Modified Effective Thermalization in Quantum Systems

• Modified effective thermalization is a framework that replaces conventional thermalization with engineered, deterministic, and algorithmic dynamical laws to control quantum equilibration.
• In Floquet and disordered systems, modified ETH scalings capture anomalous transport and subdiffusive relaxation by incorporating non-random functions and power-law corrections.
• Algorithmic and stochastic protocols enable fine-grained control of thermalization processes, enhancing performance in simulations, work extraction, and other quantum thermodynamic tasks.

Modified effective thermalization describes a broad set of mechanisms in quantum many-body dynamics, control, and simulation, in which the approach to equilibrium or canonical ensemble statistics is governed by anomalous, engineered, or explicitly modified dynamical laws. These frameworks depart from conventional textbook thermalization—typically associated with strong ergodicity, random-matrix-type ETH, or Markovian weak-coupling limits—by replacing or augmenting standard ingredients: introducing deterministic ETH, generalized dissipation, algorithmic protocols, nontrivial scaling, memory effects, and explicit microcanonical relaxation, among other phenomena. The range of "modification" includes both mathematically rigorous changes to the underlying dynamical ansatz and algorithmic or physical protocols designed to tune the convergence to equilibrium. What unites these diverse developments is the explicit quantification, and in many cases the control or characterization, of the effective approach to thermal steady-states under constraints not encompassed by conventional frameworks.

1. Deterministic Reformulations of the Eigenstate Thermalization Hypothesis

The conventional ETH asserts that for a non-integrable system, the matrix elements of a few-body observable $A$ in the eigenbasis of the Hamiltonian $H$ are given by

$$A_{ab} = A(E)\,\delta_{ab} + e^{-S(E)/2} f(E,\omega) R_{ab}$$

where $E = \frac{E_a+E_b}{2}$ is the mean energy, $S(E)$ is the thermodynamic entropy, $f$ is a smooth envelope, and $R_{ab}$ are "random" variables (usually modeled as Gaussian). This ansatz combines deterministic and random ingredients, presenting an inconsistency when applied to a fixed, non-random Hamiltonian.

A deterministic reformulation eliminates this mixing by introducing a bounded, non-random function $g_{ab}(E,\omega)$ of order unity in place of $R_{ab}$: $$A_{ab} = A(E)\,\delta_{ab} + e^{-S(E)/2} g_{ab}(E,\omega)$$ All quantities are now deterministic functionals of the system parameters and observable, resolving the logical inconsistency of the original ansatz for a particular realization of a physical system. Key structural results derived from this deterministic ETH include:

• The agreement of $A(E)$ with the equilibrium (microcanonical or canonical) average up to $O(1/N)$ corrections for large system size.
• The suppression of off-diagonal elements by $e^{-S(E)/2}$, ensuring only exponentially small temporal fluctuations in observables.
• The sufficiency of the deterministic scaling structure (rather than explicit randomness) for establishing thermalization of few-body observables in finite, non-random systems (Inozemcev et al., 2018).

2. Modified Thermalization Mechanisms in Floquet and Driven Many-Body Systems

Time-periodic (Floquet) quantum systems require modified effective thermalization frameworks due to the presence of non-trivial stroboscopic evolution and, often, absence of energy conservation. For a system with $T$-periodic Hamiltonian $H(t)$, one constructs the effective (Floquet) Hamiltonian $H_{\text{eff}}$ governing evolution over a period. The infinite-time behavior of local observables is characterized by their diagonal ensemble in the eigenbasis of $H_{\text{eff}}$: $$\overline{\langle A \rangle} = \sum_\alpha |C_\alpha|^2 \langle \phi_{\alpha 0}| A_{\text{eff}}^{[0]} | \phi_{\alpha 0} \rangle$$ where $|\phi_{\alpha 0}\rangle$ are $H_{\text{eff}}$ eigenstates, and $\overline{\langle A \rangle}$ denotes the long-time average.

If $H_{\text{eff}}$ is non-integrable and satisfies the ETH, thermalization follows automatically in the sense that long-time observables relax to a microcanonical average determined by the initial state's energy with respect to $H_{\text{eff}}$. This generalized effective ETH for periodically driven systems accommodates stroboscopic dynamics and underpins thermalization in a wide class of quantum Floquet systems (Liu, 2014).

When disorder is present, as in disordered Floquet spin chains, thermalization becomes anomalous: local observables exhibit subdiffusive relaxation and the variance of off-diagonal matrix elements acquires a power-law correction in system size beyond the standard exponential decay. The relation between the scaling exponents and dynamical transport (e.g., $z=1/\gamma$ for dynamical exponent $z$, power-law decay exponent $\gamma$) is preserved, leading to a "modified ETH" ansatz of the form

$$\mathrm{Var}(O_{\alpha\beta}) \sim e^{-S} L^{-\,\alpha(W)}$$

with $\alpha(W)$ disorder-dependent, indicating that the fundamental structure of ETH endures but with crucial modifications dictated by anomalous transport (Roy et al., 2018).

3. Algorithmic and Markovian Protocols: Controlled Modified Effective Thermalization

When aiming to realize specific population transformations or optimize thermodynamic tasks (work extraction, cooling, catalysis), one can algorithmically design Markovian thermal processes (MTPs) that implement modified effective thermalization. The key mathematical criterion is continuous thermomajorisation: the time evolution of state populations must respect monotonicity constraints for divergences $$\Sigma_a(t) = -\sum_i |p_i(t) - a\,\gamma_i/\gamma_d|$$, $a\in[0,1]$, where $\gamma$ are the populations of the Gibbs state. In this construct:

• Every MTP can be realized as a concatenation of two-level elementary thermalizations, parameterized by explicit relaxation times.
• The reachable set of populations from a given initial state is enumerated algorithmically, subject to endpoint inequalities that impose necessary and sufficient conditions for reachability within the continuous thermomajorisation framework.
• Compared to standard thermal operations, these MTPs provide strictly tighter bounds for population transfers and permit fine-grained, optimal control protocols—enabling rigorous quantification of memory effects, resource costs, and the performance limits of thermalization protocols for small quantum systems (Korzekwa et al., 2022).

4. Stochastic, Objective, and Non-Unitary Dynamical Modifications

Frameworks modifying the unitary, time-reversal symmetric foundations of quantum evolution to embed an objective, dynamical notion of quantum thermalization have been developed. The Objective Quantum Thermalization (OQT) model introduces a stochastic, fluctuation-dissipation-driven modification of the Schrödinger equation, such that

$d\ket\psi = -i\,H\,\ket\psi\,dt + \sum_{\mu,\nu} \left[ \sqrt{\Acal^\mu} (\hat L_{\mu\nu}-\langle\hat L_{\mu\nu}\rangle) dW_t^{\mu\nu} + \Acal^\mu (\ldots) dt \right] \ket\psi$

with $\Acal^\mu$ scaling as the system size and encapsulating microcanonical weights. The associated ensemble master equation relaxes any state to the microcanonical ensemble with a characteristic thermalization time $\tau_{\text{therm}}\sim 1/(\alpha\,\mathcal N)$. All non-unitary corrections vanish for small-system (atomic-scale) limits, preserving standard quantum mechanics.

The model guarantees:

• Norm preservation via an explicit fluctuation-dissipation relation.
• Conservation of average energy and approach to a unique microcanonical fixed point.
• Objective, state-by-state irreversibility and thermalization, independently of agent-defined coarse-grainings.
• Generalization to hybrid theories, resolving both measurement collapse and thermalization dynamics for macroscopic quantum systems (Mukherjee, 22 Apr 2025).

5. Anomalous and Non-Standard Scaling in Modified Thermalization

Many-body systems in which conventional diffusion or ergodicity is absent nonetheless exhibit modified effective thermalization. In subdiffusive ergodic systems, as typified by the random-field Heisenberg chain, off-diagonal elements of local observables' matrix representations scale as

$$A_{\alpha\beta} \sim e^{-S(E)/2} L^{\delta} R_{\alpha\beta}, \quad \delta = \frac{1-\gamma}{2\gamma}$$

where $\gamma$ is the long-time decay exponent of autocorrelation functions, and $L$ the system size. The envelope function $f(E,\omega)$ develops a frequency-dependent power-law, reflecting slow, anomalous thermalization dynamics. This modification is analytically tied to subdiffusive transport, breaking the direct equivalence between ergodicity (in the ETH sense) and diffusive relaxation. Empirically, the variance scaling and non-Gaussian statistics of the matrix element distributions verify this anomalous structure in numerical studies (Luitz et al., 2016, Roy et al., 2018).

6. Algorithmic and Numerical Modified Effective Thermalization in Simulation and Control

Entropic regularization approaches for rate-independent processes, and modified thermalization algorithms for wavefunction and kinetic plasma simulation, provide concrete instances of engineered effective thermalization. In entropic regularization, the deterministic, rate-independent variational dynamics are "thermalized" by introducing a small entropic penalty, yielding an effective non-linear gradient flow characterized by

$$\dot{y}(t) = -\theta D \widetilde{\Psi}^*(\nabla_x E(t, y(t)))$$

where $\widetilde{\Psi}^*$ is a logarithmic moment generating functional reflecting the "softened" dissipation due to thermal contact. In numerical plasma simulations, algorithmic (numerical) drag and diffusion coefficients, set by grid and macroparticle parameters, define a "numerically induced" effective Maxwellization time. Control strategies—particle count, grid resolution, advanced solvers—quantitatively mediate the rate of this synthetic thermalization in relation to real collisional processes (Sullivan et al., 2012, Jakucionis et al., 2023, Jubin et al., 2024).

7. Summary of Theoretical and Practical Implications

Modified effective thermalization frameworks expose the diversity of physical, mathematical, and algorithmic mechanisms underpinning equilibrium formation in closed, open, or engineered quantum systems:

• Deterministic reinterpretations clarify foundational consistency in ETH.
• Periodically driven and disordered systems require modified ETH scalings, with explicit links to anomalous transport exponents.
• Markovian process optimization and continuous thermomajorisation enable rigorous control and resource accounting in finite-dimensional thermalization, providing tight performance limits beyond traditional thermodynamic limits.
• Stochastic dynamical extensions (OQT/SUI) yield testable predictions for the scaling of relaxation and entropy production, bridging the gap between quantum microdynamics and macroscopic irreversibility.
• Engineered and algorithmic techniques in simulation and experimental platforms allow direct manipulation, acceleration, or mitigation of effective thermalization, with quantifiable impacts on observable relaxation and equilibrium statistics.

The study of modified effective thermalization thus synthesizes developments across deterministic ETH, system control, anomalous transport, algorithmic design, and foundational quantum statistical mechanics, providing a flexible and quantitative framework for understanding and exploiting equilibration in complex quantum systems (Inozemcev et al., 2018, Liu, 2014, Korzekwa et al., 2022, Mukherjee, 22 Apr 2025, Luitz et al., 2016, Roy et al., 2018).