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Internal energy $E$ is an extensive quantity for most systems. But energy extensivity is not valid in systems with long-range interactions, like gravity (e.g. in astrophysical systems). For extensive systems: $$E=TS+\bf J. \bf x +\bf{\mu}.\bf N\tag{1}$$ where $\bf J$ and $\bf x$ are intensive and extensive thermodynamic variables. (like pressure and volume, respectively)

Some other thermodynamic potentials like $G$, $F$, and $H$ are defined as : $$F=E-TS$$ $$G=E-TS-\bf J.\bf x$$ $$H=E-\bf J.\bf x$$

For extensive systems, physical meanings are obvious, as is written in Wikipedia:

we can say that ΔU is the energy added to the system, ΔF is the total work done on it, ΔG is the non-mechanical work done on it, and ΔH is the sum of non-mechanical work done on the system and the heat given to it.

Apparently these interpretations are deduced from equation $(1)$, which doesn't hold in non-extensive systems. So it seems that these potentials are ,inherently, extensive quantities, and have no meaning in non-extensive systems.

Do these potentials have a physical meaning (if applicable at all) in non-extensive systems like galaxies?

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Apparently these interpretations are deduced from equation (1), which doesn't hold in non-extensive systems. So it seems that these potentials are ,inherently, extensive quantities, and have no meaning in non-extensive systems.

I do not think that these interpretations depend on (1). They can be derived without (1) or the homogeneous property of $U(S,V,N)$. For example, the change in the Gibbs energy in a quasi-static process where $T,P$ are constant is

$$ \Delta G = \Delta U - T\Delta S + P\Delta V, $$ while the change in the energy is $$ \Delta U = T\Delta S - P\Delta V + \Delta W_{nm}, $$

where $\Delta W_{nm}$ is non-mechanical work done on the system. Putting the latter into the former, we obtain

$$ \Delta G = \Delta W_{nm}, $$

which implies the mentioned interpretation of $\Delta G$.

Systems with non-homogeneous $U$ may still be treated by thermodynamics. The definitions of $F,G,H$ remain the same, and these quantities will retain most of their usefulness. For example, the condition that in equilibrium the Gibbs energy is minimum should remain valid also for such systems.

Do these potentials have a physical meaning (if applicable at all) in non-extensive systems like galaxies?

Yes in thermodynamic systems that have non-homogeneous $U(S,V,N)$. But galaxy is not a thermodynamic system, so no for galaxies.

Ordinary (classical) thermodynamics deals with systems whose equilibrium state can be described via few thermodynamic variables. Galaxy is not such a system - it is not clear what would be "equilibrium state of galaxy". And it is not clear how temperature or entropy could be defined for such state. So for galaxy, the thermodynamic potentials are not applicable. From all the thermodynamic quantities, only energy and number of particles seem to make sense for a system like galaxy. Unfortunately, thermodynamics, and even statistical physics, can tell us very little about galaxies.

In the rest of the post, I will try to clarify the notions "extensive" and "homogeneous function of first degree" which confuse these discussions often.

Internal energy E is an extensive quantity for most systems. But energy extensivity is not valid in systems with long-range interactions, like gravity (e.g. in astrophysical systems).

I am afraid you are unfortunately making an incorrect claim here due to common confusion of the terms $extensive$ and $homogeneous~~function~~of~~first~~degree$. The adjective "extensive" was introduced to thermodynamics to delimit the subset of physical quantities, which double if we go from system to pair of such systems.

The IUPAC definition in

Quantities, Units and Symbols in Physical Chemistry (3rd edn. 2007), page 6 (page 20 of PDF file)

http://media.iupac.org/publications/books/gbook/IUPAC-GB3-2ndPrinting-Online-22apr2011.pdf

A quantity that is additive for independent, noninteracting subsystems is called extensive; examples are mass $m$, volume $V$, Gibbs energy $G$.

Very good explanation of the idea is given by Lewis and Randall in their book

G. N. Lewis, M. Randall, Thermodynamics, McGraw-Hill Book Co., 1923 :

Most of the properties which we measure quantitatively may be divided into two classes. If we consider two identical systems, let us say two kilogram weights of brass or two exactly similar balloons of hydrogen, the volume, or the internal energy, or the mass of the two is double that of each one. Such properties are called extensive.

On the other hand, the temperature of the two identical objects is the same as that of either one, and this is also true of the pressure and the density. Properties of this type are called intensive.

See also the discussion in

Redlich, Otto, Intensive and extensive properties, American Chemical Society, Journal of Chemical Education, 1970, 47 (2), p. 154

http://dx.doi.org/10.1021/ed047p154.2

If we want to use the adjective "extensive" for quantities describing galaxies, from the above discussion it follows that the mass, energy and even volume of the galaxy (if we define it somehow) $are~~extensive$. The doubling occurs by considering two independent copies of the original system, which induces no physical change in any of the two copies.

For extensive systems: $$ E=TS+\mathbf J \cdot \mathbf x + \mathbf μ\cdot \mathbf N~~~~~(1) $$

So, here the word "extensive" is not appropriate. The word "extensive" in the original sense does not apply to physical systems; it applies to quantities like $E, S, V, N$ that are (in thermodynamics) always extensive.

Your above paragraph would be better written as

-- If energy is a homogeneous function of $S, \mathbf x, \mathbf N$ of first degree, then the equation

$$ E=TS+\mathbf J \cdot \mathbf x + \mathbf μ\cdot \mathbf N~~~~~(1) $$

is valid (follows from the Euler theorem on homogeneous functions).

Function $E(S,V,N)$ is homogeneous of first degree if

$$ E(kS, kV, kN) = k E(S, V, N) $$ for any $k$.

So, the energy of any physical system is always extensive, but it does not have to be homogeneous function of first degree of variables $S,V,N$.

For example, if energy was given by the formula $E = c_1\frac{S}{N}V + c_2V^{2/3}$, it would be extensive, but not homogeneous function of $S,V,N$. Such systems may exist, for example ball of water whose energy includes its surface energy (and entropy its surface entropy). By doubling this system - we obtain two equal independent balls - the quantities $E',S',V',N'$ appropriate to the new pair system are twice the original values $E,S,V,N$ respectively, so they are all extensive variables.

However, is the energy of the ball a homogeneous function of $S, V, N$ of first degree? I do not know for sure (because I do not know the function $E(S,V,N)$ for this system), but it seems unlikely, since the surface contributes with energy and entropy that are not proportional to volume of the ball $V$ but to its surface area.

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Correct, your equation (1) does not hold in non-extensive systems. Also, none of the equations hold for a galaxy since it's not in equilibrium.

Supposing we do have equilibrium, though, equilibrium statistical mechanics is exactly the tool we need to extend these quantities to non-extensive and microscopic systems. This was one of Gibbs' main goals in his 1902 book Elementary Principles in Statistical Mechanics. To quote, "The laws of statistical mechanics apply to conservative systems of any number of degrees of freedom, and are exact," and "The laws of thermodynamics may be easily obtained from the principles of statistical mechanics, of which they are the incomplete expression".

For the canonical ensemble (a system of a given number of particles at a given temperature), Gibbs showed that free energy $A$ is its core thermodynamic potential, related to the partition function by $A = -kT\ln Z$. The reason I say this is that in the canonical ensemble one has for the expectation value of energy exactly $$\overline E = A + ST$$ where $S$ is the Gibbs entropy (aka information entropy). So, if you have an equilibrium system where you know exactly how many particles are inside and you know its temperature, the free energy is telling you exactly the partition function of the system.

As for the $J\cdot x$ terms, these can be extended as generalized coordinates and generalized forces. Neither the force nor the coordinate actually needs to be extensive. However in a non-extensive system it's not enough to just specify "the volume", we need additional coordinates to define the shape of the volume. As a result it is possible to define many Gibbs free energies and enthalpies depending on which of the degrees of freedom have constant coordinate, and which have constant force. One example though is the isothermal–isobaric ensemble where the degree of freedom is a piston that applies a constant force.

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