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Published on: December 4, 2017
Dynamical typicality in classical lattice systems
Nicolas Nessi1, Peter Reimann2
1IFLP, CONICET, Diagonal 113 y 64, La Plata, Buenos Aires, Argentina.
Physical Review. E
|July 24, 2026
Summary
In large classical lattice systems, macroscopic observables evolve predictably for most initial states. This finding applies to Hamiltonian and dissipative dynamics, provided local perturbations decay rapidly.
Area of Science:
- Statistical mechanics
- Dynamical systems theory
- Condensed matter physics
Background:
- Deterministic classical lattice systems with continuous variables are foundational in physics.
- Understanding the macroscopic behavior from microscopic initial conditions is a key challenge.
- Previous work often focused on specific models or assumptions about initial states.
Purpose of the Study:
- To demonstrate that macroscopic observables in classical lattice systems exhibit predictable temporal evolution for a vast majority of initial conditions.
- To establish general conditions under which this predictability holds, applicable to both Hamiltonian and dissipative dynamics.
- To provide a theoretical framework based on concentration of measure results.
Main Methods:
- Utilizing general concentration of measure results to bound deviations from typical behavior.
- Analyzing systems where initial conditions are sampled from a statistically independent probability distribution.
- Proving that the temporal evolution of macroscopic observables is approximately the same for most initial states.
- Establishing a condition on dynamics: the rapid decay of local perturbation influence with distance.
Main Results:
- For large classical lattice systems, macroscopic observables show remarkably consistent temporal evolution across most statistically independent initial states.
- The results hold for a broad class of dynamics, including Hamiltonian and dissipative systems.
- The proof relies on concentration of measure, providing tight bounds for large system sizes.
- A key requirement is that the influence of local perturbations must decay sufficiently fast with distance.
Conclusions:
- Macroscopic predictability emerges in large classical lattice systems from simple statistical assumptions on initial conditions.
- The findings offer a general theoretical framework applicable to diverse physical systems, including coupled map lattices and systems of coupled rotors.
- This work bridges the gap between microscopic dynamics and macroscopic behavior, highlighting the power of concentration of measure in statistical physics.
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