The time evolution of the density matrix of a relativistic quantum field theory coupled to random noise is discussed. Using path-integral techniques, we solve analytically for the density matrix of a system of free fields coupled to noise. We show that in the limit of large time the density matrix evolves to a thermal state and we compute the temperature of the state in terms of the characteristics of the noise. While the noise is coupled the temperature increases steadily in time. Also, it is necessary to choose a specific spectrum for the noise in order that all of the modes of the free field theory come to the same temperature after some given time. We then present an alternative derivation of these results using time-path techniques. In this case we consider a system where noise couples for a finite time interval and find that the propagator interpolates between finite-temperature field-theory propagators with the temperatures of the system at initial and final times. We argue that this can be used as a model of heating in an interacting field theory.