Progress in stochastic coupled Molecular Dynamics and Spin Dynamics*

Coupled Molecular Dynamics and Spin Dynamics display very attractive features that allow to get new insights of the intriguing interplay between structure and magnetism in Fe-based alloys.
Each atom is treated as a moving particle, eventually charged, supporting a classical spin.
Coordinate dependent on both magnetic exchange and magnetic anisotropic functions ensure minimal coupling between the spin and the lattice degrees of freedom to translate the well known magneto-elastic behavior of magnetic materials.
These functions need well crafted magnetic and mechanical Hamiltonian parametrized on ab-initio calculations to recover the size dependence of the energy barriers and characteristic time scales of the magnetization relaxation of condensed phases.
The equations of motion (eoms) come naturally from the Nambu dynamics that formulates covariant Poisson brackets for classical fields that encode the constrained structure of the topological space of spins.
To solve these coupled eoms in practical simulations, numerical integration algorithms based on decomposition methods have to be considered, that preserve the structure of the flow of canonical variables in such an extended phase space.
Consequences of such schemes will be discussed in details from the point of view of the known invariant quantities.
To mimic experimental conditions, various statistical ensembles have to be generated to monitor the physical properties by varying external temperature for instance, as a stochastic connection.
One consequence is that the positions, momenta and spins of all the atoms become, in return, varying quantities evolving through stochastic instead of deterministic differential equations of motion.
The challenge to compute these quantities in parallel and examples provided by the SPIN package in the open source LAMMPS code are discussed.