What Are the Limits of Pressure as a Thermodynamic Quantity?

[Raghnar]

Hallo! Ok, I know the textbook and I actually have too many degrees in "RFM"... But still I have some doubts regarding the nature of pressure, and its limit of applicability.

So Pressure is a Thermodynamical intensive quantity, thus statistically defined. That teaches us the Barometric law and that to pressure gradient correspond to a force that is consistently defined with the density of matter and the gravitational pull in an equilibrium state.

So you consider that definition in the description of the air and atmospheric pressure, you do your math and considering that the Air does not contribute substantially to the mass of the Earth (or even approximating it constant) you find the usual exponential form of the barometric law.

Microscopically you have the weight that pull down the gas, the gas compress and reach an equilibrium with the force (pushing to external layers) exerted by the gradient of pressure. The pressure is then given by the density and temperature of the gas volume as a result of this process.

In this picture you don't have the pressure "compressing" the gas, but the weight. The pressure is the results of other thermodynamic quantities.

Now let's consider another matter, for example liquid.
How does the pressure of the water sums up to this?
By the same principle the gravity pull the water to the center of the earth, and the same principle determine a different density and so on and a different pressure gradient needed.

But if we consider a body made only of omogeneus matter (e.g. liquid or gas), its gravitational field, for the Gauss Theorem, will be proportional to r
(F \sim M_{int}/ r^2 = \rho r^3/r^2 = \rho r), so also the pressure in principle should not increase dramatically going to the center.
How it is that the pressure is always considered increasing, and in many environment A LOT?

And what about a complex stellar environment?
The gravitational pull determine a density of stuff that generate a pressure gradient that counterbalance until you reach equilibrium?
And in exotic systems like neutron stars, where matter density change for tens and tens orders of magnitude in few km?
And in exploding environment like core-collapsing supernovae?

How pressure, that is defined in the familiar Barometric Law, as a counter-balancing force, is often described as the active force the determine the great densities of the neutron star or the explosing force of a core-collapsing nova?
What is pressure exactly? If its correlated to the mean free path in gases how this connect with other environments such as liquid, solid, nuclear matter, quark-gluonic matter...etc...? What are the boundaries of the representation of this thermodynamic quantity ideated to explain gas behavior and used to explain a wide variety of matter in a wide variety of environments?

I hope to receive some insight regarding this and if you can point out some reference text will be appreciated.

 

[klimatos]

In kinetic gas theory, mean gas pressures (in Pascals) are the simple product of the mean frequency of molecular impacts (in number per square meter per second) and the mean impulse transferred per impact (in Newtons).

I have always found that explanation to be simple, elegant, and phenomenologically satisfying.

 

[Raghnar]

That I know (I wrote it up), but the neutron superfluid composing the inner crust of a neutron star is definitely not a perfect gas (and there is no transfer impulse by impact inside that medium!). And still is described as "originated by the pressure of the outer layer" and "acting on the lower layers" by imposing a great pressure.

So how can we talk about pressure and still think about it in our common familiar and simplistic terms even though the environment is so different than anything we can possibly picture out in our mind?

Thermodynamic is indeed powerful and universal and Pressure as Thermodynamical intensive quantity (and thus derived from extensive and potential) is indeed applicable also to regimes far out our common perfect gas picture.

But what does it mean exactly? And what physical deduction we could do by it? And so on with the other questions...

 

[sophiecentaur]

That I know (I wrote it up), but the neutron superfluid composing the inner crust of a neutron star is definitely not a perfect gas (and there is no transfer impulse by impact inside that medium!). And still is described as "originated by the pressure of the outer layer" and "acting on the lower layers" by imposing a great pressure.

So how can we talk about pressure and still think about it in our common familiar and simplistic terms even though the environment is so different than anything we can possibly picture out in our mind?

Thermodynamic is indeed powerful and universal and Pressure as Thermodynamical intensive quantity (and thus derived from extensive and potential) is indeed applicable also to regimes far out our common perfect gas picture.

But what does it mean exactly? And what physical deduction we could do by it? And so on with the other questions...

Why should there be anything different, in principle, within a neutron star? The notion of pressure is rate of change of momentum on a given area so, despite the conditions being 'different' in extreme circumstances, you can take an elemental area and apply a similar approach.
What we each choose, to "picture out in our minds" is a personal matter. All we can do is to communicate, using common terms and models. It's a big help to avoid asking 'what is really happening?' Very few things, on scales that are widely different from our common experience, are totally like the models we hold in our heads. We just cope as best we can and accept the shortcomings. That's where the formality of Maths can help.

 

[Raghnar]

Why should there be anything different, in principle, within a neutron star? The notion of pressure is rate of change of momentum on a given area

Because for example in a superfluid there is no exchange of momentum since the time-reversal symmetry restoration.

What are the shortcoming and assumption in the definition of pressure? How can something that does not exchange momentum change the momentum (and density) of its lower layers?

 

[sophiecentaur]

Hmmm. Difficult. Are you implying that the only forces on bodies in the neutron star would be in equilibrium between gravitational attraction and intra nuclear forces and not result from the dynamics of the situation? That would be a different 'state of matter'. Fair enough, I guess.

 

[voko]

If you are not happy with the phenomenological definition of pressure, there is an abstract one. It is $$ P = - \left(\frac {\partial E} {\partial V} \right)_S, $$ where $P$ is pressure, $E$ is energy, $V$ is volume, and $S$ is entropy. See Landau & Lifschitz, Statistical Physics, for details.

 

[sophiecentaur]

Bit of a catch 22 situation there. The entropy for a superfluid is zero, I think.

 

[russ_watters]

Pressure in a closed container must be due to fluid motion, but when gravity is creating the pressure, you can set aside the fluid motion and just use weight/area.

 

[voko]

That should not matter, as long as it is held constant while differentiating.

That equation, by the way, is contained in a somewhat better known one: $$

dE = T dS - P dV. $$

 

[sophiecentaur]

I think you would be merely shifting the question about Pressure into the question of what the term Entropy represents in extreme states of matter. After all, Entropy is based, afaiaa, on thermodynamic principles, which involve a pretty classical model of particles and the way they behave.

 

[voko]

Referring again to Landau & Lifschitz, they actually define entropy for quantum statistics, and then they use the correspondence principle to deduce entropy for classical statistics.

 

[sophiecentaur]

That's good to know. I haven't a copy of L&L readily available any more, unfortunately.