Redfield—Langevin Equation for Nuclear Spin Relaxation

We present a derivation of a Langevin‐type equation for the operator $G_{\mathrm{\alpha \; \alpha }}^{\prime }\mathrm{(t)=}\exp {\mathrm{(-iHt)|\alpha }}^{\prime }\mathrm{〉〈\alpha \; |}\exp \mathrm{(–iHt)}$ , where $\mathrm{|\alpha \; 〉}$ and ${\mathrm{|\alpha }}^{\prime }〉$ are eigenstates of the subsystem Hamiltonian and H is the full Hamiltonian for subsystem plus bath. For the case in which we consider a spin system weakly coupled to a thermal bath (lattice), the equation of motion is of the form of the Redfield equation with a fluctuating term. This equation may be used to derive the standard Redfield equation for the spin density matrix as well as Bloch equations with fluctuating terms for the components of the magnetization for individual systems in the nonequilibrium ensemble. The latter presents a first principles derivation of the magnetic analog of hydrodynamic fluctuation theory. A calculation of several of the correlation functions which may be constructed from the theory is presented.