Generalized Bloch-Wangsness-Redfield Kinetic Equations

TL;DR

This paper derives a unified set of kinetic equations for spin systems using Liouville space, simplifying previous approaches and broadening applicability to heterogeneous systems without perturbation theory.

Contribution

It introduces a compact, general derivation of Bloch-Wangsness-Redfield equations using Liouville space, avoiding perturbation theory for the main derivation.

Findings

Kinetic equations can be derived without perturbation theory for the spin-lattice interaction.

Existence of a unity operator ensures convergence to thermodynamic equilibrium.

Application to spin relaxation in heterogeneous I=1/2 systems demonstrates the theory's utility.

Abstract

We present a compact and general derivation of the generalized Bloch-Wangsness-Redfield kinetic equations for systems with the static spin Hamiltonian utilizing the concept of the Liouville space. We show that the assumptions of short correlation times and large heat capacity of the lattice are sufficient to derive the kinetic equations without the use of perturbation theory for the spin-lattice interaction operator. The perturbation theory is only applied for calculation of the kinetic coefficients, for which we obtain general and compact expressions. We argue that kinetic equations for the density matrix elements are not essential for derivation of the generalized Bloch-Wangsness-Redfield equations for the expectation values of any set of physical quantities, and the latter may be obtained directly under the weak assumptions of mutual orthogonality and completeness. We show that existence of a unity operator in the algebra of spin operators is a necessary condition for convergence of the spin subsystem to thermodynamic equilibrium with the temperature equal to that of the lattice. Finally, as an application of the general theory, we discuss some features of spin relaxation in heterogeneous systems for I=1/2.