Master Equation and Two Heat Reservoirs

We analyze a simple spin-flip process under the presence of two heat reservoirs. While one flip process is triggered by a bath at temperature $T$, the inverse process is activated by a bath at a different temperature $T ^{\prime}$. The situation can be described by using a master equation approach in a second quantized Hamiltonian formulation. The stationary solution leads to a generalized Fermi-Dirac distribution with an effective temperature $T_e$. Likewise the relaxation time is given in terms of $T_e$. Introducing a spin-representation we perform a Landau expansion for the averaged spin $<\sigma>$ as order parameter and consequently, a free energy functional can be derived. Owing to the two reservoirs the model is invariant with respect to a simultaneous change $\sigma \leftrightarrow - \sigma$ and $T \leftrightarrow T ^{\prime}$. This new symmetry generates a third order term in the free energy which gives rise a dynamically induced first order transition.