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A simple model of hillslope smoothing in which sediment flux is proportional to negative slope. For constant transport coefficient $\kappa$, an ideal initial step broadens on scale $\sqrt{\kappa t}$. Real free-face collapse, nonlinear transport, changing climate and deposition can invalidate the linear diffusion equation approximation.

Solutal diffusivity is the diffusion coefficient governing transport of a dissolved constituent down its concentration gradient. For uniform $D$ and a stationary solvent, the diffusion equation is $C_t=D{\nabla }^{2}C$. It need not equal the thermal diffusivity.

For the nonlinear diffusion equation $u_t=(u{u}_x{)}_x+u/(2t)$ on $x>0,t>0$, the fixed boundary value $u(0,t)=1$ and zero slope at $x=0$ admit the displayed compact-support similarity solution, with front $x_F=2\sqrt{t}$. Substitution $u=f(x/\sqrt{t})$ reduces the equation to $(f{f}^{\prime }+\eta f/2{)}^{\prime }=0$. The boundary data make the integration constant zero. While $f>0$, division gives $f^{\prime }=-\eta /2$, hence $f=1-{\eta }^2/4$. At its first zero the diffusive flux $u{u}_x$ vanishes, so extension by zero creates no singular flux term. The solution is smooth inside its positive region and is a weak solution at the moving front.

For constant $\nu$, variation of this positive functional gives $({\partial }_t+\nu {\nabla }^2)({\partial }_t-\nu {\nabla }^2)h=0$, or $h_{tt}-{\nu }^2{\nabla }^{4}h=0$. The diffusion equation $h_t=\nu {\nabla }^{2}h$ gives zero-residual minima, but is only a subset of the stationary equations of the squared functional. Each nonzero spatial Fourier mode has both growing and decaying temporal solutions in the stationary equation. Boundary data decide which stationary paths are admissible. In two spatial dimensions, the bulk action is invariant under the anisotropic scale invariance $x\mapsto bx$, $t\mapsto b^{2}t$, $h\mapsto h$.

A thermal diffusion wave is a periodically forced solution of the heat equation, with spatially decreasing amplitude and varying wave phase. In a semi-infinite rod $x\ge 0$, assume constant thermal diffusivity, negligible side losses, a nearly uniform temperature across each cross-section, and the boundary conditions $\theta (0,t)={\Theta }_0\cos \omega t$ and $\theta (x,t)\to 0$ as $x\to \infty$. A steady periodic thermal state then satisfiesChoose the square root in the complex numbers with $a>0$ so that the amplitude decays into the rod. Substitution of the phasor into the heat equation proves the formula: $i\omega \hat{\theta }=\alpha {\hat{\theta }}^{\prime \prime }$. Taking the real part gives the displayed cosine. In this convention $a$ is the spatial attenuation slope and $b$ is the spatial lag slope, both in ${\mathrm{m}}^{-1}$; $\gamma$ combines them into one complex number. If a complex wavenumber $K$ is defined by $e^{i\omega t-iKx}$ instead, $K=b-ia$.

With $\Delta x=x_2-x_1>0$, let $A_j=|\hat{\theta }(x_j)|$ and define a positive downstream lag $\Delta \varphi$. Thenand either measurement determines thermal diffusivity:The logarithm uses an amplitude ratio, not a ratio of squared amplitudes. The lag is in radians and must be tracked continuously across cycles: a measured phase difference modulo $2\pi$ leaves an integer-cycle ambiguity. Use closer sensor spacings or vary frequency continuously to resolve that ambiguity. Equal attenuation and lag slopes provide a useful diagnostic of the no-side-loss model.

The apparent phase velocity is $\omega /b=\sqrt{2\alpha \omega }$, which depends on frequency. A thermal diffusion wave therefore does not give a material-independent travel speed for heat; it is a forced pattern in a dissipative diffusion equation. The ideal heat equation still has infinite propagation speed, and the observed lag is not the arrival time of a sharp front.

For a real scalar height field, integration by parts givesThe bulk expression is even under time reversal. The original density is not pointwise even, but changes by a total divergence; its stationary equation is time-reversal invariant. The first-order diffusion equation selected by zero residual is not invariant under ordinary time reversal at fixed positive $\nu$. Boundary conditions or the probabilistic interpretation can distinguish the two time directions even when the higher-order stationary equation does not.

With positive diffusion coefficient $a(x)$, this diffusion equation expresses conservation with flux $-a(x)u_x$. For a differentiable coefficient, its spatial operator is $a{u}_{xx}+a^{\prime }{u}_x$, rather than either $a{u}_x$ or $a{u}_{xx}$ alone. A symmetric half-grid diffusion consistency discretization uses the difference of the two adjacent interface fluxes, preserving that divergence structure. Suitable Dirichlet boundary conditions make the spatial operator dissipative in the L2 norm.

Random changes of particle velocity described by a diffusion equation in velocity space. For constant $D$, $⟨(\Delta v{)}^2⟩=2D\Delta t$. With zero boundary flux it conserves number. For a number-normalized one-dimensional distribution, its kinetic energy production is $d\mathcal{K}/dt=-m\int vD(v)f^{\prime }(v)dv$, which reduces to $mnD$ for constant $D$. Variable $D$ must remain inside the derivative.