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For positive diffusivity $D(x)$, a diffusion equation with potential-driven drift has diffusive flux $J=-D(c_x+\beta V^{\prime }c)$. On a finite interval with zero flux at both endpoints, the total amount is conserved. The substitution $c=e^{-\lambda t}{e}^{-\beta V}y$ converts the decay eigenvalue problem to the Sturm-Liouville problemThe Rayleigh quotient is $\lambda =\int p|y^{\prime }{|}^2/\int w|y{|}^2\ge 0$. Its zero eigenfunction is constant, giving the stationary solution $c_{\mathrm{eq}}=C{e}^{-\beta V}$. This is a spatial Boltzmann distribution; $D$ changes relaxation rates but cancels out of the zero-flux equilibrium profile. For a sufficiently regular positive $D$ and bounded $V$, the constant eigenfunction is the unique zero mode up to scale.

For positive thermal diffusivity $\nu$, thermal conductivity $k$ and length $L$, subtract the stationary solution $w(x)=Hx/k$. The remainder satisfies the heat equation with a homogeneous Dirichlet boundary condition at zero and a homogeneous Neumann boundary condition at $L$. The mixed Dirichlet-Neumann modes on an interval are $\sin (k_{n}x)$, where $k_n=(n+1/2)\pi /L$. Thus for an initial temperature profile $u_0$,For zero initial temperature, $b_n=-2H(-1{)}^n/(kL{k}_n^2)$. The series gives the initial profile in the usual Fourier series sense and a smooth solution for $t>0$. If $H\ne 0$, the zero initial profile does not satisfy the prescribed endpoint derivative at $t=0$: the temperature can be continuous there without its spatial derivative extending continuously to that corner. The sign $k{u}_x(L,t)=H>0$ describes incoming heat flux, since the outward conductive heat flux is $-k{u}_x$.

For $y\ne 0$, the separable differential equation gives $dy/y^2=dx$. Integrating gives $-1/y=x+C_0$, orEach solution is considered on an interval not containing its pole. The additional stationary solution $y=0$ is excluded by the stated condition.

Choose the stationary solution $w^{\prime \prime }=0$ satisfying $w(0)=0$ and $k{w}^{\prime }(L)=H$. Hence $\boxed{w(x)=Hx/k}$. Subtracting it gives the homogeneous problemThis boundary lifting leaves a Dirichlet boundary condition at one end and a Neumann boundary condition at the other, as in the fixed-temperature and fixed-flux heat equation on an interval.

For an admissible nonzero real trial function, integration by parts and the zero-flux boundary conditions reduce the Rayleigh quotient toA nonzero constant trial function attains zero. Positivity of $p$ also shows that equality requires $y^{\prime }=0$. HenceIf the conserved total amount is $M={\int }_{-L}^{L}cdx$, the normalized stationary solution isIts diffusive flux vanishes everywhere. The Boltzmann distribution fixes the spatial profile, while the diffusivity affects the relaxation toward it. Boundedness of $V$ makes the normalization finite on this finite interval.

Axisymmetry supplies conserved $p_{\theta }$ in addition to conserved energy. Consequently $\boxed{f=F(H,p_{\theta })}$ is a more general stationary solution, since both arguments are constant on every trajectory. Additional independent integrals may permit still more general distributions. This is a sufficient construction, not a claim that every stationary distribution must have exactly these two arguments.

When a dispersion relation factors as $\omega P(\omega ,k)=0$, dividing by $\omega$ discards a possible time-independent Fourier mode. This can remove a necessary stationary solution from an initial value problem. For instance, a constant-coefficient temporal equation $T^{(3)}+ia{T}^{\prime \prime }+b{T}^{\prime }=0$ has a constant solution as well as the two exponential solutions from its quadratic factor. Three initial conditions generally require all three modes. Cancellation is valid only when a nonzero frequency has already been required.