinetic term in Equation (18). The ram pressure term in Equation (19) can be linked to the boundary condition for the pressure $P_0$, of the reservoir, in the case of grand canonical ensemble. Accordingly, the boundary conditions for the macro-temperature ${\theta }_0$ (in both cases, canonical and grand canonical ensembles) and for the number density $n_0$ (canonical ensemble), of the reservoir, can be associated with the momentum flux across the cloud surface (the last term in the Equation (16)), because this flux transfers also kinetic energy, and can be caused by macro-temperature and/or number density gradients.

Finally one might conclude, although there is not a complete correspondence between our model and dynamical balance presented by the virial theorem, that the suggested thermodynamic ansatz has the potential to describe equilibrium states of hydrodynamical self-gravitating systems. For the full picture, of course, one must regard the thermodynamics of micro-particles (atoms and/or molecules), as well.

5.2. Number of Microstates

In this section, we shed light on our results for the entropy in terms of statistical mechanics. The explicit form of Equation (10) gives one an opportunity to obtain, also, formula for the number of microstates which correspond to a given macrostate, in the case of isolated system. From Boltzmann’s theory, we know that this number is

$$W=exp(Ns/\kappa ),$$

where N is the number of fluid elements in the cloud with characteristic size l.

Then, it is not difficult for one, making use of Equation (10), to obtain

$$W\left(l\right)={\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{3N\beta }{\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{(3-\gamma )N}={\left({\displaystyle \frac{l\sigma \left(l\right)}{l_{\mathrm{d}}{\sigma }_{\mathrm{d}}}}\right)}^{3N}{\left({\displaystyle \frac{l}{l_{\mathrm{d}}}}\right)}^{-\gamma N}.$$

(21)

It is interesting to note that the physical meaning of the multipliers, after the second equality, of the above equation is as follows: the first one is a number of microstates due to the chaotic turbulent motion of the fluid elements in 6D phase-space of the turbulent system and the second one comes from the fractal structure of the cloud.

Equation (21) shows that both multipliers (after the first equality) in the function $W\left(l\right)$ strictly increase with l. It can also be mentioned that $W\left(l_{\mathrm{d}}\right)=1$.

It is worth noting, also, that if we deal with canonical or grand canonical ensembles, the corresponding probability of an equilibrium macrostate of the system will be proportional to ∼$exp(-Nf(\theta ,n)/\kappa \theta )$ or to ∼$exp(-Ng(\theta ,n)/\kappa \theta )$, accordingly. The state with maximal probability minimizes the corresponding potential.

5.3. Caveats

In this section, we consider possible caveats with regard to presented model.

The first one concerns the very basis of the model. It is the assumption that we can substitute the turbulent kinetic energy for the macro-temperature of chaotic motion of fluid elements. The latter substitution is justifiable only locally, since the turbulent cascade has subscales at a given scale and according to Equation (3), the macro-temperature will be different at different subscales. Also if the turbulence is supersonic, there exist shock fronts, and the flow can be intermittent. That is why to spread the obtained macro-temperature for the whole cloud is a bit speculative. To justify this presumption we have specified that it is made in an averaged sense, i.e., we calculate the temperature at every local place, in regard to the turbulent motion of the fluid elements related to the cloud’s scale l, averaged the obtained values and attribute it as a temperature for the entire cloud, which is summarized in the Equation (2). In other words we consider our system as a physically homogeneous entity, neglecting shocks and other contrasts in density field, and intermittency as well.

The second caveat, stemming from the first one, is supposition for homogeneous medium in and outside the cloud. This is a necessary condition for one thermodynamic systems to be simple. But we know that clouds and their surroundings are strongly self-gravitating objects. Hence, the assumption for homogeneity will be broken. That is why we adopt it as a rough, but needed, at this first step approximation.

The above two caveats can be partially avoided if one considers not the whole cloud, but rather a smaller volume in the cloud, in which the inhomogeneities can easily be neglected (this approach is developed in [35]). This alternative model is simpler but does not allow one to draw clear conclusions for the whole system.

The third caveat is the hypothesis that equilibrium thermodynamics is relevant to dynamical states of MCs. Some authors ([23,36,37] and references therein) claim that MCs are dynamical self-gravitating objects that are in a state of hierarchical gravitational collapse at all scales and never achieve a steady state. However, there is a different paradigm according to which MCs are in dynamical equilibrium (for several dynamical times) expressed by the virial balance between turbulence and gravity [3,4,10,38]. So the equilibrium thermodynamics is appropriate to MCs during these periods, for the turbulent velocity scaling exponent $0\le \beta <0.45$, and our model matches this scenario.

At the end of this section, we conclude that the novel approach to studying possible equilibrium states of hydrodynamical turbulent isothermal self-gravitating systems that we have presented in this work has its reasons, although there are valid caveats that one can raise.

6. Conclusions

In the present paper, we suggest a model for studying equilibrium dynamical states of an hydrodynamical isothermal turbulent self-gravitating system, presented by a molecular cloud, making use of the principles and tools of equilibrium thermodynamics. On this basis, we present the simple idea that the local turbulent motion of fluid elements is purely chaotic and can be regarded as an ideal macro-gas. Starting from this point, we write down explicit formulae for the internal and the total energy and for the first principle per fluid element. Using them, we obtain expressions for the entropy, the free energy, and the Gibbs potential per fluid element. Then, we study two possible thermodynamic ensembles, describing our cloud: canonical, and grand canonical. Exploring their use for equilibrium states, we find that the model of canonical ensemble does not have an extremum for the free energy. The model of the grand canonical ensemble exhibits a local minimum of the Gibbs energy and is appropriate for describing a stable state of our cloud. Although there exist several valid caveats against the model, we consider it as an original approach for studying the equilibrium states of the considered systems.