A measure of how quickly cosmic rays spread through a medium due to scattering.The diffusion coefficient quantifies how quickly particles or energy spread through a medium due to random motion. In the context of cosmic rays, it measures how fast charged particles like protons and electrons scatter off magnetic field irregularities in space and thereby diffuse through the interstellar or intergalactic medium. A higher coefficient means faster spreading, while a lower one means the particles remain more localized.
The concept of diffusion was formalized by Adolf Fick in 1855, and its application to cosmic-ray propagation was developed in the mid-20th century by physicists like Ginzburg and Syrovatskii.
Diffusion is described by Fick's law, where the particle flux is proportional to the gradient of density, with the diffusion coefficient as the proportionality constant. In cosmic-ray transport, the coefficient often depends on particle energy and the properties of the magnetic turbulence. It is typically derived from the mean free path of scattering: the longer the mean free path, the larger the diffusion coefficient. The resulting random walk leads to a Gaussian spread of particles over time, with the variance growing linearly with time.
It is essential for modeling how cosmic rays propagate through the galaxy, predicting their arrival times and spectra at Earth, and interpreting data from observatories like Fermi and IceCube.
In the interstellar medium, the diffusion coefficient for cosmic rays is typically about 10²⁸ cm²/s — roughly a million times larger than that of heat in copper, reflecting the vast distances between scattering events.
| J | variable | Diffusive flux (amount per unit area per unit time) — diffusive flux mole per square metre per second |
| c | variable | Concentration of the diffusing species — concentration mole per cubic metre |
| x | variable | Spatial coordinate along the diffusion direction — position metre |
| D | constant | Material-specific proportionality constant relating flux to concentration gradient — diffusion coefficient = varies square metre per second |
| ∂/∂x | operator | Partial derivative with respect to position — partial derivative |
| D | variable | Diffusion coefficient — diffusion coefficient metre squared per second |
| E | variable | Particle energy — energy joule |
| b | variable | Energy scaling index |
| r_d | variable | Diffusion length scale — diffusion length metre |
| D | variable | Diffusion coefficient — diffusion coefficient metre squared per second |
| t | variable | Time elapsed — time second |
| P | variable | Probability density function — probability density |
| t | variable | Time — time second |
| D | variable | Diffusion coefficient — diffusion coefficient metre squared per second |
| ∇ | operator | Spatial gradient — nabla operator |
| ∇· | operator | Spatial divergence — divergence operator |
| ∂/∂t | operator | Partial derivative with respect to time — partial derivative |
| \phi_i | variable | vertical flux of species i — particle flux metre per second per square metre |
| n_i | variable | number density of species i — number density per cubic metre |
| v | variable | bulk vertical velocity — velocity metre per second |
| K_{zz} | variable | eddy diffusion coefficient — eddy diffusion coefficient square metre per second |
| n_{\text{tot}} | variable | total number density — number density per cubic metre |
| X_i | variable | mole fraction of species i — mole fraction |
| z | variable | altitude — altitude metre |
| D_i | variable | molecular diffusion coefficient of species i — diffusion coefficient square metre per second |
| H_i | variable | scale height of species i — scale height metre |
| \alpha_T | variable | thermal diffusion factor — thermal diffusion factor |
| T | variable | temperature — temperature kelvin |
| \frac{\partial}{\par | operator | derivative with respect to altitude — partial derivative |
| \frac{d}{dz} | operator | total derivative with respect to altitude — total derivative |
| D | variable | Diffusion coefficient — diffusion coefficient metre squared per second |
| E | variable | Particle energy — energy joule |
| B | variable | Magnetic field strength — magnetic field strength tesla |
| R_inj | variable | Injection scale (turbulence outer scale) — length metre |
