Limits of Information Flow Between Classically Interacting Particles

TL;DR

This work defines a principled, physics-aware measure of information flow between a particle and its environment by casting it as a saddle-point mutual information problem under energy and power constraints in a zero-momentum frame. The core result yields an information-flow rate ${\cal R} = {\cal P}_0/(2{\cal E}_0)$, established via Gaussian priors and an additive-channel viewpoint, and shown to persist in both the simplified momentum model and the full state-space treatment. Extending to a springlike two-particle coupling, the authors derive a time-dependent mutual information between the particles, identify conditions under which information transfer spikes, and demonstrate that the small-time behavior matches the same rate ${\cal R}$ calculated earlier. The work connects information flow to thermodynamic quantities, suggesting a non-equilibrium temperature interpretation and offering a framework for analyzing information exchange in far-from-equilibrium classical systems, with implications for stochastic thermodynamics and quantum information paradigms.

Abstract

Pinning down a precise understanding of information flow within physical interactions remains a central concern to fields like stochastic thermodynamics and quantum information science. In both spheres a careful accounting of bits (or qubits) enables a deeper understanding of the physical nature of information. In this work we propose a measure of information flow as a saddle-point solution of the mutual information. This approach places a lower bound on the channel capacity between a particle and an interacting environment. The measure is given by P/2E in nats/sec, with P the average power flux between the particle and its environment, and E the initial average energy of the particle, all computed in a frame where the particle has zero average momentum. We use a communication theory lens to suggest an associated channel analogy, in which this bound is interpreted as a signal-to-noise ratio. We find that this measure can also quantify early-time information flow for a particle interacting with a thermal bath.

Figures (3)

• Figure 1: (a) A classical particle in one dimension under the influence of a random environment (analyzed in sections \ref{['sect:II']} and \ref{['sect:III']}). $X_0$ and $P_0$ are the position and momentum respectively at time $0$, and $X_t$ and $P_t$ give the position and momentum at $t$. $\Delta P_t$ is the change in momentum. (b) Two particles interacting with a springlike interaction (analyzed in section \ref{['sect:IV']}).
• Figure 2: Determinant of ${\bf L}_t$: We plot $|{\bf L}_t|=(m^2 + M^2 + 2mM\cos\omega t + mM\omega t\sin\omega t)/(m+M)^2$ as a function of $\omega t$ to show how the mass ratio, $M/m$, determines the first zero-crossing for the determinant, and thus the first spike for the mutual information (see FIGURE \ref{['fig:mutualInfo']}).
• Figure 3: Mutual Information: We plot the mutual information (in nats) for the springlike model $I(\vec{X}_t;\vec{Y}_0)$ for two mass ratios. Parameters are set so that ${\bf K}_{X_0}={\bf K}_{Y_0}=\left(1001\right)$. Spikes indicate divergence toward $+\infty$.

Theorems & Definitions (2)

• Lemma 1
• Lemma 2