Thermalization in classical systems with discrete phase space

We study the emergence of statistical mechanics in isolated classical systems with local interactions and discrete phase spaces. We establish that thermalization in such systems does not require global ergodicity; instead, it arises from effective local ergodicity, where dynamics in a subsystem may appear pseudorandom. To corroborate that, we analyze the spectrum of the unitary evolution operator and propose an ansatz to describe statistical properties of local observables expanded in the eigenfunction basis - the classical counterpart of the Eigenstate Thermalization Hypothesis. Our framework provides a unified perspective on thermalization in classical and quantum systems with discrete spectra.

💡 Research Summary

The paper investigates how statistical mechanics emerges in isolated classical many‑body systems whose phase space is discrete and whose interactions are local. The authors challenge the traditional view that global ergodicity—i.e., a single trajectory uniformly exploring the entire accessible phase space—is required for thermalization. Instead, they propose that “effective local ergodicity” suffices: a subsystem can appear pseudorandom even when the full system’s dynamics are highly structured.

To formalize this idea, the authors work with one‑dimensional lattice models where each site can take one of q discrete values, so the configuration space is X = ℤ_q^L. Time evolution is given by an invertible map F : X → X, i.e., a permutation of the finite set of configurations. Observables are functions on X and form a finite‑dimensional Hilbert space Fun(X) equipped with the inner product (A|B) = Σ_x A*_x B_x, the discrete analogue of the L² inner product with Liouville measure. The Koopman operator U, defined by (U A)(x) = A(F⁻¹x), is unitary on this space, allowing the authors to treat classical dynamics with the same spectral tools used for quantum unitary evolution.

The spectrum of U is completely determined by the periodic orbits (cycles) of the permutation F. For each orbit γ of length T_γ, one obtains T_γ orthonormal eigenfunctions ϕ(γ)_k (k = 0,…,T_γ−1) with eigenvalues e^{-i ω(γ)_k} where ω(γ)_k = 2πk/T_γ. The k = 0 mode corresponds to the uniform distribution on the orbit; all other modes encode fluctuations around it. Any observable A can be expanded in this eigenbasis, yielding Fourier‑like coefficients A(γ)_k = (ϕ(γ)_k|A).

The long‑time average of A for an initial probability distribution ρ₀ is given by the projection onto the zero‑frequency sector:
A_{ρ₀} = Σ_γ T_γ (ρ₀|ϕ(γ)_0) A(γ)_0.
Thus thermalization reduces to the question of whether the orbit‑averaged values A(γ)_0 approach the microcanonical average A_mc = (1/|X|) Σ_x A_x.

The authors introduce a “random‑orbit” model as a benchmark. If one samples a long orbit by drawing each configuration independently from a uniform Bernoulli distribution, the central limit theorem implies that for large T_γ
A(γ)k = A_mc δ{k,0} + T_γ^{-1/2} R(γ)_k,
where the R(γ)_k are independent Gaussian variables with zero mean and variance σ_A² = ⟨A²⟩_mc – A_mc². This reproduces the structure of the Eigenstate Thermalization Hypothesis (ETH) in quantum systems, where matrix elements of observables in the energy eigenbasis consist of a smooth diagonal part plus a random off‑diagonal part suppressed by the Hilbert‑space dimension.

In deterministic, locally interacting cellular‑automaton‑type models, exact randomness is absent, yet the authors argue that the effective statistics of the coefficients A(γ)_k should be the same, provided the mean orbit length ⟨T⟩ grows exponentially with system size (as is typical for reversible cellular automata). They define the mean deviation MD