Non-relativistic vs Ultra-relativistic

  • Thread starter Piano man
  • Start date
In summary, the usual values for internal energies of gas depend on whether it is non-relativistic or ultra-relativistic. For non-relativistic, the energy is E=3/2 kT for three degrees of freedom, while for ultra-relativistic, it is E=3kT. The mathematical explanation for this involves using statistical mechanics to average the energy over possible states with a Boltzmann weighting factor. This can be done through integration, with the help of modified Bessel functions of the second kind and the general solution for pressure being nkT. More information on this topic can be found at the Partition function (statistical mechanics) on Wikipedia.
  • #1
Piano man
75
0
Just a quick question re internal energies for gas.

For non-relativistic, the usual value is E=3/2 kT for three degrees of freedom.
For ultra-relativistic, it is E=3kT.

Does anyone have a mathematical explanation for that, or a link to a derivation?
Thanks.
 
Physics news on Phys.org
  • #2
You'll have to be familiar with statistical mechanics. You'll solve the problem by averaging the energy over possible states with a Boltzmann weighting factor.

[itex]<X> = \frac{\int X e^{-E/kT} d^3 p}{\int e^{-E/kT} d^3 p}[/itex]

Energy E, momentum p, etc.

You can do this integral by going into spherical coordinates for the momentum, integrating out the angles, and using the appropriate expression for the total energy. In the general case, it's rather complicated, but in the nonrelativistic and ultrarelativistic limits, it's easier.

With the help of page 376 of Abramowitz and Stegun - Page index, I was able to find the appropriate integrals in the general case. For that, one substitutes
p = m*sinh(u)
E = m*cosh(u)
and then integrates with (9.6.24), giving

[itex]<E> = m\frac{K_4(x) - K_0(x)}{2(K_3(x) - K_1(x))}[/itex]

where x = m/kT and the K's are modified Bessel functions of the second kind. A&S also gives some formulas for the K's for the large and small limits of x.
 
  • #3
It's also interesting to calculate the pressure. In general,

[itex]P_{ij} = n <v_i p_j> = n<\frac{p_i p_j}{E}>[/itex]

for number density n, which becomes

[itex]P_{ij} = P \delta_{ij} ,\ P = n<\frac{p^2}{3E}>[/itex]

One can solve this case by integration by parts:

[itex]\int e^{-E/kT} \frac{p^2}{3E} p^2 dp = \frac13 \int e^{-E/kT} p^3 dE = kT \int e^{-E/kT} p^2 dp[/itex]

yielding P = nkT as the general solution.

I've been working in the classical limit, of course.

You can find more details at Partition function (statistical mechanics) (Wikipedia)
 
  • #4
Thank you very much. That gives me something to work from.
 
  • #5


The difference between non-relativistic and ultra-relativistic internal energies for a gas can be explained by the different assumptions made about the particles in the gas. In non-relativistic systems, the particles are assumed to have low velocities compared to the speed of light, and their kinetic energy is given by the classical formula E=1/2 mv^2. In this case, the average kinetic energy of the particles is directly proportional to the temperature, and the factor of 3/2 comes from the three different directions in which the particles can move.

On the other hand, in ultra-relativistic systems, the particles are assumed to have velocities close to the speed of light, and their kinetic energy is given by the relativistic formula E=mc^2, where m is the mass of the particle and c is the speed of light. In this case, the average kinetic energy of the particles is still directly proportional to the temperature, but the factor of 3 is due to the fact that the particles can move in three spatial dimensions, as well as in the time dimension.

A more detailed derivation of these formulas can be found in textbooks on thermodynamics and statistical mechanics. I would recommend checking out "An Introduction to Thermal Physics" by Daniel V. Schroeder or "Thermodynamics and Statistical Mechanics" by Walter Greiner for a more comprehensive explanation.

I hope this helps clarify the difference between non-relativistic and ultra-relativistic internal energies for a gas.
 

What is the difference between non-relativistic and ultra-relativistic?

Non-relativistic refers to a system where the speeds of particles are much lower than the speed of light, while ultra-relativistic refers to a system where particles are moving at speeds close to the speed of light.

How does the theory of relativity apply to these two systems?

Einstein's theory of relativity states that the laws of physics are the same for all observers, regardless of their relative motion. This means that the laws of physics will behave differently in non-relativistic and ultra-relativistic systems.

What are some examples of non-relativistic systems?

Everyday objects and phenomena, such as cars, airplanes, and sound waves, can be considered non-relativistic as their speeds are significantly lower than the speed of light.

Can ultra-relativistic speeds be achieved in our universe?

According to the theory of relativity, it is impossible for an object with mass to reach the speed of light. However, certain subatomic particles, such as cosmic rays, can reach speeds close to the speed of light.

What are some practical applications of understanding the difference between non-relativistic and ultra-relativistic systems?

Understanding the principles of relativity and the differences between non-relativistic and ultra-relativistic systems is crucial in many fields of science, such as astrophysics, particle physics, and engineering. It also plays a role in the development of technologies such as particle accelerators and GPS systems.

Similar threads

  • Quantum Physics
3
Replies
87
Views
4K
  • Special and General Relativity
Replies
15
Views
1K
  • Special and General Relativity
Replies
10
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
Replies
9
Views
1K
Replies
18
Views
921
  • Special and General Relativity
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
Replies
3
Views
1K
Replies
7
Views
967
Back
Top