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An adiabatic equilibrium is obtained for an intense, axisymmetric charged-particle beam propagating through a periodic solenoidal focusing field. The thermal beam distribution function is constructed. The beam root-meansquare (rms) envelope equation is derived, and the selfconsistent nonuniform density profile is calculated. Other statistical properties such as flow velocity, temperature, total emittance and rms thermal emittance, equation of state, and Debye length are discussed. Good agreement is found between theory and a recent high-intensity beam experiment.