Mesoscopic non-equilibrium thermodynamics for quantum systems

An extension of the scheme of non-equilibrium thermodynamics developed previously is given for quantum mechanical systems. A master equation for the density matrix of the system follows from this scheme. Onsager relations are given and derived. Application to a spin system gives the Bloch equations. The application to a one-dimensional damped harmonic oscillator results in equations, which enable us to calculate the usual Green functions. For the last case we derive, as a new alternative, quantum mechanical Langevin equations. A new element, compared to the classical Langevin equations, is a random velocity. The correlation of the random velocity with the random force then results from the zero point motion of the oscillator. The application of mesoscopic non-equilibrium thermodynamics to these wellknown problems illustrates the usefullness of this method.