Information Storage in the Stochastic Ising Model

Most information storage devices write data by modifying the local state of matter, in hope that sub-atomic interactions stabilize for sufficiently long time, thereby allowing later recovery. Motivated to explore how temporal evolution physical states magnetic media affects their capacity, this work initiates study retention locally-interacting particle systems. The system dynamics follow stochastic Ising model (SIM) over a 2-dimensional $\sqrt {{n}}\times \sqrt {{n}}$ grid. initial spin configuration $ {X}_{0}$ serves as user-controlled input. output ${X}_{{t}}$ is produced running $t$ steps Glauber dynamics. Our main goal evaluate capacity ${I}_{{n}}({t}):=\max _{{p}_{{X}_{0}}}{I}({X}_{0};{X}_{{t}})$ when time scales with system’s size $n$ . While positive (but low) temperature regime our interest, we start exploring simpler zero-temperature We first show at zero temperature, order bits can be stored indefinitely coding stable, striped configurations. optimal infinite backing off ${t} , higher orders ${I}_{{n}}({t})$ are achievable. First, via linear arguments imply ${I}_{{n}}({t}) = \Theta ({n})$ ${t}={O}({n})$ To go beyond scale, develop droplet-based achievability scheme reliably stores $\Omega \left ({{n}/\log {n}}\right)$ ${t}={O}({n}\log {n})$ ( $\log {n}$ replaced any ${o}({n})$ function). Moving but low regime, two results provided. an drawn from Gibbs measure cannot retain more than single bit ${t}\geq \exp ({C}\beta {n}^{1/4+\epsilon })$ time. On other hand, scaling inverse $\beta stripe-based (that temperature) shown its ${e}^{{c}\beta }$