Weak Ergodicity Breaking Overview

• Weak ergodicity breaking is a phenomenon where time averages along individual trajectories do not converge to the ensemble average, despite the phase space being fully accessible.
• It arises from mechanisms like heavy-tailed waiting times and spatial heterogeneity, leading to non-self-averaging behavior even at infinite time.
• In quantum many-body systems, effects such as quantum scars and Hilbert space fragmentation result in persistent revivals and anomalous dynamics.

Weak ergodicity breaking (WEB) is a class of nonergodic stochastic or quantum dynamical behavior characterized by the equivalence of time averages and ensemble averages failing, even in processes where the trajectory can in principle explore the entire phase or Hilbert space. Unlike strong ergodicity breaking, in which the system is partitioned into dynamically disconnected sectors (e.g., spin glasses or many-body localized systems), in WEB the phase space is connected but the exploration is so inefficient—typically due to heavy-tailed residence or recurrence time statistics—that time averages computed along a generic single trajectory remain random variables in the infinite time limit, rather than self-averaging to the ensemble average.

1. Definitions, Conceptual Distinctions, and Mathematical Criteria

A process is ergodic in the weak sense if for any observable $O$,

$$\overline{O}_T = \frac{1}{T} \int_{0}^{T} O(X(t))\,dt \xrightarrow{T\to\infty} \langle O \rangle_{\rm st} = \int O(x) P_{\rm st}(x) dx$$

almost surely. WEB arises when this convergence fails: the limiting distribution of $\overline{O}_T$ possesses a nonzero variance, and observables computed along single trajectories do not self-average. In quantum many-body systems, the analogous criterion is that the fraction of ETH-violating (“nonthermal”) eigenstates vanishes in the thermodynamic limit while their existence manifests in starkly anomalous dynamics from specially prepared initial conditions, such as persistent oscillations or revivals (Serbyn et al., 2020, Ramesh et al., 2024, Buijsman et al., 2024, Giachetti et al., 11 Nov 2025, Chen et al., 2022).

The ergodicity breaking (EB) parameter $J$ for trajectory-dependent observables, such as the time-averaged mean-squared displacement (TA MSD), quantifies this effect (Thiel et al., 2013): $$J(\tau,T) = \frac{ \langle [\overline{\Delta X^2}(\tau;T)]^2\rangle - \langle \overline{\Delta X^2}(\tau;T) \rangle^2 }{ \langle \overline{\Delta X^2}(\tau;T) \rangle^2 }$$ For ergodic dynamics, the limit $T\to\infty$ yields $J\to0$; for WEB, $J$ saturates to a positive value determined by renewal or trapping statistics.

2. Paradigmatic Dynamical Mechanisms of Weak Ergodicity Breaking

2.1 Classical and Stochastic: Power-law Trapping, Subdiffusion, and Temporal Disorder

• Continuous-time random walks (CTRW) with heavy-tailed waiting times $\psi(\tau)\sim \tau^{-1-\alpha}$, $0<\alpha<1$, produce subdiffusive ensemble-averaged MSDs $\langle x^2(t)\rangle\propto t^\alpha$. Time-averaged MSDs scale as $\propto\Delta$ and display nontrivial random amplitudes even as $T\to\infty$, a benchmark of WEB (Jeon et al., 2010, Thiel et al., 2013, Carmi et al., 2011).
• First-passage, recurrence, and occupation-time statistics in such processes are governed by distributions (e.g., Lévy-Smirnov, Lamperti, arcsine laws) with diverging means and broad support. In applications, the distribution of occupation fractions or time-averaged observables corresponds to non-delta, U-shaped or broad curves indicating that for long but finite $T$, most single trajectories are anomalous (0704.1769, Carmi et al., 2011, Ramesh et al., 2024).

2.2 Markovian Models with Spatial Heterogeneity

• Heterogeneous Diffusion Processes (HDP): Diffusion with space-dependent diffusivity $D(x)\sim|x|^\alpha$ yields anomalous diffusion and WEB even in absence of temporal memory: the ensemble and time-averaged MSDs have different scaling in time, and TAMSD amplitudes remain broadly distributed (Cherstvy et al., 2013).

2.3 Quantum Many-Body: Constraint-Induced Hilbert Space Structure and Quantum Many-Body Scarring

• Quantum Many-Body Scars: Nonthermal eigenstates (scars) appear in otherwise chaotic many-body spectra due to symmetry, kinematic constraints, or embedded algebraic structures (e.g., emergent or approximate SU(2)). Such eigenstates violate ETH, possess low entanglement entropy, and dominate the dynamics from special initial states (such as product state density waves), giving rise to long-lived revivals or persistent oscillations, distinct from both integrable and strongly localized regimes (Serbyn et al., 2020, Chen et al., 2022, Buijsman et al., 2024, Schecter et al., 2019, Desaules et al., 2022, Mukherjee et al., 2021, Russomanno et al., 2022).
• Hilbert Space Fragmentation: Emergent (often kinematic) constraints isolate small dynamically disconnected sectors (“fragments”), and generic initial states evolved in such subspaces do not thermalize (Katsura et al., 2024, Papić, 2021, Serbyn et al., 2020).

3. Mathematical Structures and Physical Diagnostics

3.1 Diagnostics for Classical/Stochastic Processes

• Occupation fraction distributions: Lamperti’s law, arcsine law, and their generalizations, characterize the statistical distribution of time spent in a given region. For example, the occupation fraction $\lambda = t_+ / T$ in subdiffusive CTRW obeys

$$G(\lambda) = \frac{\sin(\pi\alpha)}{\pi} \frac{ \lambda^{\alpha-1} (1-\lambda)^{\alpha-1} } { \lambda^{2\alpha} + (1-\lambda)^{2\alpha} + 2\cos(\pi\alpha) \lambda^{\alpha}(1-\lambda)^\alpha }$$

(Carmi et al., 2011, 0704.1769).

• Power spectrum statistics: Even binned estimates from a single realization remain random (variance does not vanish as $T\to\infty$); e.g., for 1/f noise from heavy-tailed CTRW, the spectral exponent β can be robustly extracted but the amplitude fluctuates universally (Niemann et al., 2010).
• First-passage time distributions: Exhibiting power-law ($t^{-3/2}$) tails, with divergent means, causing extreme disparity between time and ensemble averages (Ramesh et al., 2024).

3.2 Quantum/Many-body Diagnostics

• Entanglement entropy scaling: Scar states exhibit area-law or logarithmic scaling $S_A\propto \log L$, sitting well below the “thermal” Page curve for typical eigenstates. This is a diagnostic of failure of ETH in these states (Serbyn et al., 2020, Chen et al., 2022, Buijsman et al., 2024).
• Dynamics and fidelity revivals: Persistent, high-amplitude revivals (quasiperiodic or exact) in the fidelity $F(t) = |\langle\psi(0)|\psi(t)\rangle|^2$, and local observables for scar-initialized quenches, in stark contrast to rapid thermalization from generic states (Serbyn et al., 2020, Schecter et al., 2019, Desaules et al., 2022).
• Energy-level statistics: Wigner-Dyson statistics in the thermalizing bulk (chaotic), Poisson or semi-Poisson statistics within embedded integrable or scar sectors. This transition is diagnostic in both numerical and analytical settings (Chen et al., 2022, Katsura et al., 2024).
• Hilbert/Krylov subspace localization: Dynamics from LLS (long-lived states) explores only a small Krylov subspace; its dimension grows only polynomially with system size, in contrast to the full exponentially large Hilbert space (Deger et al., 2023).

4. Exemplary Models and Empirical Realizations

System Type	Principal WEB Mechanism	Diagnostic Signature
Classical CTRW	Power-law trapping times	$J>0$, broad $\overline{O}_T$ distr., Lamperti/arcsine laws (Jeon et al., 2010, Carmi et al., 2011)
Heterogeneous Diffusion	Spatially variable $D(x)$	Linear TAMSD, nontrivial amplitude scatter (Cherstvy et al., 2013)
Many-body scars	Embedded nonthermal subspace	Low-entropy, equally spaced nonthermal eigenstates, persistent oscillations (Serbyn et al., 2020, Schecter et al., 2019, Chen et al., 2022)
Hilbert space fragmentation	Kinematic or local constraints	Many disconnected fragments, slow or non-thermal dynamics in "frozen" or small fragments (Katsura et al., 2024, Mukherjee et al., 2021)
Supersymmetric models	SUSY-algebra induced towers	Scarred eigenstates, perfect revivals in special initial states (Buijsman et al., 2024)
Optical sensing	Diffusive Lévy/Wiener process, arcsine laws	Time vs. ensemble averaging yields dramatically different variances for finite time (Ramesh et al., 2024)

Additionally, lattice discretization effects can robustly yield undamped long-lived oscillations after a quantum quench in O(n) models, with persistent failure of dephasing in the presence of a sharp spectral edge, a further generic route to WEB not reliant on fine-tuned algebraic embeddings (Giachetti et al., 11 Nov 2025).

5. Robustness, Universality, and Physical Implications

• Robustness: In some models, such as the randomly-constrained spin chains (Deger et al., 2023), WEB is sharply robust to spatial disorder or constraint randomness above a threshold. In contrast, fine-tuned scars (as in the PXP chain) can be fragile to local perturbations, with revivals and eigenstate overlaps quickly destroyed.
• Universality and Lattice Effects: The universal presence of persistent oscillations in discrete lattice systems underlies much of the observed WEB in quenched quantum many-body systems (Giachetti et al., 11 Nov 2025).
• Implications for Sensing and Computation: In optical sensing, exploiting the presence of WEB enables precision gains by moving from time- to ensemble-averaging strategies: dividing a fixed energy budget over parallel shorter measurements, the variance is dramatically reduced for short $T$. As $T$ grows, the system reverts to ergodic statistics (Ramesh et al., 2024).
• Biological Relevance: In cellular contexts, weak ergodicity breaking underlies the large variability in TF binding/unbinding sojourns, illustrated by Lamperti/arcsine occupation statistics, and may buffer low copy-number noise (0704.1769, Jeon et al., 2010).

6. Mathematical and Physical Distinctions from Strong Ergodicity Breaking

• Nature of Nonergodicity: In WEB, all regions of phase/Hilbert space remain in principle accessible; the phase space or spectrum is not partitioned into strictly disjoint sectors. Instead, the inefficiency or slowness of exploration due to heavy tails, constraints, or embedded fragments causes time and ensemble statistics to diverge.
• Thermalization: The thermal eigenstate majority persists; the nonthermal subspace is subextensive, and generic initial conditions thermalize (with characteristic entanglement and observable scaling). In strong ergodicity breaking (e.g., many-body localization), the breakdown is extensive: almost all eigenstates (or initial conditions) fail to thermalize (Papić, 2021, Katsura et al., 2024).

7. Open Issues and Extensions

• Dynamics in the Thermodynamic Limit: The fraction of nonthermal eigenstates in weakly broken ergodic systems generally vanishes as system size grows; yet for experimentally accessible sizes, dynamics from special initial conditions can display long-lived coherent behavior.
• Role of Memory and Nonlocality: Global memory effects can induce WEB even in finite-state random walks with finite mean waiting times, via reinforcement/contagion processes as in generalized urn models (Budini, 2016).
• Breakdown and Crossover: For certain mean-field spin glass models, the classic “aging with loss of memory” scenario may not yield to full WEB; persistent memory of the initial condition can result in strong, not weak, ergodicity breaking, suggesting model-dependence of the generic scenario (Folena et al., 2023).

8. Representative Examples

Class	Model/Mechanism	Key Reference
Subdiffusive CTRW	Power-law waiting times	(Jeon et al., 2010, Carmi et al., 2011, Thiel et al., 2013)
Heterogeneous diffusion	Space-dependent $D(x)$	(Cherstvy et al., 2013)
Quantum scars	Constrained PXP, spin-1 XY, Schwinger QLM	(Serbyn et al., 2020, Schecter et al., 2019, Desaules et al., 2022)
Supersymmetric lattice fermions	Embedded SUSY algebra	(Buijsman et al., 2024)
Random constraints	Disorder-driven transition	(Deger et al., 2023)
Optical sensing	Arcsine law in time-integrated intensity	(Ramesh et al., 2024)
Embedded integrable fragments	Nonproduct MPO projectors	(Katsura et al., 2024)

• "Weak Ergodicity Breaking in Non-Hermitian Many-body Systems" (Chen et al., 2022)
• "Weak Ergodicity Breaking and Quantum Many-Body Scars in Spin-1 XY Magnets" (Schecter et al., 2019)
• "Weak ergodicity breaking in an anomalous diffusion process of mixed origins" (Thiel et al., 2013)
• "Constraint-induced breaking and restoration of ergodicity in spin-1 PXP models" (Mukherjee et al., 2021)
• "Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary" (0704.1769)
• "In vivo anomalous diffusion and weak ergodicity breaking of lipid granules" (Jeon et al., 2010)
• "Weak Ergodicity Breaking in the Schwinger Model" (Desaules et al., 2022)
• "Weak ergodicity breaking in Josephson-junction arrays" (Russomanno et al., 2022)
• "1/f^beta noise in a model for weak ergodicity breaking" (Niemann et al., 2010)
• "Weak ergodicity breaking with isolated integrable sectors" (Katsura et al., 2024)
• "Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes" (Cherstvy et al., 2013)
• "Weak ergodicity breaking transition in randomly constrained model" (Deger et al., 2023)
• "On weak ergodicity breaking in mean-field spin glasses" (Folena et al., 2023)
• "Weak ergodicity breaking induced by global memory effects" (Budini, 2016)
• "Weak ergodicity breaking from supersymmetry in a fermionic kinetically constrained model" (Buijsman et al., 2024)
• "Universality and weak-ergodicity breaking in quantum quenches" (Giachetti et al., 11 Nov 2025)
• "Quantum Many-Body Scars and Weak Breaking of Ergodicity" (Serbyn et al., 2020)
• "Weak ergodicity breaking through the lens of quantum entanglement" (Papić, 2021)
• "A fractional Feynman-Kac equation for weak ergodicity breaking" (Carmi et al., 2011)
• "Weak Ergodicity Breaking in Optical Sensing" (Ramesh et al., 2024)

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This compendium reflects the rich structure and universality of weak ergodicity breaking across classical, stochastic, and quantum many-body domains, and elucidates the precise dynamical, spectral, and statistical mechanisms and diagnostic protocols by which it arises and is quantified.