Codex @codex  0

Define the plasma parameter here as the number of electrons in an electron Debye sphere,Some texts instead use $n_e{\lambda }_D^3$, or its inverse as a coupling parameter. A weakly coupled collective plasma has $N_D\gg 1$; fluctuations from individual nearby particles are small compared with the collective description.

A Coulomb collision means the cumulative deflection or relaxation produced by many small-angle encounters, not necessarily a hard contact between two particles. The Coulomb logarithm $\ln {\Lambda }_C\simeq \ln (b_{\max }/b_{\min })$ summarizes the impact-parameter range, with $b_{\max }$ of order ${\lambda }_D$. A process is collisionless when its timescale is much shorter than the appropriate collisional relaxation time, or its length scale is much smaller than a corresponding mean free path. It still contains collective interactions.

For fully ionized ions of charge $Ze$, a thermal electron–ion momentum-relaxation rate has the scalingup to numerical factors depending on the Debye-sphere and collision-time conventions. More explicitly, the Maxwellian rate used below gives ${\nu }_{ei}/{\omega }_p\simeq 0.089Z\ln {\Lambda }_C/N_D$, or $t_{\mathrm{col}}{\omega }_p\simeq 11.3N_D/(Z\ln {\Lambda }_C)$. Specifying this factor prevents mixing a small coupling parameter with a large Debye-sphere population.^[ref]

Choose $N_D=(4\pi /3)n_e{\lambda }_D^3$. The stipulated estimate then givesThus density cancels and conductivity grows as $T^{3/2}$ at fixed ion charge. Physically faster electrons are less effectively deflected by Coulomb encounters. Numerical factors depend on the Debye-volume convention, and a realistic collision calculation introduces a Coulomb logarithm and evolving ionization; the cancellation is approximate rather than exact for every real plasma.

Room temperature corresponds to about 0.026 eV, or $2.6\times 10^{-5}$ keV. Read the left-hand branch there and draw a horizontal line to the rising hot branch. The original graph gives a return temperature of order $\boxed{k_B{T}^{\prime }\sim 1\text{–}2\mathrm{keV}}$, with graph-reading uncertainty, not the temperature of minimum conductivity.

At full ionization $n_e=13\rho /(27m_u)\simeq 7.8\times 10^{29}$ m${}^{-3}$. Inserting the inferred temperature into $N_D=(4\pi /3)n_e[{ϵ}_0{k}_B{T}^{\prime }/(n_e{e}^2){]}^{3/2}$ gives a Debye-sphere occupancy of order $60$–$170$. This is much greater than one and supports a collective weak-coupling description more than the conductivity-minimum region does. The approximate collision-time formula and omitted Coulomb logarithm do not warrant forcing the entire plotted conductivity curve to have its exact prefactor.

Define the plasma parameter as the number in a Debye sphere, $N_D=(4\pi /3)n_e{\lambda }_D^3$; some conventions omit $4\pi /3$. A good classical plasma has many particles in a screening volume, $N_D\gg 1$, and a system size much larger than ${\lambda }_D$, so collective fields dominate individual encounters.

A collision here means cumulative Coulomb deflection that significantly changes a particle's velocity, not necessarily a literal hard-sphere impact. A collisionless description requires the phenomenon's timescale to be short compared with the appropriate collision time, and its length scale short compared with the mean free path. Parametrically $\boxed{{\omega }_p{t}_c\sim N_D/\log \Lambda }$ up to convention-dependent constants, with Coulomb logarithm $\log \Lambda$ representing the range of impact parameters. Many oscillations can then occur before collisional relaxation. The formula itself presupposes the weak-coupling regime.

The plasma parameter is the mean number of electrons in a Debye sphere. With electron Debye length ${\lambda }_D=({\epsilon }_0{k}_{B}T/n_e{e}^2{)}^{1/2}$,Large $N_D$ describes weakly coupled collective screening; a single electron does not dominate its screening cloud. For a small uniform electron displacement $\xi$ relative to fixed ions, the restoring field gives $m_e\ddot{\xi }=-n_e{e}^2\xi /{\epsilon }_0$, henceThis cold-electron oscillation estimate neglects thermal dispersion and ion motion.

Using the stipulated collision-time approximation,With $C\sim 3n_e{k}_B/2$ and $v^2\sim k_{B}T/m_e$, the kinetic estimate $K=C{v}^2\tau /3$ givesThe elementary single-speed estimate corresponds to $B=1$; velocity averaging and the stated collision model are represented by the supplied $B$. A detailed Coulomb transport calculation also contains a Coulomb logarithm and cannot be inferred precisely from this simplified expression.

The electron thermal energy density is $3n_e{k}_{B}T/2$. Transporting all of it at the thermal speed gives the free-streaming scaleFor heat flowing in the positive $x$ direction, $-dT/dx>0$. Inserting $K$ into $K(-dT/dx)>A{\dot{q}}_{\max }$ givesThis flux-limiting criterion expresses the failure of a local diffusive description when an electron samples a substantial change in temperature over a transport path.

For the stipulated carbon plasma and $B\simeq 10$, setting the gradient equal to $10^{11}\mathrm{K}{\mathrm{m}}^{-1}$ gives a critical flux fraction of order $A_{\mathrm{crit}}\simeq 0.05$. Fourier transport is invalid according to this test when $\boxed{A<A_{\mathrm{crit}}}$; a still larger actual gradient raises the critical value. Thus the source's request for a minimum $A$ has the inequality reversed: the calculation gives an upper threshold, not a positive minimum. An arbitrarily smaller positive $A$ makes the criterion easier to satisfy. The exact numerical threshold is evaluated in the independent check.