### 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 . 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 ,
stress-energy-momentum tensor , and associated conserved particle number density current
(starting with some fundamental Lagrangian, say; cf. [115, 48, 116]). 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 (85)). If the reader likes, the application of at a spacetime
point means that has been determined with respect to a flat tangent space at that
point.
Once 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

where
At sufficiently small temperatures, will just be a function of the number density of particles
at the spacetime point in question, i.e. . Similarly, the pressure is obtained as
and it will also be a function of .
One must be very careful to distinguish from . 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.