Maxwell Demon Dynamics: Deterministic Chaos, the Szilard Map, and the Intelligence of Thermodynamic Systems

We introduce a deterministic chaotic system—the Szilard map—that encapsulates the measurement, control, and erasure protocol by which Maxwellian demons extract work from a heat reservoir. Implementing the demon’s control function in a dynamical embodiment, our construction symmetrizes the demon and the thermodynamic system, allowing one to explore their functionality and recover the fundamental trade-off between the thermodynamic costs of dissipation due to measurement and those due to erasure. The map’s degree of chaos—captured by the Kolmogorov-Sinai entropy—is the rate of energy extraction from the heat bath. Moreover, an engine’s statistical complexity quantifies the minimum necessary system memory for it to function. In this way, dynamical instability in the control protocol plays an essential and constructive role in intelligent thermodynamic systems.

Figure 1

Szilard engine as a deterministic dynamical system: The Szilard map ${\mathcal{T}}_{\text{Szilard}}={\mathcal{T}}_{\text{Erase}}^A\circ {\mathcal{T}}_{\text{Control}}\circ {\mathcal{T}}_{\text{Measure}}$. Regions left and right of $\delta$ are colored to aid visually tracking particle ensemble history. Rightmost panel: Action of ${\mathcal{T}}_{\text{Szilard}}^3$ resulting in self-similar (fractal) structure in density $\hat{\rho }$; uniform $\hat{\rho }$ requires $\eta =\gamma \delta$. These assume the demon’s reset memory state is $A$. (The Supplemental Material includes an animation [15].)

Figure 2

$\epsilon$-Transducers for the symbolic dynamics of the Szilard information engine from the Markov partition of its joint state space. (a) The $\epsilon$-transducer for ${\mathcal{T}}_{\text{Szilard}}$ that reads in the periodic control signal for measure ($M$), control ($C$), and erase ($E$). (b) ${\mathcal{T}}_{\text{Szilard}}$ single-state $\epsilon$-transducer: Memoryless over the full measure-control-erase protocol. Transitions $\beta |\alpha$: $p$ denote reading protocol symbol $\alpha$, taking the transition with probability $p$, and emitting symbol $\beta$. Asymptotic state probabilities are given in parentheses underneath state names.

Figure 3

Beyond Landauer’s principle: Thermodynamic costs (energy dissipation $Q_{\text{diss}}$) for measurement, control, and erasure in Szilard’s information engine as a function of $\gamma$ (demon partition) with SUS barrier at $\delta =1/2$. Landauer’s principle applies only at $\gamma =1/2$ (vertical yellow band at the middle): Measurements are thermodynamically free, but erasure costs because heat is dissipated as a result of demon resetting. Costs exactly flip at $\gamma =4/5$, though.