Calculating the magnetic permeability of a known plasma I have a plasma with known composition, temperature, and exact electron/ion/particle densities.
How can I then calculate a theoretical permeability of the plasma? Is this even possible? I assume that if I know the composition, I can work it out. Any help is very much appreciated.
 A: You know you are asking a good question when it was the thesis question of Niels Bohr and Hendrika Johanna van Leeuwen. The relative magnetic permeability is $\mu_r= \left(1+\chi_m\right)$, where $\chi_m=\left(\frac{\partial M}{\partial H}\right)_{H=0}$ is the magnetic susceptibility. The magnetization $M$, i.e. the magnetic dipole moment per unit volume, in a non-rotating plasma of uniform density is the statistical average of the dipole moments of each charge multiplied with the number density of dipoles, $M=n_0 \langle \mu \rangle$. 
If the charges are spinless, they don't have intrinsic magnetic moments but could create magnetic moments by circular motion in a uniform magnetic field.  The kinetic dipole moment of a charge is $\boldsymbol{\mu} = \frac{q}{2 }\mathbf{r}\times\mathbf{\dot{r}}$. The conserved Hamiltonian is
\begin{equation}
\mathcal H = \frac{\left(\mathbf{p}-q \mathbf{A}\right)^2}{2 m}+ q \phi\,.
\end{equation}
From Hamiltonian mechanics we get $\mathbf{\dot{r}}=d\mathcal H /d \mathbf{p}=\left(\mathbf{p}-q \mathbf{A}\right)/m$.
Substituting $\mathbf{p}_{m}=\left(\mathbf{p}-q \mathbf{A}\right)$, where $\mathbf{p}_m$ stands for mechanical momentum and $\mathbf{p}$ is the canonical momentum, we get $\boldsymbol{\mu}=\frac{q}{2m}\mathbf{r}\times \mathbf{p}_{m}$. We assume the magnetic field is $\mathbf{B}=B_z \hat{\mathbf{z}}$. The average kinetic dipole moment is
\begin{equation}
\langle \mu \rangle = \frac{\int \mu e^{-\mathcal H/k_B T}\, \frac{d\mathbf{p}\,d\mathbf{r}}{h^3}}{\int e^{-\mathcal H/k_B T}\, \frac{d\mathbf{p}\,d\mathbf{r}}{h^3}}\propto\int_{-\infty}^{\infty} p_{m,x} e^{-\frac{p_{m,x}^2}{2mk_B T}}\, dp_{m,x} =0\,.
\end{equation}
This results in the Bohr-van Leeuwen Theorem, which states that the magnetization of a non-rotating system of spinless charges in thermal equilibrium is zero. 
If the charges have spin, they have intrinsic magnetic moments $\mu_1$, with average value
\begin{equation}
\langle \mu_1\rangle = \frac{\int_{-1}^{1} \cos\theta\mu_1 e^{{\cos\theta\mu_1 H }/{k_BT}}\,d(\cos\theta)}{\int_{-1}^{1} e^{{\cos\theta\mu_1 H }/{k_BT}}\,d(\cos\theta)} = \mu_1 \left[ \coth\left(\frac{\mu_1 H}{k_B T}\right)-\frac{k_B T}{\mu_1 H}\right]\,.
\end{equation}
For plasmas we have $\mu_1 H \ll k_B T$, such that $\langle \mu_1\rangle \approx \mu_1 \frac{\mu_1 H}{3 k_B T}$ and $M=n_0 \langle \mu_1\rangle$. Therefore, the magnetic susceptibility is $\chi_m=\left(\frac{\partial M}{\partial H}\right)_{H=0}=\frac{\mu_1^2 n_0 }{3 k_B T}$, giving magnetic permeability
\begin{equation}
\mu_r=1+\frac{\mu_1^2 n_0 }{3 k_B T}\,.
\end{equation}
For typical plasmas the intrinsic magnetic moment of charges is very small compared to the temperature, so that plasmas have effectively the permeability of vacuum.
