Quantifying non-Markovianity in magnetization dynamics via entropy production rates

TL;DR

The paper addresses non-Markovianity in ultrafast magnetization dynamics by comparing LLG, inertial LLG (iLLG), and open-system LLG (os-LLG) through entropy-production rates tied to the relative entropy $D[p(t)\|\pi_\beta]$ with $\dot{\Sigma}(t) = -k_{\mathrm{B}}\partial_t D[p(t)\|\pi_\beta]$. Analytically, LLG is contractive with $\\dot{\\Sigma}(t)\\ge 0$, while iLLG and especially os-LLG can exhibit negative $\\dot{\\Sigma}(t)$ due to memory effects, signaling non-Markovianity. Numerical simulations for cobalt-like parameters show os-LLG produces the largest non-Markovianity across initial conditions and field orientations, with $\\mathcal{N}_{A}\\sim 30$ compared to $\\sim 2$ for iLLG, highlighting the critical role of memory kernels in ultrafast magnetization modeling. These findings advance understanding of memory effects in magnetic dynamics and suggest experimental avenues to detect entropy production changes, informing more accurate descriptions of ultrafast spin dynamics.

Abstract

Magnetization dynamics is commonly described by the stochastic Landau-Lifshitz-Gilbert (LLG) equation. On picosecond timescales, inertial and open-system extensions of the LLG equation are necessary to interpret recent experiments. We show analytically and numerically that the standard LLG equation exhibits strictly positive entropy production rates, while inertial and open-system LLG dynamics display temporarily negative entropy production rates indicating non-Markovianity. Here we quantify the degree of non-Markovianity using established measures. Our numerical calculations show that the open-system LLG equation consistently exhibits the highest magnitude of non-Markovianity for different initial conditions and magnetic field orientations.

Figures (8)

• Figure 1: Probability distributions of the magnetization vector in spherical coordinates for LLG, iLLG, and os-LLG: (a) For all three equations, the magnetization is initialized in the parallel configuration. (b) The steady state distribution is obtained by time-averaging the distribution once the system has equilibrated, here exemplarily shown for the LLG equation. (c) Analytically calculated Gibbs state $\pi_\beta(\mathbf{m})$, which matches the numerically obtained steady-state of each of the magnetization equations. (d), (e), (f) Shows the magnetization distributions after $t = 100$ ps for the LLG, iLLG, and os-LLG equations, respectively, evidencing differences in the temporal evolution. All six panels show the distribution of the magnetization in spherical coordinates $(\varphi,\vartheta)$ for a fixed magnetization magnitude of $\left|\mathbf{m} \right| = 1.0$.
• Figure 2: Relative entropy of the LLG, iLLG, and os-LLG equations initialized in the parallel configuration as a function of time. The damping parameter is always $\alpha=0.15$ and the inertial parameter is $\tau = 120$ fs for the simulations of iLLG and os-LLG (with Lorentzian parameters fixed as $\nu_0 = 4.2$ THz, $\Gamma = 0.2$ THz and ${\cal A} = 242\,\mathrm{THz}^3$, see Ref. Hartmann_2025). For all $N_{\mathrm{traj}} = 20000$ trajectories the temperature is $T_{\mathrm{sim}} = 0.1$.
• Figure 3: Entropy production rate of the LLG, iLLG, and os-LLG equation. (a) Entropy production rates of the LLG equation for parallel and canted configurations stay positive for all times, while (b) for the iLLG equation, the canted configuration leads to temporarily negative entropy production rates. (c) The entropy production rates of the os-LLG equation oscillate quickly between positive and negative for both, parallel and canted configurations. The grey region highlights violations of contractivity. Traces shown are after taking a moving average over five data points that suppresses the fluctuations of the limited statistics (50 000 trajectories). The remaining parameters correspond to those in Fig. \ref{['fig:EP_example']}.
• Figure 4: Magnitude of non-Markovianity. For the three different magnetization dynamics equations and the two different configurations (triangles stand for parallel and squares for canted configuration), we calculate the non-Markovianity measures $\mathcal{N}_A$ Eq. \ref{['eq:nonMarkov_measure_area']} (filled symbols) and $\overline{ \mathcal{N}}_A$ Eq. \ref{['eq:nonMarkov_measure_area_norm']} (empty symbols). For both measures, they are zero for both configurations in the LLG case. In the iLLG case, they are zero for the parallel configuration but non-zero for the canted configuration. In the os-LLG case, both non-Markovianity measures are non-zero for both configurations and significantly larger than in the iLLG case. The difference between the configurations is larger in the iLLG case, but smaller in the os-LLG case. The measures $\mathcal{N}_{\mathrm{A}}$ and $\overline{\mathcal{N}}_{\mathrm{A}}$ are evaluated over the first $12$ ps of the dynamics for the trajectories shown in Fig. \ref{['fig:EPR_comparison']}.
• Figure 5: Relative entropy for different levels of coarse graining. Left panel: The relative entropy over time until the system has arrived at equilibrium (i.e. $D(t) = 0$ and $\dot{D}(t) = 0$). The system evolves via the standard LLG equation, but the resolution of the coarse-graining is increased from 50 (orange solid line) to 200 (blue dashed line) to 450 (green dash-dotted line) bins. Center panel and right panel: The same as in the left panel, but the system evolves via iLLG and os-LLG equation, respectively. The initial condition and the magnetic field are fixed by the parallel parameter configuration. The relative entropy is calculated for $N_{\mathrm{trac}} = 2000$ trajectories.