A cosmic-ray spectrum includes its propagation history

Let n(E,position,t) be particle number per volume per energy in the local medium frame. A scalar, spatially uniform diffusion coefficient D is an approximation to transport in magnetic turbulence; a coherent ordered field can require a diffusion tensor. The diffusion limit needs many scattering events and scales longer than a scattering mean free path and time. Its formal Green function has a tail at arbitrarily large distance immediately, so the approximation cannot describe a relativistic causal front or very early nearly ballistic transport.

For a point impulse injecting N particles in a fixed energy bin, with no boundaries or energy losses, the three-dimensional solution is a normalized Gaussian. Its mean squared radial displacement is 6Dt, whereas one Cartesian component has variance 2Dt. The root-mean-square distance is a population spread, not the path length traveled by a particle. In particular, dividing the spread by time does not produce the particle speed, which can remain very close to c throughout the random walk.

A finite confinement region loses particles through its boundaries. For one-dimensional diffusion starting at the center between absorbing surfaces at ±H, the mean first exit time is H²/(2D). This particular coefficient is a geometric first-passage result, not a universal Galactic residence-time formula. A steady population supplied at rate Q and lost on a characteristic residence time has total number approximately Q times that time. If diffusion rises as rigidity to a power δ, escape becomes faster at higher rigidity.

At fixed charge and ultrarelativistic energy, rigidity is proportional to energy. A source rate proportional to E⁻ᵖ then produces a simple loss-free reservoir spectrum proportional to E⁻⁽ᵖ⁺δ⁾. The conclusion concerns a specified steady escape model. A local density from a thin source disk and an extended halo has additional geometry-dependent factors; doubling the halo size does not universally quadruple a measured local flux. Winds, reacceleration, nearby discrete sources and radiative electron losses can change the slope or introduce breaks without changing the injection mechanism.

Spallation changes nuclear composition, while synchrotron and Compton cooling can reshape electron spectra. For an energy-loss rate b(E)=−dE/dt>0, a transport equation adds the term ∂[bn]/∂E; catastrophic removal requires a separate sink. Secondary-to-primary abundance ratios constrain traversed material rather than directly naming a unique source distance. At ultrahigh energies, proton photopion losses, nuclear photodisintegration and cosmic expansion also limit propagation. Arrival direction alone is especially ambiguous for charged nuclei because magnetic deflection depends on rigidity and the intervening field. Air-shower energy and composition are inferred through shower and detector-response models, not read directly from the source’s injected spectrum.

The heat lab isolates one diffusion mechanism. Reset and set both controls to mode one and diffusivity 0.5; Step twice to time one, predicting amplitude exp(−0.5)≈0.607. Reset and set mode two and diffusivity 0.5; Step twice, predicting exp(−2)≈0.135. Shorter spatial structure fades faster at the same diffusivity. This analytic periodic one-dimensional Fourier mode is not the three-dimensional point-source Green function, a random particle realization, an energy-loss calculation or a detector response model.