Information Storage in the Stochastic Ising Model

Abstract

Most information storage devices write data by modifying the local state of matter, in the hope that sub-atomic local interactions stabilize the state for sufficiently long time, thereby allowing later recovery. Motivated to explore how temporal evolution of physical states in magnetic storage media affects their capacity, this work initiates the study of information retention in locally-interacting particle systems. The system dynamics follow the stochastic Ising model (SIM) over a 2-dimensional $\sqrt{n}\times\sqrt{n}$ grid. The initial spin configuration $X_0$ serves as the user-controlled input. The output configuration $X_t$ is produced by running $t$ steps of Glauber dynamics. Our main goal is to evaluate the information capacity $I_n(t):=\max_{p_{X_0}}I(X_0;X_t)$ when time $t$ scales with the system's size $n$. While the positive (but low) temperature regime is our main interest, we start by exploring the simpler zero-temperature dynamics. We first show that at zero temperature, order of $\sqrt{n}$ bits can be stored in the system indefinitely by coding over stable, striped configurations. While $\sqrt{n}$ is order optimal for infinite time, backing off to $t<\infty$, higher orders of $I_n(t)$ are achievable. First, linear coding arguments imply that $I_n(t) = \Theta(n)$ for $t=O(n)$. To go beyond the linear scale, we develop a droplet-based achievability scheme that reliably stores $\Omega\left(n/\log n\right)$ for $t=O(n\log n)$ time ($\log n$ can be replaced with any $o(n)$ function). Moving to the positive but low temperature regime, two main results are provided. First, we show that an initial configuration drawn from the Gibbs measure cannot retain more than a single bit for $t\geq \exp(C\beta n^{1/4+\epsilon})$ time. On the other hand, when scaling time with the inverse temperature $\beta$, the stripe-based coding scheme is shown to retain its bits for $e^{c\beta}$.