Thermal transistor behavior of a harmonic chain

Thermal transistor behavior of a harmonic chain with three heat reservoirs is explicitly analyzed. Temperature profile and heat currents of the rather general system are formulated and then heat currents for the simplest system are exactly calculated. The matrix connecting the three temperatures of the reservoirs and those of the particles comprises a stochastic matrix. The ratios R (1) and R (2) between heat currents, characterizing thermal signals can be expressed in terms of two external variables and two material parameters. It is shown that the ratios R (1) and R (2) can have wide range of real values. The thermal system shows a thermal transistor behavior such as the amplification of heat current by appropriately controlling the two variables and two parameters. We explicitly demonstrate the characteristics and mechanisms of thermal transistor with the simplest model.