Transfer entropy and flow of information in two-skyrmion system

TL;DR

This study analyzes information flow between two interacting skyrmions confined in a box at finite temperature using Thiele-Langevin dynamics and information-theoretic metrics. It shows that the skyrmion dynamics cannot be captured by a master equation and that chiral gyrotropic motion breaks detailed balance, producing asymmetric information circulation in equilibrium. Transfer entropy $T_{Y\rightarrow X}(\Delta t)$ exhibits a pronounced peak whose delay is largely set by the confinement scale, scaling as $d/v$ with $v=\sqrt{2k_B T/m}$, and is largely insensitive to the interaction range $\xi$; this peak represents the information transmission time required for state changes driven by inter-skyrmion repulsion. The results clarify the physical meaning of transfer entropy in a simple, controllable system and point to potential applications in ultralow-power Brownian computing and stochastic learning, where directed information flow can be harnessed.

Abstract

We theoretically investigate the flow of information in an interacting two-skyrmion system confined in a box at finite temperature. Using numerical simulations based on the Thiele-Langevin equation, we demonstrate that the skyrmion motion cannot be fully described by the master equation, highlighting the system's simplicity with its nontrivial dynamics. Particularly, due to the chiral motion of skyrmion, we find asymmetric flow of information even in equilibrium, which is demonstrated by the violation of the detailed balance condition. We analyze this novel system using information-theoretical quantities including Shannon entropy, mutual information, and transfer entropy. By the analyses, the physical significance of transfer entropy as a measure of flow of information, which has been overlooked in previous studies, is elucidated. Notably, the peak position of the transfer entropy, as a function of time delay, is found to be independent of the interaction range between the two skyrmions yet dependent on the box size. This peak corresponds to the characteristic time required for changing the skyrmion state: the box size divided by the average velocity of skyrmions. We can understand that the information transmission time consists of the time to obtain mutual information and the time to write the information. Since the unusual asymmetric circulation of information is revealed in this two-skyrmion system, it can be a unique platform for future applications to natural computing using flow of information, including more efficient machine learning algorithm.

Figures (9)

• Figure 1: (a) Snapshot of the two skyrmions drawn by the orange and green circles with radius $R$, in a square box. The box is divided into the four cells (cell 0, 1, 2, and 3). (b) Trajectories of the two skyrmions. For the orange (green) skyrmion, the initial and final point is shown by the filled and open red (blue) circle, respectively. We see the clockwise cyclotron motion (black arrow) and the counterclockwise skipping trajectory (red arrows).
• Figure 2: Time evolution of the occupation probabilities $q_t(x)$ for $x=0,1,2,3$ (red, black, green, and blue, respectively) for (a) single skyrmion, (b) single skyrmion with $G=0$, and (c) single moving skyrmion and fixed skyrmion at the origin, with $G=0$. The dots are the simulated results, and the solid curves are the analytical solutions of the master equation. The gray horizontal line indicates $q=0.25$, to which all probabilities converge in $t \rightarrow \infty$ limit.
• Figure 3: Probability ratio $r_{0\rightarrow1}(\Delta t)$ as a function of the time delay $\Delta t$. (a) and (b) is for the one-body and two-body cases, respectively. The red horizontal lines indicate $r_{0\rightarrow1} = 1$, or the detailed balance condition.
• Figure 4: For the two-skyrmion system, time evolution of the occupation probabilities $q_t(x)$ for $x=0,1,2,3$ (red, black, green, and blue, respectively) are shown. The dots are the result of the simulation, and the solid lines are given from the analytical solutions of the master equation ($w=0.02$ and $w'=0.012$).
• Figure 5: (a) Shannon entropy $H(X_t)$ of the skyrmion in units of $\ln{4}$. The different colors (black, red, blue, green, and yellow) correspond to several interaction ranges $\xi/\xi_0 = 2.0, 2.5, 3.0, 3.5$, and $4.0$, respectively. The gray plots are for the non-interacting case (free skyrmion). (b) Mutual information $I(X_t:Y_{t-\Delta t})$ is plotted as a function of the time delay $\Delta t$, in the same manner as (a).