We describe how AGN-produced cosmic rays form large X-ray cavities and radio lobes in the hot diffuse gas in galaxy groups and clusters. Cosmic rays are assumed to be produced in a small shocked region near the cavity center, such as at the working surface of a radio jet. The coupled equations for gasdynamics and cosmic ray diffusion are solved with various assumptions about the diffusion coefficient. To form large, long-lived cavities similar to those observed, the diffusion coefficient must not exceed κ ∼ 10 28 cm 2 s −1 in the hot gas, very similar to values required in models of cosmic ray diffusion in the Milky Way. When κ < ∼ 10 28 cm 2 s −1, cosmic rays are confined within the cavities for times comparable to the cavity buoyancy time, as implied by observations of X-ray cavities and their radio synchrotron emission. Collisions of proton cosmic rays with thermal plasma nuclei followed by π 0 decays can result in enhanced gamma ray emission from the cavity walls
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