On the basis of a small‐signal, one‐dimensional analysis, a set of basic macroscopic differential equations, governing the fluctuations in quantities such as the electron‐beam temperature, the mean velocity, and the current density, has been derived by taking moments of the Liouville equation with respect to the velocity variable. This set of differential equations expresses the conservations of charge, momentum, and energy, and is valid for an arbitrary amount of velocity spreading and includes the effect of heat conduction.A system of differential equations, governing the correlation among the fluctuations in the mean velocity, current density, and beam temperature, is also derived. The relationship among the various noise parameters along the electron beam is obtained in the form of a system of differential equations whose solution gives detailed information on the variation of the noisiness parameter along the beam. The solution of the system of differential equations thus derived is also discussed.