Stochastic thermodynamics of micromagnetics

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 the Landau–Lifshitz (LL) equation, which describes the irreversible and deterministic dynamics of magnetic moments. The resulting stochastic LL 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 the Crooks fluctuation theorem, and verify it using numerical simulations.