# Temperature anisotropy in plasmas

If a species of a plasma is described by a distribution function $$f(\mathbf{r},\mathbf{v},t)$$, is it possible to have different temperature (thermal energy) values for different directions (x, y and z)?

The definition of kinetic temperature is given by

$$\frac{3}{2}k_BT = \frac{1}{2}m\langle c^2\rangle = \frac{1}{2}m\langle c_x^2\rangle + \frac{1}{2}m\langle c_y^2\rangle + \frac{1}{2}m\langle c_z^2\rangle,$$

where $$c_\alpha=v_\alpha-\langle v_\alpha\rangle$$, and $$\alpha=x, y,z$$.

From this, as described in Bittencourt's book (chapter 6, section 6.4, eq. 6.12), it is possible to define a certain $$T_\alpha = \frac{m}{k_B} \langle c_\alpha^2\rangle$$. With this if $$\langle c_x^2\rangle = \langle c_y^2\rangle = \langle c_z^2\rangle$$, $$T=T_\alpha$$, and if not, $$T=\frac{1}{3}(T_x + T_y + T_z)$$.

If this is correct, what is the meaning and application of $$T_\alpha$$? It would be great to have other references about the treatment of temperature in different directions in a plasma. It is hard to find books with good discussions on the subject.

Note that some of this answer is taken from https://physics.stackexchange.com/a/488052/59023 and What is the correct relativistic distribution function?.

In plasmas that are weakly collisional or collisionless, it is perfectly okay to have anisotropic velocity distribution function (VDFs). These plasmas are neither in thermodynamic or thermal equilibrium.

The most common anisotropic VDF is the bi-Maxwellian given by: $$f_{s}\left( v_{\parallel}, v_{\perp} \right) = \frac{ n_{s} }{ \pi^{3/2} \ V_{T \parallel, s} \ V_{T \perp, s}^{2} } \ exp\left[ - \left( \frac{ v_{\parallel} - v_{o, \parallel, s} }{ V_{T \parallel, s} } \right)^{2} - \left( \frac{ v_{\perp} - v_{o, \perp, s} }{ V_{T \perp, s} } \right)^{2} \right] \tag{0}$$ where $$\parallel$$($$\perp$$) refer to directions parallel(perpendicular) with respect to a quasi-static magnetic field, $$\mathbf{B}_{o}$$, $$V_{T_{j, s}}$$ is the $$j^{th}$$ thermal speed (actually the most probable speed), $$v_{o, j, s}$$ is the $$j^{th}$$ component of the bulk drift velocity of the distribution (i.e., from the 1st velocity moment), and $$n_{s}$$ is the number density or zeroth velocity moment of species $$s$$.

It is now commonly observed and accepted that a more appropriate model is the bi-kappa VDF, given by: $$f_{s}\left( v_{\parallel}, v_{\perp} \right) = A_{s} \left[ 1 + \left( \frac{ v_{\parallel} - v_{o, \parallel, s} }{ \sqrt{ \kappa_{s} - 3/2 } \ \theta_{\parallel, s} } \right)^{2} + \left( \frac{ v_{\perp} - v_{o, \perp, s} }{ \sqrt{ \kappa - 3/2 } \ \theta_{\perp, s} } \right)^{2} \right]^{- \left( \kappa_{s} + 1 \right) } \tag{1}$$ where the amplitude is given by: $$A_{s} = \left( \frac{ n_{s} \ \Gamma\left( \kappa_{s} + 1 \right) }{ \left( \pi \left( \kappa_{s} - 3/2 \right) \right)^{3/2} \ \theta_{\parallel, s} \ \theta_{\perp, s}^{2} \ \Gamma\left( \kappa_{s} - 1/2 \right) } \right) \tag{2}$$ and where $$\theta_{j, s}$$ is the $$j^{th}$$ thermal speed (also the most probable speed), $$\Gamma(x)$$ is the complete gamma function, and $$\kappa_{s}$$ is the kappa index and can be any value larger than 3/2.

If this is correct, what is the meaning and application of $$T_{\alpha}$$?

In non-equilibrium systems with anisotropic VDFs, the components of the temperature pseudotensor are just measures of mean kinetic energy (i.e., the 2nd velocity moment) in the bulk flow rest frame along any given direction (e.g., see https://physics.stackexchange.com/a/218643/59023). This generally requires an assumption about the relationship between pressure and temperature and the most commonly used form is given by: $$P_{s, j} = n_{s} \ k_{B} \ T_{s, j} \tag{3}$$

It would be great to have other references about the treatment of temperature in different directions in a plasma. It is hard to find books with good discussions on the subject.

There are lots of good books like Gurnett and Bhattacharjee, 2005 among others that have lots of detailed discussions on the topic.