Examination of the "Leakage-Lifetime" Approximation in Cosmic-Ray Diffusion

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Abstract
The diffusion of cosmic-ray particles in a finite volume of space with a simultaneous diffusion and/or transport in energy is considered. The solution of the appropriate differential equation may be expressed as an expansion in eigenfunctions of the differential operator. If one approximates the solution by keeping only the lowest or fundamental eigenfunction, one obtains the common "leakage-lifetime" approximation. In some situations this approximation can be justified, but in others (e.g., synchrotron or inverse-Compton losses) it cannot. The reason for the failure in this case can be seen from the point of view of the expansion. The solution of the general case of Fermi acceleration, synchrotron losses, and energy fluctuation acting together is also obtained by this method.