Lecture 17: Ultra-relativistic gas

Note

Mirrors should think longer before they reflect. – Jean Cocteau

Warning

This lecture corresponds to Chapter 25 of the textbook.

Summary

Attention

Because the dispersion relation for an ultra-relativistic particle is different than that of a free particle (like in ideal gas), the thermodynamic properties are markedly different.

Students will remember that the partition function for a gas is calculated using the density of states, which is itself dependent on the dispersion relation. In an ideal gas, the dispersion relation is quadratic (see box below) while for a gas of ultra-relativistic particles, the dispersion relation is linear. Therefore, we find:

\begin{tabular}{lll} \hline Property & Non-relativistic & Ultrarelativistic \\ \hline$Z_{1}$ & $\frac{V}{\lambda_{\text {th }}^{3}}$ & $\frac{V}{\Lambda^{3}}$ \\ & $\lambda_{\text {th }}=\frac{h}{\sqrt{2 \pi m k_{ B } T}}$ & $\Lambda=\frac{c \pi^{2 / 3}}{k_{ B } T}$ \\ \hline$U$ & $\frac{3}{2} N k_{ B } T$ & $3 N k_{ B } T$ \\ & $\frac{5}{2} N k_{ B } T$ & $4 N k_{ B } T$ \\ & $\frac{N k_{ B } T}{V}$ & $\frac{N k_{ B } T}{V}$ \\ & $=\frac{2 u}{3}$ & $=\frac{u}{3}$ \\ \hline$U$ & $N k_{ B } T\left[\ln \left(n \lambda_{ th }^{3}\right)-1\right]$ & $N k_{ B } T\left[\ln \left(n \Lambda^{3}\right)-1\right]$ \\ \hline Adiabatic expansion & $V T^{3 / 2}=$ constant & $V T^{3}=$ constant \\ & $p V^{5 / 3}=$ constant & $p V^{4 / 3}=$ constant \\ \hline \end{tabular}

Warning

Equipartition theorem for ultra-relativistic particles

We saw in Lecture 11: Equipartition of energy that the equipartition theorem established that for each degree of freedom in a system, the internal energy is

U=\frac{1}{2} k_B T

Does that apply to a gas of ultra-relativistic particles? No!

Why? Because it is essential to remember that the equipartition theorem is only valid if the energy contributions are quadratic. While this is true for kinetic energy of non-relativistic particle or for the harmonic approximation, it is no longer true for an ultra-relativistic particle since, in that case:

E=\hbar k c.

For such a linear relationship the corresponding result for the equipartition theorem is

U=N k_B T

for N particles.

What’s a dispersion relation?

Simply put, a dispersion relation relates the wavelength or wave-number of a wave to its frequency. Formally:

\omega(k)\quad {\rm or} \quad k(\omega)

Often, boundary conditions are imposed to the wave-number (or quasi-momentum). In this course, we examined specifically that the boundary conditions imposed to a wave confined to a box of size L imposes only some wavelength to fit in the box (so the amplitude of the wave disappears at the edges of the box – since the potential there is formally infinite).

We have encountered a number of different dispersion relations in this course:

  1. Photons

    \omega=ck

    where c is the speed of light.

  2. Acoustic phonons (Debye)

    \omega=v_sk

    where v_s is the speed of sound.

  3. Phonons (general)

    The relationship is usually more complicated and there are both acoustic and optical phonons. Each system has a unique set of phonons and specific calculations must be performed to establish the vibrational dispersion relation.

  4. Free particles (e.g., electrons; non-relativistic particles)

    \omega=\frac{\hbar k^2}{2m}

    This relationship is just a way to say that the energy is determined by kinetic energy only.

  5. Ultra-relativistic particles (very large velocity, comparable to speed of light)

    \omega= k c

Photons are examples of ultra-relativistic particles.

Learning Material

Copy of Slides

The slides for Lecture 17 are available in pdf format here: pdf

Screencast

Test your knowledge

  1. A gas is composed of particles with velocity close to the speed of light. Among the following assertions, which one is true?

    1. Because it is universal, the equipartition theorem applies to this system.

    2. The pressure of this gas is smaller than that of a non-relativistic gas with identical energy density.

    3. There is not enough information to provide a conclusive answer.

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!

Homework Assignment

Solve the following problems from the textbook: