Pressure tensor ($P_{ij}$)

In the local rest frame of a gas, the pressure tensor is the directional momentum flux $P_{ij}=\int p_i{v}_{j}f(\mathbf{p})d^{3}p$, where $f$ is the phase-space distribution function per unit momentum volume and $\mathbf{v}$ is particle velocity. For an isotropic distribution, $P_{ij}=P{\delta }_{ij}$, recovering the kinetic pressure of an isotropic gas. An anisotropic distribution can exert different normal pressures along different directions.

For a central differentiable pair potential, an exact local representation of the interaction momentum flux isHere $n_2$ is the two-particle number density. The line integral assigns each pair's exchanged momentum along the segment joining the particles. Translating both endpoints gives ${\partial }_s{n}_2=\mathbf{R}\cdot {\nabla }_{\mathbf{x}}{n}_2$. Integrating over $s$ and using pair-exchange symmetry proves that $-\nabla \cdot {\mathbf{P}}^{\varphi }$ is the interaction force density. Its homogeneous isotropic specialization gives the interaction part of the virial pressure from pair correlations. The kinetic pressure tensor accounts separately for random particle motion.

Directional dependence of particle pressure. In a gyrotropic plasma, only parallel plasma pressure and perpendicular plasma pressure are independent. Their difference changes the effective magnetic tension and can drive firehose instability or mirror instability.

In Gaussian units, the gyrotropic pressure tensor adds directional stress to the magnetic tension. A transverse displacement of a uniform magnetic field obeys $\rho {\partial }_t^2{\mathbf{\xi }}_{\perp }=\mathcal{T}{\partial }_{\parallel }^2{\mathbf{\xi }}_{\perp }$. Positive $\mathcal{T}$ gives a restoring force; negative $\mathcal{T}$ gives the fluid firehose instability. This follows by combining $-\nabla \cdot \mathsf{P}$ with the divergence of the Maxwell stress tensor. In SI replace $B^2/(4\pi )$ by $B^2/{\mu }_0$.

A compressive kinetic plasma instability generally driven by sufficiently large perpendicular plasma pressure relative to parallel plasma pressure. Perturbed field-strength minima trap particles and alter perpendicular pressure-balanced plasma perturbation. Its threshold depends on species distributions and plasma beta; isotropic equilibrium alone does not imply this instability.

Parallel particle pressure can overcome magnetic tension. In the long-wavelength gyrotropic plasma limit, the Alfvén wave has ${\omega }^2=k_{\parallel }^2[B^2/(4\pi )+p_{\perp }-p_{\parallel }]/\rho$, so the condition in the title makes it unstable. The detailed kinetic threshold at shorter scales requires additional physics.

The normal stress along the local magnetic field, evaluated using the random parallel velocity. At zero equilibrium mean flow, its linear perturbation is obtained from $m\int v_{\parallel }^2\delta f{d}^{3}v$ together with the perturbation of the velocity-space measure.

The equal normal stresses in the two directions perpendicular to the local magnetic field for a gyrotropic phase-space distribution function. The factor one half averages the two perpendicular velocity components. With first adiabatic invariant $\mu =v_{\perp }^2/(2B)$, the integrand is $m\mu Bf$.