Generally speaking, the total energy density can be a function of independent parameters
other than the particle number density
, such as the entropy density
, assuming that
the system scales in the manner discussed in Section 5 so that only densities need enter the
equation of state. (For later convenience we will introduce the constituent indices
,
,
etc. which range over the two constituents
, and do not satisfy any kind of summation
convention.) If there is no heat conduction, then this is still a single fluid problem, meaning that
there is still just one unit flow velocity
[36]. This is what we mean by a two-constituent,
single fluid. We assume that the particle number and entropy are both conserved along the
flow, in the same sense as in Equation (126
). Associated with each parameter there is thus a
conserved current density four-vector, i.e.
for the particle number density and
for the entropy density. Note that the ratio
is comoving in the sense that
In terms of constituent indices ,
, the associated combined first and second laws can be written in
the form
Given that we only have one four-velocity, the system will still just have one fluid element
per spacetime point. But unlike before, there will be an additional conserved number, ,
that can be attached to each worldline, like the particle number
of Figure 7
. In order
to describe the worldlines we can use the same three scalars
as before. But how
do we get a construction that allows for the additional conserved number? Recall that the
intersections of the worldlines with some hypersurface, say
, is uniquely specified by
the three
scalars. Each worldline will have also the conserved numbers
and
assigned to them. Thus, the values of these numbers can be expressed as functions of the
. But most importantly, the fact that each
is conserved, means that this function
specification must hold for all of spacetime, so that in particular the ratio
is of the form
. Consequently, we now have a construction whereby this ratio identically satisfies
Equation (154
), and the action principle remains a variational problem just in terms of the three
scalars.
The variation of the action follows like before, except now a constituent index must be attached to
the particle number density current and three-form:
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