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The Debye length is the distance over which a small, slowly varying electrostatic potential is screened by mobile thermal charges. For classical electrons in a weak potential $\varphi$, their Boltzmann distribution givesTake fixed singly charged ions of density $n_0$. Then $\rho =e(n_0-n_e)\simeq -n_0{e}^2\varphi /(k_{\mathrm{B}}T)$, and Poisson's equation outside a test charge becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$, withThe screened point-charge potential is proportional to $e^{-r/{\lambda }_D}/r$. The derivation assumes nondegenerate statistics and $|e\varphi |\ll k_{\mathrm{B}}T$. Mobile ions would add their own contribution to ${\lambda }_D^{-2}$; the expression here is the electron screening length.

The electron Debye length is the scale over which a weak electrostatic disturbance is screened. Take the ions as an immobile neutralizing background and the electrons as an isothermal Boltzmann distribution. For a small potential $\varphi$,Poisson's equation then becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$ away from the perturbing source, withA test charge has a screened potential proportional to $e^{-r/{\lambda }_D}/r$. For fully static screening with mobile ions, inverse squared Debye lengths from all species add; the displayed expression is the electron scale used below.

The electron plasma frequency is the natural collective oscillation frequency when electrons are displaced from a nearly fixed ion background. Displace a uniform electron fluid by $\xi$ along a slab normal. The charge separation produces the restoring electric force $-n_e{e}^2\xi /{ϵ}_0$ on an electron. Hence $m_e\ddot{\xi }=-n_e{e}^2\xi /{ϵ}_0$ andIt sets a response time $1/{\omega }_p$ and the cold unmagnetized electromagnetic cutoff frequency. These collective effects arise from long-range electric fields, even when individual binary collisions are infrequent.

Use collisional inverse bremsstrahlung in a weakly coupled, unmagnetized plasma. The specified density is $n_e=10^{26}{\mathrm{m}}^{-3}$, and $k_{B}T=1000\mathrm{eV}$. The electron Debye length is $2.35\times 10^{-8}\mathrm{m}$, the plasma frequency is $5.64\times 10^{14}{\mathrm{s}}^{-1}$, and the laser angular frequency is $\omega =2\pi c/\lambda =1.79\times 10^{15}{\mathrm{s}}^{-1}$. Thus $X={\omega }_p^2/{\omega }^2\simeq 0.099<1$: the plasma is underdense and a propagating wave exists.

Estimate $b_{\min }$ as the larger of the classical Coulomb-deflection length $Z{e}^2/(4\pi {ϵ}_0{k}_{B}T)$ and the electron quantum length $\hslash /\sqrt{m_e{k}_{B}T}$. Both are about $9\times 10^{-12}\mathrm{m}$ here, giving $\ln {\Lambda }_C\simeq 8$. With the usual thermal collision convention,The $Z$ factor follows from $n_i{Z}^2=n_{e}Z$.^[ref]

The weakly collisional Drude model gives ${ϵ}_r=1-{\omega }_p^2/[\omega (\omega +i{\nu }_{ei})]$ for fields proportional to $e^{-i\omega t}$. Expanding at ${\nu }_{ei}\ll \omega$ yields a real refractive index $n_r=\sqrt{1-X}\simeq 0.949$ and intensity attenuation coefficientConsequently the intensity e-folding absorption length isi.e. of order a centimetre. Taking a Coulomb logarithm between about 5 and 10 gives roughly $5$–$10\mathrm{mm}$. The electric-field amplitude decay length is twice this intensity length. This estimate assumes a uniform slab thick enough to define an absorption length and neglects gradients, nonlinear quiver heating and other absorption mechanisms.

A weak positive potential perturbs Boltzmann electrons as $n_e\simeq n_0[1+e\varphi /(k_{B}T)]$ against nearly fixed ions. Poisson's equation outside the test charge becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$, withThe resulting potential is proportional to $e^{-r/{\lambda }_D}/r$. The Debye length is the screening distance over which charge imbalance is suppressed, within this classical weak-perturbation model.

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.

Write $\epsilon ={\epsilon }_0{\epsilon }_r$ and take $n_0$ to be the reservoir density of each monovalent ion species. Equal electrochemical potentials in equilibrium give the Boltzmann ion densities in a symmetric electrolyte,Combining this with the electrostatic Poisson equation gives the planar Poisson-Boltzmann equation,The Debye length is ${\kappa }^{-1}$. If the symbol $n_0$ instead denotes the combined density of both species, the factor two is absent; the per-species convention above makes the normalization explicit. At $|e\psi |\ll k_{B}T$, the linearized planar screening solution obeys ${\psi }^{\prime \prime }={\kappa }^2\psi$. With $\psi (0)=\zeta <0$ and $\psi (\infty )=0$,

The positive ions are counterions and the negative ions are co-ions. At weak potential,They approach $n_0$ from above and below respectively, with equal first-order excess and deficit. For stronger negative potential, interpret the source's strong-potential comparison as a statement about $|e\psi |$, since $\psi$ itself is negative. Counterions become strongly concentrated near the wall, while co-ions are exponentially depleted and remain nonnegative. Linearization misses this asymmetry and would eventually predict unphysical negative co-ion density. Point-ion mean-field theory itself eventually needs finite-size and correlation corrections; it provides no steric cap at arbitrarily large counterion concentration.

For steady tangential electro-osmosis, with no imposed pressure gradient and negligible inertia, the Stokes equation is $\eta v^{\prime \prime }+{\rho }_{e}E=0$. Since ${\rho }_e=-\epsilon {\psi }^{\prime \prime }$,The conditions $v(0)=0$ and $v^{\prime }(\infty )=0$ giveThe printed constant speed is the bulk velocity outside the double layer, not the velocity at the no-slip wall. With negative $\zeta$ the positive counterion excess drives fluid along the applied field. The far-field value is the Helmholtz-Smoluchowski electro-osmotic velocity.^[ref]

For the finite channel, first model the added wall as an ideally uncharged insulating boundary with no normal field on its outer side: ${\psi }^{\prime }(d)=0$, while $\psi (0)=\zeta$. The one-charged-wall electro-osmotic channel then hasIntegrating the same force balance and enforcing $v(0)=v(d)=0$ givesFor $U>0$, the maximum occurs atIn that limit $v/U\simeq 1-x/d-e^{-\kappa x}$: the velocity rises rapidly from zero at the charged wall, reaches a maximum a few screening lengths away, then falls approximately linearly to zero at the neutral wall. One charged wall does not give a symmetric plug across a channel with no-slip at the other wall.

An alternative electrode condition $\psi (d)=0$ means a wall held at reservoir potential; it is not the same as zero surface charge. Under that convention,with $\cosh [\kappa (d-x_{\max })]=\sinh (\kappa d)/(\kappa d)$. Both boundary conventions have the same leading thin-double-layer profile above. An externally imposed pressure gradient would add a pressure-driven profile; none is specified here.

For dilute nonadsorbing uncharged polymers in equilibrium or sufficiently weak flow, the entropic polymer depletion near nonadsorbing walls suppresses chain configurations near each wall. A depletion layer of order the coil radius of gyration is expected at both walls, with a bulk-like central region when $d$ is much larger than the coil size. Strong confinement rounds this into a central enrichment. Neutrality removes a direct electrophoretic force, not excluded-volume entropy. Adsorption, concentration and strong-flow migration were not specified, so this is a conditional equilibrium sketch rather than a universal flowing-polymer profile.

On spatial scales much larger than the Debye length, a plasma may have nearly cancelling positive and negative electric charge densities while still carrying an electric current and having a nonzero electric field. Quasineutrality is an approximation on appropriate slow macroscopic scales, not an assertion that each fluid element has exactly zero charge or that charge separation can be ignored at every boundary.