Weak and Strong Thermalization

• Weak and strong thermalization are classifications of dynamical relaxation in many-body systems, differentiating how observables converge to equilibrium.
• Weak thermalization involves time-averaged convergence with persistent fluctuations, while strong thermalization requires pointwise convergence and small fluctuations at nearly all times.
• Experimental and theoretical studies in quantum chains, stochastic models, and holographic frameworks reveal that state complexity and spectral structure govern thermalization behavior.

Weak and strong thermalization classify the dynamical relaxation properties in isolated quantum and classical many-body systems, and in classical and stochastic models with long memory, according to how observables approach their equilibrium (thermal) values under non-equilibrium evolution. The distinction is foundational to modern nonequilibrium statistical physics, and appears in quantum quenches, ergodic dynamics of large systems, generalized Langevin descriptions, and holographic approaches to field theory and cosmology. Weak thermalization denotes convergence of time-averaged observables to equilibrium with persistent or large instantaneous fluctuations, while strong thermalization requires pointwise or trace-norm convergence at nearly all times. These regimes, rigorously and operationally defined in a variety of frameworks, are sharply distinguished in specific quantum models, yet are increasingly recognized as the extremes of a continuous crossover parameterized by effective state complexity and spectrum structure.

1. Formal Definitions and Criteria

Quantum Many-Body Systems:

• Strong thermalization is defined by the requirement that, for any local observable $O$ or subsystem $A$, the reduced state (or observable average) converges to the thermal (e.g., Gibbs) value at all long times:

$$\lim_{t\to\infty}\langle O(t)\rangle = \langle O\rangle_{\rm th}, \qquad \lim_{t\to\infty} \|\rho_A(t) - \rho_{A, \rm th}\| = 0$$

This is equivalent to trace-norm convergence for the reduced density matrix of any finite subsystem.

• Weak thermalization is characterized by the approach of time-averaged observables to their equilibrium value, while instantaneous values may exhibit large, persistent oscillations:

$$\lim_{T\to\infty} \frac{1}{T} \int_0^T \langle O(t) \rangle dt = \langle O \rangle_{\rm th}$$

but $|\langle O(t)\rangle - \langle O\rangle_{\rm th}|$ may not vanish as $t \to \infty$ (Bañuls et al., 2010, Chen et al., 2021, Prazeres et al., 2023).

Discrete-Time Quantum Dynamics (Clifford-Floquet, Quantum Cellular Automata):

• Strong thermalization: Convergence of expectation values for all local observables at all late times:

$$\lim_{n\to\infty} \omega_n(a) = \omega_\infty(a),\ \forall\ a\ \text{local}$$

• Weak thermalization: Convergence on a density-one subsequence:

$$\exists\ J_a\ (\text{density one}):\ \lim_{n\in J_a,\ n\to\infty} \omega_n(a) = \omega_\infty(a)$$

(Kapustin et al., 1 Jan 2026)

Stochastic/Memory Kernel Dynamics (Generalized Langevin Equation, GLE):

• Weak thermalization: The system relaxes (in the absence of external forces) to the stationary Maxwellian or Gibbs distribution, independent of initial conditions, though no steady response is possible for applied forces if dissipation is absent.
• Strong thermalization: With any bounded force (including constant), the system achieves a finite steady state (terminal velocity or equivalent), not possible in purely dissipationless GLEs with long memory (Procopio et al., 14 May 2025).

2. Physical and Mathematical Origins

Quantum Many-Body Systems

In closed quantum chains, which display chaotic dynamics, the distinction arises from the structure of the initial state projected onto the energy spectrum:

• Strong thermalization is seen for initial states with broad overlap on many mid-spectrum eigenstates (maximal participation ratio), leading to rapid, monotonic decay to thermal equilibrium with only small fluctuations (typically $\sim O(e^{-\alpha N})$).
• Weak thermalization arises when the initial state overlaps only a narrow band of eigenstates (typically near the spectral edge), resulting in large, non-decaying oscillations of observables about their equilibrium values, with thermalization only when averages over long times are taken (Bañuls et al., 2010, Sun et al., 2020, Chen et al., 2021, Prazeres et al., 2023).

The effective dimension $D_{\rm eff}$ of the initial state, as defined by

$D_{\rm eff} = \left( \sum_n |c_n|^4 \right)^{-1}$

(where $c_n$ are energy coefficients), controls the amplitude and timescale of fluctuations. Small $D_{\rm eff}$ leads to weak thermalization, while $D_{\rm eff} \sim \exp(\alpha N)$ implies strong thermalization (Prazeres et al., 2023).

Classical/Stochastic Models

In non-Markovian systems with long-range memory (GLEs with $k(t) \sim t^{-\kappa}$), weak thermalization emerges for $0 < \kappa < 1$, where the velocity distribution relaxes to equilibrium for all initial data, but the lack of true dissipation precludes steady-state responses under constant forcing (strong thermalization is absent). For $\kappa > 1$, ergodicity is broken and no thermalization occurs (Procopio et al., 14 May 2025).

Unitary Lattice Automata and Floquet Systems

In translationally-invariant Clifford quantum cellular automata, "strong diffusivity" of the Floquet map $L$ leads to strong thermalization, while "weak diffusivity" implies only weak thermalization, with possible rare recurrences or sparse fluctuations at zero-density times (Kapustin et al., 1 Jan 2026).

3. Typical Diagnostic Frameworks and Metrics

Framework	Strong Thermalization	Weak Thermalization
Quantum chains	Pointwise observable convergence; small $\Delta_O^2$	Large long-time fluctuations, time-average converges
Krylov basis	Maximal infinite-time-average Krylov complexity $\overline{C} \sim O(N)$	$\overline{C} \ll O(N)$, persistent oscillations
GLEs (memory kernels)	Steady-state under arbitrary forcing	Equilibrium distribution only without forcing
Floquet/Clifford automata	Observable convergence for all times	Convergence along density-one subsequence

Key quantitative diagnostics include:

• Infinite-time-average Krylov complexity $\overline{C}$: maximal $\overline{C}$ indicates strong thermalization, smaller $\overline{C}$ indicates weak (Alishahiha et al., 2024).
• Effective dimension $D_{\rm eff}$: exponential in system size for strong, small for weak (Prazeres et al., 2023, Sun et al., 2020).
• Fluctuation bounds: $\Delta_O^2 \sim \|O\|^2/D_{\rm eff}$ (Prazeres et al., 2023).
• Participation ratios and entanglement entropy: rapid entropy growth and large participation ratio for strong, slow entropy growth for weak (Chen et al., 2021).
• Matrix-element statistics (ETH): vanishing support $s_A$ and variance $\sigma_A^2$ of diagonal elements for strong ETH, only vanishing variance for weak ETH (Dabelow et al., 2022, Ikeda et al., 2012).

4. Relation to the Eigenstate Thermalization Hypothesis (ETH)

The strong/weak distinction is mirrored in ETH:

• Strong ETH (sETH): All energy eigenstates in an energy window have expectation values of local observables close to microcanonical average ($s_A \rightarrow 0$ as system size grows) (Dabelow et al., 2022, Ikeda et al., 2012).
• Weak ETH (wETH): Only the variance of these expectation values vanishes ($\sigma_A\rightarrow0$), permitting rare outliers (Ikeda et al., 2012).

Strong ETH is sufficient for strong thermalization, but not necessary: Weak ETH suffices for weak thermalization, and for typical observables and initial states, the distinction may be negligible in the thermodynamic limit—fluctuations vanish exponentially in $N$. Integrable systems often violate sETH while satisfying wETH, leading to generic weak but not strong thermalization (Dabelow et al., 2022, Ikeda et al., 2012).

5. Experimental and Numerical Signatures

• Superconducting qubit arrays and spin-chain simulations have unambiguously observed both strong and weak thermalization within the same Hamiltonian by varying initial states. Strong thermalization is typified by fast, monotonic decay of local observables/traces to thermal values, and saturation of entanglement entropy to the Page value. In weak thermalization, persistent oscillations and reduced entropy growth are observed, but time-averaged observables nonetheless match thermal predictions (Chen et al., 2021, Sun et al., 2020, Bañuls et al., 2010).
• Disordered quantum chains and many-body localization (MBL) realize additional regimes where only part of the system thermalizes, or thermalization is sample-dependent, illustrating the nuanced role of disorder in the crossover between thermalizing and non-thermalizing phases (Pal et al., 16 May 2025).
• Hydrodynamic and holographic models also reproduce the strong-to-weak thermalization dichotomy: at infinitely strong coupling (AdS/CFT), relaxation times are $O(1/T)$ (strong), while at weak coupling, timescales scale parametrically with the coupling (e.g., $\tau_{\rm therm} \propto \alpha_s^{-13/5}$ in QCD kinetic theory), corresponding to weak thermalization dominated by classical Boltzmann or kinetic descriptions (Grozdanov et al., 2016, McDonough, 2020, Berges et al., 2020).

6. Crossover, Scaling, and Theoretical Synthesis

The transition between weak and strong thermalization is not generically associated with a sharp phase transition. Rigorous dephasing bounds show that the long-time fluctuations of observables (quantified by $\Delta_O^2$) scale as $1/D_{\rm eff}$, and $D_{\rm eff}$ itself varies smoothly with the choice of initial state. Consequently, as one continuously changes the energy density or the structure of the initial state, the system interpolates between weak and strong thermalization regimes without singularity or criticality (Prazeres et al., 2023, Alishahiha et al., 2024).

In all standard scenarios (non-integrable, translation-invariant lattices with unique Gibbs state), almost all pure states in a microcanonical shell are strongly locally thermal (canonical typicality), and strong dynamical thermalization holds for large-population-entropy (highly delocalized) initial conditions (Mueller et al., 2013). In typical experiments and numerics, the difference becomes physically relevant only for observables or initial states specifically engineered to have narrow energy support or low effective dimension.

7. Broader Implications and Context

• Quantum information scrambling is intimately related but distinct: strong and weak thermalization set the dynamics of local observables, while scrambling (e.g., tripartite mutual information) can reach maximal values even when local observables do not thermalize (as in weak thermalization) (Sun et al., 2020).
• Holographic duality and QCD cosmology connect the strong/weak dichotomy to the coupling dependence of relaxation rates and hydrodynamization times in strongly interacting quantum fields, with implications for heavy-ion phenomenology and early-universe reheating (McDonough, 2020, Berges et al., 2020, Grozdanov et al., 2016).
• Disordered and inhomogeneous systems reveal that weak and strong thermalization can coexist spatially, or be sample-dependent, depending on energy scales and disorder realization, highlighting the phase-space complexity of real many-body systems (Pal et al., 16 May 2025).

In summary, the distinction between weak and strong thermalization encapsulates fundamental differences in how closed or stochastic systems approach equilibrium, as determined by spectral properties, initial-state complexity, and system-specific dynamical rules. These concepts, now rigorously defined and experimentally realized, provide the foundation for ongoing research into ergodicity, chaos, and the emergence of statistical mechanics in quantum systems (Bañuls et al., 2010, Chen et al., 2021, Alishahiha et al., 2024, Prazeres et al., 2023, Mueller et al., 2013, Procopio et al., 14 May 2025, Kapustin et al., 1 Jan 2026).