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5.2 From microscopic models to the fluid equation of state

Let us now briefly discuss how an equation of state is constructed. For simplicity, we focus on a one-parameter system, with that parameter being the particle number density. The equation of state will then be of the form ρ = ρ(n). In many-body physics (such as studied in condensed matter, nuclear, and particle physics) one can in principle construct the quantum mechanical particle number density n QM, stress-energy-momentum tensor QM Tμν, and associated conserved particle number density current μ nQM (starting with some fundamental Lagrangian, say; cf. [11548116]). But unlike in quantum field theory in curved spacetime [13], one assumes that the matter exists in an infinite Minkowski spacetime (cf. the discussion following Equation (85View Equation)). If the reader likes, the application of T QM μν at a spacetime point means that T QM μν has been determined with respect to a flat tangent space at that point.

Once T QμMν is obtained, and after (quantum mechanical and statistical) expectation values with respect to the system’s (quantum and statistical) states are taken, one defines the energy density as

ρ = uμu ν⟨TQM ⟩, (87 ) μν
where
u μ ≡ 1⟨n μ ⟩, n = ⟨n ⟩. (88 ) n QM QM
At sufficiently small temperatures, ρ will just be a function of the number density of particles n at the spacetime point in question, i.e. ρ = ρ(n). Similarly, the pressure is obtained as
1 ( ) p = -- ⟨TQM μμ⟩ + ρ (89 ) 3
and it will also be a function of n.

One must be very careful to distinguish T QM μν from Tμν. The former describes the states of elementary particles with respect to a fluid element, whereas the latter describes the states of fluid elements with respect to the system. Comer and Joynt [35] have shown how this line of reasoning applies to the two-fluid case.


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