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5.1 Fundamental, or Euler, relation

Consider the standard form of the combined First and Second Laws3 for a simple, single-species system:
dE = T dS − p dV + μdN. (72 )
This follows because there is an equation of state, meaning that E = E (S, V,N ) where
∂E || ∂E || ∂E || T = ---||, p = − ----||, μ = ---||. (73 ) ∂S |V,N ∂V |S,N ∂N |S,V
The total energy E, entropy S, volume V, and particle number N are said to be extensive if when S, V, and N are doubled, say, then E will also double. Conversely, the temperature T, pressure p, and chemical potential μ are called intensive if they do not change their values when V, N, and S are doubled. This is the additivity property and we will now show that it implies an Euler relation (also known as the “fundamental relation” [96]) among the thermodynamic variables.

Let a tilde represent the change in thermodynamic variables when S, V, and N are all increased by the same amount λ, i.e. 

˜S = λS, V˜ = λV, N˜ = λN. (74 )
Taking E to be extensive then means
˜E (˜S, ˜V ,N˜) = λE (S,V, N ). (75 )
Of course we have for the intensive variables
˜T = T, ˜p = p, ˜μ = μ. (76 )
dE˜ = λ dE + E dλ = ˜T dS˜− p˜dV˜+ ˜μd ˜N = λ (T dS − pdV + μdN ) + (TS − pV + μN )dλ, (77 )
and therefore we find the Euler relation
E = TS − pV + μN. (78 )
If we let ρ = E ∕V denote the total energy density, s = S ∕V the total entropy density, and n = N∕V the total particle number density, then
p + ρ = Ts + μn. (79 )

The nicest feature of an extensive system is that the number of parameters required for a complete specification of the thermodynamic state can be reduced by one, and in such a way that only intensive thermodynamic variables remain. To see this, let λ = 1 ∕V, in which case

S˜= s, V˜ = 1, ˜N = n. (80 )
The re-scaled energy becomes just the total energy density, i.e. ˜ E = E ∕V = ρ, and moreover ρ = ρ (s,n ) since
ρ = E˜(˜S,V˜,N˜) = E˜(S∕V, 1,N ∕V ) = E˜(s,n ). (81 )
The first law thus becomes
˜ ˜ ˜ ˜ ˜ d E = T d S − ˜pd V + ˜μ dN = T ds + μ dn, (82 )
d ρ = T ds + μ dn. (83 )
This implies
| | ∂-ρ|| ∂-ρ|| T = ∂s ||, μ = ∂n ||. (84 ) n s
The Euler relation (79View Equation) then yields the pressure as
∂ ρ|| ∂ρ || p = − ρ + s---||+ n ---||. (85 ) ∂s|n ∂n |s

We can think of a given relation ρ(s,n) as the equation of state, to be determined in the flat, tangent space at each point of the manifold, or, physically, small enough patches across which the changes in the gravitational field are negligible, but also large enough to contain a large number of particles. For example, for a neutron star Glendenning [48Jump To The Next Citation Point] has reasoned that the relative change in the metric over the size of a nucleon with respect to the change over the entire star is about 10−19, and thus one must consider many internucleon spacings before a substantial change in the metric occurs. In other words, it is sufficient to determine the properties of matter in special relativity, neglecting effects due to spacetime curvature. The equation of state is the major link between the microphysics that governs the local fluid behavior and global quantities (such as the mass and radius of a star).

In what follows we will use a thermodynamic formulation that satisfies the fundamental scaling relation, meaning that the local thermodynamic state (modulo entrainment, see later) is a function of the variables N ∕V, S∕V, etc. This is in contrast to the fluid formulation of “MTW” [80Jump To The Next Citation Point]. In their approach one fixes from the outset the total number of particles N, meaning that one simply sets dN = 0 in the first law of thermodynamics. Thus without imposing any scaling relation, one can write

1 dρ = d(E ∕V ) = T ds + --(p + ρ − Ts) dn. (86 ) n
This is consistent with our starting point for fluids, because we assume that the extensive variables associated with a fluid element do not change as the fluid element moves through spacetime. However, we feel that the use of scaling is necessary in that the fully conservative, or non-dissipative, fluid formalism presented below can be adapted to non-conservative, or dissipative, situations where dN = 0 cannot be imposed.
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