Solution
= Solution
\b[Molecular transport across gradients.] A <gas> can transport <momentum>, thermal <energy> and species abundance even when its average bulk <velocity> is zero. <Molecules> arriving at a point carry properties acquired near their preceding <collisions>. A small gradient makes the populations arriving from opposite sides slightly different, producing transport from larger to smaller values of the carried property. A continuum transport law requires a <mean free path> $\lambda$ small compared with the gradient <length>, and relaxation times short compared with the observation <time>.
For a dilute <hard-sphere gas>,
$$
\overline v=\sqrt{\frac{8k_BT}{\pi m}},\qquad
\lambda=\frac1{\sqrt2\,n\pi d^2},
$$
where $d$ is molecular <collision> diameter. Both <speed> and <collision> distance matter: high <speed> alone does not determine a transport coefficient.
A simple kinetic estimate makes the common structure explicit. For a quantity $a(y)$ carried per <particle>, a <molecule> moving at angle $\theta$ to $+y$ brings approximately $a(y-\lambda\cos\theta)$. Expanding to first order and averaging an isotropic <velocity> distribution gives
$$
J_a\sim n\overline v\left\langle\cos\theta\,[a(y)-\lambda\cos\theta\,a'(y)]\right\rangle
=-\frac13n\overline v\lambda\frac{da}{dy},
$$
since $\langle\cos\theta\rangle=0$ and $\langle\cos^2\theta\rangle=1/3$. This mean-free-path argument estimates the coefficients; a detailed <collision> calculation changes their numerical prefactors and distinguishes different relaxation rates.
\b[<Momentum> transport and <viscosity>.] For a <shear flow> with <mean> <velocity> $U_x(y)$, the transported directed <momentum> per <molecule> is $mU_x$. The net flux of $x$ <momentum> in the $+y$ direction is
$$
\Pi_{yx}=-\eta\frac{dU_x}{dy},\qquad
\boxed{\eta\sim\frac13nm\overline v\lambda.}
$$
Here $\eta$ is <dynamic viscosity> in $\mathrm{Pa\,s}$. The associated shear <stress> on a material surface has the corresponding traction sign convention; the flux carries <momentum> from the faster-moving layer towards the slower-moving one. The <kinematic viscosity> is $\nu=\eta/(nm)$, with units $\mathrm{m^2\,s^{-1}}$. <Viscosity> smooths <velocity> gradients, and the lost ordered mechanical <energy> becomes <heat>.
\b[<Heat> transport.] Let $c_v=d\varepsilon/dT$ be the thermal <heat capacity> \i[per <molecule>], in $\mathrm{J\,K^{-1}}$. Substituting $a=\varepsilon(T)$ gives <Fourier's law>,
$$
\mathbf J_Q=-\kappa\nabla T,\qquad
\boxed{\kappa\sim\frac13n c_v\overline v\lambda.}
$$
<Thermal conductivity> has units $\mathrm{W\,m^{-1}K^{-1}}$. For a <monatomic gas>, translational equipartition gives $c_v=3k_B/2$. Rotational or vibrational excitations in other <gases> can increase the <energy> carried per <molecule>, while <collisions> govern how those modes exchange <energy>. A <temperature gradient> can therefore be transported differently from directed <momentum> even when the dimensional estimates look similar.
\b[Species transport.] For a dilute tracer or a mixture with prescribed total density and negligible bulk flow, <molecules> diffuse down concentration gradients. The first-gradient number flux is <Fick's first law>,
$$
\mathbf J_s=-D\nabla n_s,\qquad
\boxed{D\sim\frac13\overline v\lambda.}
$$
The <diffusivity> $D$ has units $\mathrm{m^2\,s^{-1}}$; multiplying number flux by the species <mass> gives <mass> flux. More generally one uses <diffusion> flux relative to the mixture's bulk <velocity> and specifies <mass> or mole fractions. The approximation is not a claim that an arbitrary pure-gas <pressure> gradient can remain at rest while only a Fick flux acts. In a mixture, <collision> cross-sections and relative molecular <speeds>, including the <reduced mass>, determine the species diffusivities.
\b[Dependence on density and <temperature>.] The <kinetic estimates of gas transport coefficients> predict, for fixed <collision> diameter and molecular <heat capacity>,
$$
\eta\propto\frac{\sqrt{m k_BT}}{d^2},\qquad
\kappa\propto\frac{c_v}{d^2}\sqrt{\frac{k_BT}{m}},\qquad
D\propto\frac1{nd^2}\sqrt{\frac{k_BT}{m}}.
$$
Thus <dynamic viscosity> and <thermal conductivity> are approximately density independent in the dilute continuum regime. Reducing density reduces the number of carriers but increases their <mean free path> in the same proportion. At fixed density all three estimates grow as $T^{1/2}$; at fixed <pressure>, $n=p/(k_BT)$ and
$$
\boxed{D\propto\frac{T^{3/2}}p,\qquad
\eta\propto T^{1/2},\qquad \kappa\propto T^{1/2}.}
$$
<Kinematic viscosity> and <thermal diffusivity> also grow roughly as $T^{3/2}/p$ at fixed <pressure> in this model. Real interaction potentials make effective <collision> cross-sections <temperature> dependent, and internal excitations change $c_v$. Dense-gas correlations and boundaries invalidate the simple cancellation. In particular, when the <mean free path> approaches the apparatus scale, transport becomes free molecular; it cannot be described by extending the same local, pressure-independent Fourier and <viscosity> coefficients indefinitely.
\b[<Heat> <diffusion equation>.] For a stationary <gas> with constant <volumetric heat capacity> $c=n c_v$, constant <thermal conductivity> and negligible compression <work>, the <energy> balance in any fixed <volume (thermodynamics)> is
$$
\frac{d}{dt}\int_V cT\,dV=-\oint_{\partial V}\mathbf J_Q\cdot\widehat{\mathbf n}\,dA+\int_VH\,dV.
$$
Here $H$ is a possible <heat> generation rate per <volume (thermodynamics)>. The <divergence theorem> and <Fourier's law> yield
$$
\boxed{c\frac{\partial T}{\partial t}=\kappa\nabla^2T+H,
\qquad \frac{\partial T}{\partial t}=\alpha\nabla^2T+\frac Hc,
\qquad \alpha=\frac\kappa c.}
$$
<Thermal diffusivity> $\alpha$ has units $\mathrm{m^2\,s^{-1}}$ and is of order $\overline v\lambda/3$. In a moving or compressing <gas>, the <material derivative>, pressure-volume <work> and viscous heating must also enter the internal-energy equation. The displayed pure <diffusion equation> describes the approximation in which those effects are absent or negligible.
\b[Representative solutions.] With no source, a steady one-dimensional slab between <temperatures> $T_a,T_b$ at $x=0,L$ has $T=T_a+(T_b-T_a)x/L$ and uniform <heat flux> $-\kappa(T_b-T_a)/L$. Thus a constant spatial gradient transports <heat> without changing local <temperature> in <time> when the boundaries continually supply and remove it.
For an infinite medium and an initial sinusoidal <temperature> perturbation,
$$
\boxed{T(x,t)=T_b+A\cos(kx)e^{-\alpha k^2t}.}
$$
Short spatial wavelengths decay faster, and the characteristic smoothing <length> after <time> $t$ is $\sqrt{\alpha t}$. A finite slab with both ends held at $T_b$ has a <sine> expansion with factors $e^{-\alpha(n\pi/L)^2t}$; its slowest relaxation <time> is $L^2/(\pi^2\alpha)$.
An instantaneous <heat> input $Q_A$ per unit transverse <area> at $x=0$ in an otherwise uniform infinite medium gives the <Gaussian> solution
$$
\boxed{T(x,t)-T_b=\frac{Q_A}{c\sqrt{4\pi\alpha t}}\exp\left(-\frac{x^2}{4\alpha t}\right),\quad t>0.}
$$
Its <integral> $\int c(T-T_b)dx=Q_A$ conserves the deposited <heat>. The peak falls as $t^{-1/2}$ while its width grows as $\sqrt t$; <diffusion> redistributes <energy> instead of transporting a sharp front at a single <velocity>.
Finally, a semi-infinite body initially at $T_i$ and held at surface <temperature> $T_s$ from $t=0$ has
$$
\boxed{T(x,t)=T_i+(T_s-T_i)\operatorname{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right),\quad x\geq0.}
$$
It has $T(0,t)=T_s$ and approaches $T_i$ far from the boundary at finite <time>. These solutions demonstrate steady conduction, mode decay, spreading of localized <heat> and boundary-driven <diffusion>. Their idealized behaviour at extremely short times or arbitrarily long distances must be interpreted within the continuum regime rather than as an assertion of instantaneous microscopic communication.