Transfer Entropy and Flow of Information in Two-Skyrmion System

We theoretically investigate the flow of information in an interacting two-skyrmion system confined in a box at finite temperature. By numerical simulations based on the Thiele-Langevin equation, we demonstrate that the skyrmion motion cannot be fully described by the master equation, highlighting the nontrivial dynamics. Particularly, due to the chiral motion of skyrmion, we find asymmetric flow of information with violating the detailed balance condition. We analyze this system using information-theoretical quantities including Shannon entropy, mutual information, and transfer entropy. The physical significance of transfer entropy, which has been overlooked in previous studies, is elucidated. Notably, the peak position of the transfer entropy, as a function of time delay, is independent of the interaction range yet dependent on the box size. This peak corresponds to the characteristic time required for changing the skyrmion state. Due to the unusual asymmetric circulation of information, the two-skyrmion system can be a unique device for future applications to the natural computing.

💡 Research Summary

The paper investigates how information flows between two interacting magnetic skyrmions confined in a square box at finite temperature, using a combination of stochastic dynamics and information‑theoretic analysis. The authors model the skyrmion motion with the Thiele–Langevin equation, which includes inertial mass, Gilbert damping, a gyrotropic term (G × v), repulsive skyrmion‑skyrmion forces, wall confinement forces, and thermal noise satisfying the fluctuation‑dissipation theorem. Parameters are chosen from realistic experimental values (mass ≈10⁻²² kg, damping α = 0.02, gyrotropic constant G≈5×10⁻¹⁴ kg s⁻¹, diffusion D≈7×10⁻¹⁴ kg s⁻¹). The repulsive interaction is modeled as an exponential decay for separations larger than the skyrmion radius R and as a constant force for contact, while the walls are represented by an exponential potential. Simulations use a time step of 1 ns, total duration 1 µs, and 10⁵ independent trajectories to obtain reliable statistics.

First, a single skyrmion’s trajectory is coarse‑grained into four spatial cells (0–3). The occupation probabilities qₜ(x) evolve in time, showing that the gyrotropic term induces a clockwise cyclotron motion together with a counter‑clockwise “skipping” motion along the walls. This combination yields asymmetric transition probabilities: the forward transition ratio r₀→₁(Δt) exceeds unity, violating detailed balance even though the system is in thermal equilibrium. When the gyrotropic term is switched off (G = 0), detailed balance is restored and the dynamics are well described by a master equation with constant transition rates. This demonstrates that the chiral nature of skyrmion dynamics creates a non‑reversible information flow that cannot be captured by a simple Markovian description.

For the two‑skyrmion case, the authors compute three information‑theoretic quantities: Shannon entropy, mutual information, and transfer entropy. Transfer entropy T_{Y→X}(Δt) quantifies the directed information flow from skyrmion Y at time t − Δt to skyrmion X at time t. The key finding is that 2 T_{Y→X}(Δt) exhibits a pronounced peak at a delay τₚₑₐₖ that depends only on the box size d and not on the interaction range ξ. The peak time matches the characteristic transmission time d/⟨v⟩, where ⟨v⟩ is the average skyrmion speed. Hence, the peak marks the minimal time required for one skyrmion to influence the other via repulsive interaction, i.e., the physical information‑transmission delay. The authors argue that this delay consists of two contributions: the time needed to acquire mutual information and the time needed to “write” that information into the partner’s state.

The master‑equation analysis, performed with fitted transition rates, reproduces the single‑skyrmion statistics only when the spatial discretization is coarse (two cells) or when one skyrmion is artificially fixed at the origin. With four cells, the master equation fails to capture overshooting behavior and the asymmetric transition ratios, underscoring the loss of non‑Markovian, chiral information in coarse‑grained models.

Overall, the study reveals that the intrinsic chirality of skyrmion dynamics generates a persistent, directional information flow even in equilibrium, and that transfer entropy provides a unique quantitative probe of this effect. The dependence of the transfer‑entropy peak on system size suggests a route to engineer information‑processing delays in skyrmion‑based devices, positioning the two‑skyrmion system as a promising platform for natural computing and stochastic information processing. Future work may explore larger skyrmion networks, external driving fields, and integration with machine‑learning algorithms to harness these asymmetric information channels for practical computation.