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The collisionless equation in the title transports a gyrophase-independent phase-space distribution function at fixed first adiabatic invariant. Its leading guiding center position has $\dot{\mathbf{R}}={\mathbf{u}}_{\perp }+v_{\parallel }\mathbf{b}$. The parallel acceleration includes $(q/m)E_{\parallel }-\mu \mathbf{b}\cdot \nabla B$ and, when required, terms from motion of the local frame. Both $v_{\parallel }$ and the parallel derivative refer to the exact perturbed magnetic field.

The transverse guiding center drift produced by a magnetic field strength gradient, with first adiabatic invariant $\mu =v_{\perp }^2/(2B)$ and signed cyclotron frequency ${\Omega }_s$. Together with magnetic curvature drift, it produces velocity-dependent transverse transport.

Use Gaussian units and the kinetic magnetohydrodynamics equationstogether with species continuity equations, quasineutrality and the drift-kinetic equation. Here $p_{\perp },p_{\parallel }$ sum the species pressures. Interpret the thermal speed denominator in the source as the species mass $m_{\alpha }$, so $v_{\mathrm{th}\alpha }^2=2T_{0\alpha }/m_{\alpha }$ and the first adiabatic invariant is $\mu =v_{\perp }^2/(2B)$.

The specified Maxwell-Boltzmann distribution has zero mean motion and isotropic pressure, $p_{\perp 0}=p_{\parallel 0}=p_0$. All equilibrium scalars depend only on $x$, while ${\mathbf{b}}_0\cdot \nabla x=0$ and ${\mathbf{b}}_0\cdot \nabla B_0=0$. Therefore the equilibrium streaming and mirror terms of the drift-kinetic equation vanish. The remaining terms vanish because ${\mathbf{u}}_0=0$ and $E_{\parallel 0}=0$; the parallel electric constraint is also satisfied because the equilibrium parallel pressure gradient is zero. Species continuity equations and the induction equation are stationary, and quasineutrality holds by assumption.

It remains to check the perpendicular force balance. The radial magnetic pressure and magnetic tension giveThis is preciselyThus the Z-pinch is a steady solution of the stated kinetic magnetohydrodynamics system. Its equilibrium electric current is supplied by the small species drifts implicit in this approximation; zero leading common flow does not require zero electric current.

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 $m\mu Bf$.

For a gyrophase-independent phase-space distribution function, $d^{3}v=2\pi v_{\perp }d{v}_{\perp }d{v}_{\parallel }$. With first adiabatic invariant $\mu =v_{\perp }^2/(2B)$, this gives the measure in the title. When the magnetic field changes, both the measure and any explicit factor $\mu B$ in a moment must be perturbed.