# Relativistic maxwell-boltzmann-distribution

• magicfountain
In summary, in thermodynamics, the maxwell-boltzmann-distribution can be used to find the average speed of gas particles, regardless of relativistic effects. However, at high temperatures, the average speed may exceed the speed of light. To describe gases with an average speed of 0.5 relativistically, the nonrelativistic formula must be replaced with the relativistic one, which involves integrating over the Boltzmann factor and normalizing the distribution to N particles per volume. The Boltzmann factor can also be included in the partition function by using the Maxwell-Juttner expression, which consists of the Boltzmann factor and a function of temperature.
magicfountain
In thermodynamics (ignoring relativistic effects) you can use the maxwell-boltzmann-distribution to find the average speed of the gas particles.
$v^2=\frac{8kT}{\pi m}$

But there are high Temperatures that would have average speeds > c.
Are there distributions that describe gases with an average speed of 0.5 relativistically?

Well, you have to replace the nonrelativistic formula with the relativistic one.

The average squared velocity will go towards c^2, and all particles are always slower than c for every temperature.

mfb said:
Well, you have to replace the nonrelativistic formula with the relativistic one.

...which to me is not very illuminating, since it's given without any derivation and written in terms of a goofy choice of variable.

Can't you just take the partition function and put in the relativistic expression for the energy? I.e.:

$$e^{-\beta\sqrt{m^2+p^2}}$$

This is in units with c=1, and beta is the inverse temperature.

Yes, I think so. And I see the Maxwell–Juttner expression does consist of this Boltzmann factor, plus a function of T in front. To get the factor in front you have to normalize the distribution to N particles per volume, which involves integrating over the Boltzmann factor. Nonrelativistically the integral leads to (m/2πkT)3/2. But here we have to integrate over the relativistic Boltzmann factor, and that's where the K2(T) Bessel function comes from.

@bcrowell
@Bill_K
that helped a lot. i guessed that i had to do lagrange multipliers with relativistic expressions, but i was too lazy to really think about it. thanks for reminding me that it actually just leads to the partition function and you have to plug in the rel. terms there.

## What is the Relativistic Maxwell-Boltzmann Distribution?

The Relativistic Maxwell-Boltzmann Distribution is a statistical distribution that describes the distribution of velocities for a large number of particles in a system at a given temperature. It takes into account the relativistic effects of high speeds on the distribution, unlike the classical Maxwell-Boltzmann Distribution.

## What assumptions are made in the derivation of the Relativistic Maxwell-Boltzmann Distribution?

The derivation of the Relativistic Maxwell-Boltzmann Distribution assumes that the particles are non-interacting, the system is in thermal equilibrium, and the particles follow a Boltzmann distribution in energy.

## How does the Relativistic Maxwell-Boltzmann Distribution differ from the classical Maxwell-Boltzmann Distribution?

The classical Maxwell-Boltzmann Distribution does not take into account the effects of relativity on the distribution of velocities, while the Relativistic Maxwell-Boltzmann Distribution does. This means that at high speeds, the classical distribution underestimates the number of particles with high velocities compared to the relativistic distribution.

## What is the significance of the Relativistic Maxwell-Boltzmann Distribution in physics?

The Relativistic Maxwell-Boltzmann Distribution is important in many areas of physics, such as thermodynamics, statistical mechanics, and astrophysics. It helps to describe the behavior of particles in a system at high speeds, which is crucial for understanding phenomena such as black holes, neutron stars, and supernovae.

## How is the Relativistic Maxwell-Boltzmann Distribution used in practical applications?

The Relativistic Maxwell-Boltzmann Distribution is used in various practical applications, such as in the design of nuclear reactors, particle accelerators, and high-speed aircraft. It is also used in the study of cosmic rays and the behavior of particles in the early universe.

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