special relativity - What is the equation of state for a relativistic fluid/gas?

Say we have a relativistic fluid/gas, as we have in some astrophyical systems.

Now let us write:

• $e$ - energy density in the fluid's rest frame.




• $P$ - pressure in the fluid's rest frame.




• $n$ - number density in the fluid's rest frame.




• $m$ - mass of the particles.

I know that for the non-relativistic case we have:

$$e=nmc^2+frac{1}{hat{gamma}-1}P$$

where $hat{gamma}$ is the adiabatic index. $hat{gamma}=1+frac{2}{f}$ for a gas with $f$ degrees of freedom.

For the ultra-relativstic case we have:

$$e=3P$$

My question is what is $e(P,n)$ for a relativstic case (which is the general case of the 2 limits shown above)? I would also like to know how to derive it.

---

Is the following way the correct way to do it ? :

The number density of particles is:
$$n=int_{0}^{infty} n_p(p) dp $$

The pressure is:
$$P=int_{0}^{infty} frac{1}{3} p v(p) n_p(p) dp $$

The energy density is:
$$e=int_{0}^{infty} epsilon(p) n_p(p) dp $$

where:

$$n_p(p)= (2s+1)frac{1}{ e^{({epsilon(p)-mu})/{k_B T}}+(-1)^{2s+1} } frac{4pi p^2}{h^3}$$

Here $s$ is the spin of the particles, for electrons $s=frac{1}{2}$.

$$ epsilon(p)=(m^2c^4+p^2c^2)^{frac{1}{2}} $$

$$ v(p)= frac{depsilon}{dp}=frac{p}{m}left(1+left(frac{p}{mc}right)^2right)^{-frac{1}{2}} $$

From calculating the three integrals above we can finally obtain $e(P,n)$.

• Can anyone confirm this is the proper way to do it, or am I missing something here?




• It seems as if those integrals cannot be solved analytically - is this
  true?




• Perhaps in this case there is no explicit formula for $e(P,n)$?

If we found evidence of life on Mars, how would we know that it originated on Mars rather than Earth?

Why would we assume that the early martian life originated on Mars,
rather than Earth?

There's still a whole lot we don't know. As Wayfaring stranger points out in the comments, Origin is a whole different question. It's possible that life originated outside our solar system and came to either Mars and/or Earth from outside the solar-system. I don't think anyone who studies this idea is "Assuming" life originated on Mars, only that the idea has a chance of being true.

Is it not likely that somewhere between 2.5-3 billion years ago, a
major volcanic eruption or meteor impact could have hit an earth
teeming with simple-life and sent it hurdling through space to Mars?

As I understand it, volcanic eruptions are unlikely to send anything into space unless it's a smaller sized moon. The escape velocity (young earth, maybe 10 km/s, young mars, maybe 3-4), volcanic eruptions, as far as I know, don't shoot things out at 10,000 - 20,000 MPH. But meteor impacts of sufficient size can do that.

Mars is a better meteor debris making target than the Earth cause it's smaller, so the gravity is lower and presumably it's mostly had a thinner atmosphere too. We've found martial meteors on Earth. We might not find any Earth meteors on mars cause it takes a much bigger impact to knock bits of rock off Earth and because the Atmosphere slows objects down, both coming in and going out.

I understand that my theory has the gravity of the sun working against
it, but I also see it not being entirely impossible.

The sun isn't as big a factor as you might think. Once something is knocked off a planet and it gets into orbit around the sun, gravitational assists can move it around further out or further inside the solar system. What presumably happens is that, with a big enough impact, many thousands if not millions of bits of debris get into solar-system orbit and from there, some of them land on other planets - probably much less than 1% of those hit Earth, but if it carries life that can survive the trip, all you need is one rock.

I mean, 100 years ago, one could assume that life originating on Mars
and then being sent this was was ridiculous and impossible but now
it's a rational, viable theory.

While that's true, the "People used to think this was impossible" isn't a scientific approach for what might be true. We should determine what is possible and/or thought to be likely, based on physical evidence, not what wasn't understood 100 years ago. Your example is a good point on why it's important to keep an open mind about the unknown. You can still make theories based on evidence, and keep an open mind on the unknown. There's really no conflict between the two.

Considering such a scenario, how would scientist verify that these
life-forms didn't originate from Earth, rather than Mars? It seems to
me that this question is the first question that would need to be
asked and answered after the fossil-records were found and verified.

It's a good question.

The simple answer is that Mars cooled first and Mars (likely) had oceans first, so it's a better candidate to have developed life first though extremophiles can live in hot oceans, so . . . time will tell.

bioinformatics - Can I compare Shannon indices of metagenome gene data?

I'm not an expert on Shannon-Weaver Index, but according to wikipedia it is the same as exponentially transformed Renyi entropy. If it is the case, you can compare them since they are scale invariant summary statistics. If you want error bars, you can always try resampling methods such as bootstrapping. Hypothesis testing can also be done with bootstrapping, although the power of the statistical test may not be very strong and fail to reject the null.

temperature - Which moons have cold traps? (i.e. low ecliptic inclination in orbital and rotational axes)

Many of the moons of Jupiter and Saturn are tidally locked and probably move ice from their equatorial regions to their polar regions. Ganymede is the best example where you see bright polar ice caps and a dark equatorial zone. A migration process occurs on all of these moons in which light from the sun gets absorbed by water ice molecules and sends them on random jumps. After a long random walk, they eventually end up near the pole where they stay for a long time even if they are not in a perpetual shadow, and they stay there for nearly forever if they are.

But, if the moon has a very thick coating of ice everywhere, this process may not have dug down deep enough, so it is still covered with ice everywhere, in which case it is hard to see the effect happening.

bioinformatics - Generating custom human DNA sequences based on traits such as eye colour?

I'm wondering if it would be possible to create software (unless some already exists, but I couldn't find any) to generate human DNA (the base pairs on the double helix) containing genes representing specific permutations (eye colour, hair colour, etc.)?

Basically, something like the "character builder" from those "Saints Row"-style video games, except with actual human chromosomes, enabling you to essentially create a 'custom' human.

Of course, that is assuming that all human DNA has a common structure, and that the entire sequence can either be assembled from individual chromosomes, or by using a reference genome and modifying specific genes/chromosomes according to user input. Is this the case?

One setback here, of course, could be the number of bases in each chromosome, which ranges between 100 million and 250 million, and the (approximately) 23000 human genes - a lot of data to manipulate.

cell biology - How Do Large Ocean Viruses Form Their Own Organelles?

Several large viruses (Arslan 2011) form their own organelles within the amoebae they invade.

How do these organelles form?

---

Reference:

Arslan, D., Legendre, M., Seltzer, V., Abergel, C., Claverie, J-M. (2011) Distant Mimivirus relative with a larger genome highlights the fundamental features of Megaviridae. PNAS 108(42): 17486-17491 [DOI]

botany - Why do plants have pith and how is it useful to them?

The pith (medulla) forms part of the ground tissue system of a plant, and specifically it is the ground tissue which lies interior to a plants vascular tissues (xylem, phloem etc.)

The ground tissue system is responsible for much of a plants metabolic functioning, and contains various specialized cell types which aid in photosynthesis and storage of photosynthesis products. The pith is made from parenchyma cells.