Typicality from concentration of measure in models with binary state variables

Area of Science:

• Statistical mechanics
• Complex systems theory
• Dynamical systems

Background:

• Classical systems with numerous binary variables can exhibit complex behaviors.
• Understanding macroscopic properties from microscopic states is a fundamental challenge.
• Typicality, or predictable macroscopic behavior, has been observed in quantum systems.

Purpose of the Study:

• To analytically demonstrate the emergence of static and dynamic typicality in classical systems.
• To establish a connection between static and dynamic typicality.
• To explore the implications of typicality in classical systems, drawing parallels with quantum dynamics.

Main Methods:

• Analytical derivation of typicality for large subsets of the state space.
• Concentration of macroscopic observables around typical values.
• Analysis of dynamical trajectories from initial conditions.
• Illustration using one-dimensional cellular automata models.

Main Results:

• Macroscopic observables (e.g., density, energy) are sharply concentrated around typical values for large system subsets.
• Dynamical trajectories of macroscopic observables are approximately the same for similar initial conditions.
• The phenomenon of dynamical typicality is shown to be analogous to that in isolated quantum systems.
• Analytical results are validated through the analysis of cellular automata.

Conclusions:

• Static typicality in classical systems implies predictable macroscopic behavior.
• Dynamical typicality ensures consistent macroscopic evolution from similar initial states.
• The findings highlight a unifying principle of typicality across classical and quantum physics.