Heat flux structure for Ornstein–Uhlenbeck particles of a one-dimensional harmonic chain

A one-dimensional harmonic chain of $N$ particles is considered, located between two thermal reservoirs (Ornstein–Uhlenbeck particles). An exact solution is constructed for the system of equations describing the dynamics of the system. On the basis of this solution, an analytical expression is obtained for the discrete expression of the heat flux of the model under study, when the time $t \to \infty$, which corresponds to the consideration of stationary transport conditions. It is shown that the heat flux includes two physically different components. The first of them is proportional to the temperature difference between the reservoirs and characterizes the heat transfer along the chain. The second determines the initial value of the flow when the temperatures of the tanks are equal.