A circuit is presented which may be used to solve the differential equations describing the populations of a multi-level system as it approaches thermal equilibrium. The system may consist of any number of energy levels connected by simple transitions (representing the thermal contact with a reservoir) in which one member of the total population is transferred from one level to another. The circuit is designed so that the populations and population differences in the energy level system are proportional to steady state and transient voltage differences which are easily measured. The analog has been used to analyze the spin-lattice relaxation mechanisms in a 36 level electron and nuclear spin system in which electron spin resonance measures the population differences of adjacent levels. © 1965 The American Institute of Physics.