The Maxwell-Boltzmann ("thermal") distribution constitutes the natural thermodynamic equilibrium state for a charged particle beam, and knowledge of its properties is therefore of fundamental importance. The Boltzmann relation for the particle density has a nonanalytic form when the space-charge force is included. We use numerical integration to determine the transverse and longitudinal density profiles for a relativistic beam in a linear focusing system at different temperatures T± and 7V The calculated profiles are related to space-charge tune depression, rms width, perveance, and emittance of the beam.PACS numbers: 41.85.Ew, 52.25.Wz Many advanced charged particle beam experiments and applications, such as high-power microwave sources, free electron lasers, linear accelerators for heavy-ion inertial fusion, spallation neutron sources, radioactive waste transmutation, high-energy colliders, and other uses, require very high beam intensity so that the beam dynamics depend strongly on the particle density profile. It is therefore of fundamental interest to know the equilibrium state of the charged particle beam for a given situation. Thermodynamically, this equilibrium state is best described by a Maxwell-Boltzmann ("thermal") distribution with different transverse and longitudinal temperatures (T ± and T\\) since in practice many beams are not equipartitioned. Many effects lead to coupling between T± and T\\ [1]; we deal here with cases where the coupling is small. When space-charge forces are significant the equilibrium density profiles have a nonanalytic form and must be found numerically, which explains why the thermal distribution has received less attention in the literature on beam theory than it deserves. Lawson [2] has published numerical results for the radial density profiles of a continuous nonrelativistic thermal beam in a linear focusing channel. In our work reported here we extend Lawson's results by including the relativistic factor y 2 , correcting an error, and correlating the density profiles with space-charge tune depression, perveance, and emittance of the rms equivalent uniform (K-V) beam. In addition, we determine the line-charge density profiles for a bunched beam with linear longitudinal focusing forces for different longitudinal temperatures and relate the results to the tune depression and other parameters of the rms equivalent parabolic bunch.The equilibrium distribution /of a group of charged particles in a focusing channel can be found from the Vlasov equation. We assume that the potential can be written as the sum of a transverse potential >±(r) and a longitudinal potential >\\(z) in cylindrical coordinates, and that the beam and focusing system are uniform, or "smooth." Each of these potentials is the sum of a selfcomponent [0j_ 5 (r) and >\\ s (z)] and an external focusing component [(/>± e (r) and (j>\\ e (z)].The Maxwell-Boltzmann distribution has the form f^foQxpi -H/kBT), where H is the single-particle Hamiltonian, kg is Boltzmann's constant, and T is the temperature. I...