1 2 Ju l 2 00 5 Symplectic integrators for classical spin systems Robin Steinigeweg and Heinz - Jürgen Schmidt ∗

Symplectic integrators for classical spin systems

We suggest a numerical integration procedure for solving the equations of motion of certain classical spin systems which preserves the underlying symplectic structure of the phase space. Such symplectic integrators have been successfully utilized for other Hamiltonian systems, e. g. for molecular dynamics or non-linear wave equations. Our procedure rests on a decomposition of the spin Hamiltoni...

0 40 70 95 v 1 2 6 Ju l 2 00 4 Lectures on Mathematical Cosmology

On the Applicability of Constrained Symplectic Integrators in General Relativity

The purpose of this note is to point out that a naive application of symplectic integration schemes for Hamiltonian systems with constraints such as SHAKE or RATTLE which preserve holonomic constraints encounters difficulties when applied to the numerical treatment of the equations of general relativity. It is well known that the equations of General Relativity (GR) can be derived from a variat...

Projection Operator Approach to Transport in Complex Single-Particle Quantum Systems

We discuss the time-convolutionless (TCL) projection operator approach to transport in closed quantum systems. The projection onto local densities of quantities such as energy, magnetization, particle number, etc. yields the reduced dynamics of the respective quantities in terms of a systematic perturbation expansion. In particular, the lowest order contribution of this expansion is used as a s...

Geometric Integrators for Classical Spin

Practical, structure-preserving methods for integrating classical Heisenberg spin systems are discussed. Two new integrators are derived and compared, including (1) a symmetric energy and spin-length preserving integrator based on a Red-Black splitting of the spin sites combined with a staggered timestepping scheme and (2) a (Lie-Poisson) symplectic integrator based on Hamiltonian splitting. Th...