  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 $e^{c\beta}$.
