The Diffusion Equation

A parabolic partial differential equation describing how a conserved quantity spreads down its own gradient over time.

Statement

The diffusion equation is dn/dt = div(D grad n), where n is a density (particles, heat, or magnetic flux recast as a field), D is the diffusion coefficient, div is divergence and grad is gradient. For constant scalar D in one dimension it reduces to dn/dt = D d2n/dx2. It is the canonical parabolic PDE: local in time, smoothing in space, and irreversible.

The equation follows from a conservation law dn/dt + div(flux) = 0 combined with a Fick's-law closure flux = -D grad n. The minus sign encodes that transport carries material from high to low concentration. In plasmas D is rarely a simple constant; it depends on collisionality, magnetic geometry, and turbulence, and it is generally a tensor with very different values along and across the magnetic field.

Solutions and Scaling

A point source in free space spreads as a Gaussian whose width grows as sqrt(D t), so the characteristic diffusion time across a system of size L is tau ~ L^2 / D. This quadratic scaling is why confinement improves faster than linearly as a device grows, and why cross-field diffusion coefficients are the central figure of merit for magnetic confinement. Eigenmode solutions on a bounded domain decay as exp(-t/tau_k) with the shortest wavelengths dying first.

Anisotropy in Magnetized Plasma

In a strong magnetic field, parallel transport is fast (free streaming limited only by collisions) while perpendicular transport is suppressed because particles are tied to field lines within a gyroradius. Classical theory gives cross-field D of order the gyroradius squared times the collision frequency. Observed transport is usually much larger, driven by turbulence rather than collisions, which motivates the anomalous-transport and gyrokinetic programs.

For the Hyperion breeder concept, the relevant quantity is the effective cross-field thermal and particle diffusivity that sets energy confinement; because the diffusion time scales as L^2/D, the spherical-tokamak geometry and its transport assumptions are load-bearing inputs to any confinement estimate. All such estimates here are simulation-stage, not measured hardware.