Directional information transfer between interacting Brownian particles

Abstract

We theoretically investigate how information flows when two particles interact with each other. Understanding the physical mechanisms of directional information flow is crucial for advancing information thermodynamics and stochastic computing. However, the fundamental connection between mechanical motion and causal information transfer remains elusive. To focus only on essential effects of physical dynamics, we examine two interacting Brownian particles confined in a one-dimensional potential. By simulating their Langevin dynamics, we quantify the causal information exchange using transfer entropy. We demonstrate that a mass asymmetry inherently breaks the symmetry of information flow, inducing a net directional transfer from the heavier to the lighter particle. Physically, the heavier particle, possessing larger inertia and higher active information storage, retains the memory of its trajectory longer against thermal fluctuations, thereby acting as a source of information. We analytically clarify that this net transfer is governed by a competition between the difference in memory capacity and the predictability of the particle trajectories. Furthermore, we reveal that the net information flow scales logarithmically with the mass ratio. These findings provide essential insights into the physical significance of transfer entropy and the nature of information flow in general physical systems.

Figures (9)

• Figure 1: (a) Snapshot of two interacting Brownian particles ($X$ and $Y$) in a wall potential. The curve illustrates the potential from the two walls, which vanishes in $1 \leq x/\xi \leq 3$. (b) Trajectories of the two particles, which are well confined between the walls. It is seen that particle $X$ ($Y$) predominantly exists in the left (right) region because of the repulsive interaction.
• Figure 2: (a) Time evolution of Shannon entropy $H(X_t)$ (particle $X$) and $H(Y_t)$ (particle $Y$) for mass parameter $\mu = 1$ (identical particles). The vertical line, $\tau_\mathrm{th} = \xi / v_\mathrm{th}$, represents the characteristic time of thermal motion (see the text). (b) Similar plots for an asymmetric case $\mu=2$ ($Y$ is twice heavier than $X$).
• Figure 3: Mutual information $I(X_t : Y_{t-\Delta t})$ at the reference time $t=t_\mathrm{f}$, illustrated as a function of the time lag $\Delta t$. The mass ratio $\mu$ varies from $\mu=1$ to $5$.
• Figure 4: Active information storage of particle $X$ [panel (a), lighter] and $Y$ [panel (b), heavier], as a function of the time lag $\Delta t$. The mass ratio $\mu$ is varied from 1 to 5.
• Figure 5: Transfer entropies $T_{Y\to X}$, $T_{X\to Y}$, and their difference $T_\mathrm{net}$ for (a) $\mu=1$, (b) $\mu=2$, and (c) $\mu=3$. The shaded region illustrates the statistical noise level $\pm 1/\sqrt{N}$. The vertical dashed lines represent the characteristic time scale of thermal motions, $\tau_\mathrm{th}$ (see the text).