  In this work, we study the stochastic thermodynamics of micro-magnetic
systems. We first formulate the stochastic dynamics of micro-magnetic systems
by incorporating noises into Landau-Lifshitz (LL) equation, which describes the
irreversible and deterministic dynamics of magnetic moments. The resulting
stochastic Landau-Lifshitz (sLL) equation obeys detailed balance, which
guarantees that, with the external field fixed, the system converges to
thermodynamic equilibrium with vanishing entropy production and with
non-vanishing probability current. We then discuss various thermodynamic
variables both at the trajectory level and at the ensemble level, and further
establish both the first and the second laws of thermodynamics. Finally, we
establish fluctuation theorems, and verify them using numerical simulations.
