Go to previous page Go up Go to next page

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
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.

  Go to previous page Go up Go to next page