# What is a “perfect fluid”, really?

In Statistical Mechanics, unless I'm making a confusion mistake, a perfect fluid is defined as a large collection of particles without any "internal" interactions (except maybe from point collisions). Long and short range forces are all neglected. This gives the usual perfect gas law: $$\begin{equation}\tag{1} p V = N k T. \end{equation}$$ Assuming adiabatic irreversibility, this implies the polytrope relation, which is a special case of a barotrope state relation : $$\begin{equation}\tag{2} p = \kappa \, \rho_{\text{mass}}^{\gamma}, \end{equation}$$ where $$\kappa$$ is a constant and $$\gamma$$ is the adiabatic index of the fluid. Of course, $$\rho_{\text{mass}}$$ is the proper mass density of the fluid. We could also find $$\begin{equation}\tag{3} p = (\gamma - 1) \, \rho_{\text{int}}, \end{equation}$$ where $$\rho_{\text{int}}$$ is the internal energy density, defined as $$\rho_{\text{int}} = \rho - \rho_{\text{mass}}$$ if $$\rho$$ is the total energy density (I'm using natural units so $$c \equiv 1$$).

Now, in Special (and General) Relativity, a perfect fluid is defined as any substance that doesn't show any macroscopic viscosity and shear (this $$\underline{\text{suggest}}$$ no internal microscopic interactions, but this isn't obvious), and such that the energy-momentum of the fluid is diagonal and isotropic in the proper reference frame: $$\begin{equation}\tag{4} T_{ab} = \begin{bmatrix} \rho & 0 & 0 & 0 \\ 0 & p & 0 & 0 \\ 0 & 0 & p & 0 \\ 0 & 0 & 0 & p \end{bmatrix}. \end{equation}$$ Here, a barotropic relation may admit any function $$p(\rho)$$, and not just (2) or (3) above. For example, we may admit the Van der Waals law which isn't a perfect fluid in statistical mechanics (there are some short range forces in action) : $$\begin{equation}\tag{5} p = \frac{c \, \rho_{\text{mass}} \, T}{1 - a \, \rho_{\text{mass}}} - b \, \rho_{\text{mass}}^2. \end{equation}$$ This special relativistic definition isn't the same as the statistical mechanics definition, since it may admit fluids with some internal interactions (yet without showing any macroscopic shear and viscosity).

Now, I'm finding myself irritated by these two definitions, which aren't exactly equivalent. So what really is a "perfect fluid"? The $$\underline{\text{statistical}}$$ one (without any microscopic internal interactions), or the $$\underline{\text{relativistic}}$$ one (which may admit internal interactions)?

Or is there two different "names" that may differentiate the two definitions, something like "ideal fluid" and "perfect fluid" or something else?

I don't like the two inequivalent definitions having the same name, since it opens the door to confusion. I don't want a sloppy use of the same name, just because we are working in different fields (classical statistical mechanicians, or general relativists, or fluid dynamicians, ...).

I have the impression that the statistical definition is the right one, from an historical perspective, and that the relativists should call their perfect fluid as "ideal fluid", instead. Is that right? Or maybe it should be the reverse??

The sloppy Wikipedia appears to reverse the names: https://en.wikipedia.org/wiki/Perfect_fluid, which apparently was written by a relativist! And https://en.wikipedia.org/wiki/Ideal_gas which calls "ideal gas" the perfect fluid of statistical mechanics. Now I'm all confused! Wikipedia isn't a good reference for physics definitions, since there are frequently many inconsistencies.

• I'm not sure to understand your previous comment. The statistical definition of an ideal gas (what I wrongly called "perfect fluid") is a collection of particles without any internal interaction. It's a bunch of free particles. This is how we derive the formula $p V = N k T$, and so the polytrope formula (2). – Cham Jan 30 at 21:42