Entropy Relaxation Approach

• Entropy relaxation approach is a framework that combines entropy functionals with relaxation dynamics to model the evolution of complex physical and information-theoretic systems.
• It integrates methods ranging from many-body quantum dynamics and nonequilibrium thermodynamics to scaling laws in materials science and stable numerical schemes.
• The approach provides actionable insights into irreversible processes by quantifying information loss and guiding robust simulation techniques across diverse scientific domains.

The entropy relaxation approach encompasses a diverse array of methods and theoretical frameworks that combine entropy functionals with relaxation dynamics for modeling, analyzing, and simulating the evolution of complex physical, stochastic, and information-theoretic systems. Across quantum, statistical, materials, and computational mathematics domains, entropy relaxation serves both as a tool for quantifying loss of information under non-equilibrium evolution and as a principle for designing stable, structure-preserving numerical schemes.

1. Information-Theoretic Entropy Relaxation in Many-Body Quantum Dynamics

In closed quantum systems, entropy relaxation is used to quantify the approach of a many-body wave function to a stationary, highly entropic state after a non-equilibrium perturbation. Notably, Bera et al. introduced a many-body Shannon entropy for bosonic systems, defined in terms of the time-dependent amplitudes $C_{\vec{n}}(t)$ of Fock states in a multiconfigurational expansion: $$S^{\mathrm{info}}(t) = -\sum_{\vec{n}} |C_{\vec{n}}(t)|^2 \ln |C_{\vec{n}}(t)|^2$$ This entropy directly measures the delocalization of the full many-body state in Fock space and is sensitive to genuine many-body correlations beyond mean-field (Bera et al., 2019). Following a sudden quench of the interaction strength, $S^{\mathrm{info}}(t)$ exhibits a rapid, nearly linear growth followed by saturation to a plateau (the 'maximum entropy state'), especially for non-local, strongly-coupling dipolar interactions.

The entropy relaxation is inseparably linked to the loss of spatial coherence in first-order Glauber functions and the simultaneous build-up of anti-bunching in second-order correlations. In numerical implementations, the time evolution is governed by the Multiconfigurational Time-Dependent Hartree for Bosons (MCTDHB) method, ensuring self-consistent propagation of both the orbitals and occupation amplitudes. The entropy-relaxed state at long times is characterized by the broadest possible occupation distribution, complete loss of off-diagonal coherence, and pronounced anti-bunching, indicating strongly correlated, highly mixed states inaccessible via mean-field approximations.

2. Entropy Relaxation in Nonequilibrium Thermodynamics and Stochastic Processes

From the perspective of nonequilibrium thermodynamics, entropy relaxation is formalized as a monotonic decrease of the Kullback-Leibler (KL) divergence between a time-dependent probability distribution $p(x,t)$ and the maximal entropy equilibrium distribution $p^*(x;\lambda)$. Altaner defines strongly relaxing dynamics as any evolution satisfying

$\frac{d}{dt} D[p(t) \| p^*(t)] \leq 0$

for all $t$, where $D[p\|p^*]=\int p(x) \ln\frac{p(x)}{p^*(x)} dx$ (Altaner, 2017). This is equivalent to the Second Law of thermodynamics, ensuring irreversibility as a continuous loss of information about the microstate. For Markov processes with detailed balance, this monotonic KL contraction is guaranteed by the properties of the infinitesimal generator.

Within this framework, the entropy production (or relative entropy dissipation) quantifies the rate at which the system approaches equilibrium and is a Lyapunov functional under broad dynamical classes. In open or driven systems, generalized bounds such as the non-isothermal Landauer principle naturally arise through the same KL-based entropy relaxation formalism.

3. Entropy Relaxation and Scaling Laws in Materials Science

In complex amorphous materials and soft matter, where microscopic configurations determine macroscopic response, entropy relaxation provides direct empirical and theoretical connections between structure and dynamical relaxation. Under oscillatory shear, the relaxation rate of plastic flow (measured from the decay of self-intermediate scattering functions) is observed to scale exponentially with the system's instantaneous two-body excess entropy, $S_2$: $$\tau_{\alpha}(t-t_d) \sim \exp\left(-c \frac{S_2(t)}{k_B}\right)$$ The lag $t_d$ is proportional to the relaxation time itself, such that static structural measures at time $t$ predict dynamical relaxation at $t-t_d$ (Galloway et al., 2021). This entropy-relaxation scaling is robust across mono- and bidisperse colloidal samples and provides a practical method for inferring dynamic properties (e.g., viscosity, yield) from single-frame structural data, bypassing the need for time-resolved trajectory analysis. The approach generalizes to contexts where local shear-induced rearrangements are the primary ingredients of relaxation, subject to the dominance of two-body correlations in defining $S^{\mathrm{ex}}$.

4. Entropy-Relaxation in Computational and Numerical Methods

A distinct, yet mathematically related, strand of entropy relaxation appears in numerical analysis for hyperbolic conservation laws and viscous PDEs. The central construct is the design of fully discrete time integration schemes (such as explicit, implicit, or multirate Runge-Kutta methods) that guarantee discrete entropy dissipation or conservation.

The relaxation Runge-Kutta (RRK) paradigm introduces a relaxation parameter $\gamma_n$ at each step: $$u^{n+1}_\gamma = u^n + \gamma_n \sum_{i=1}^s b_i f_i$$ where $\gamma_n$ is chosen via a scalar nonlinear equation to satisfy

$\eta(u^{n+1}_\gamma) - \eta(u^n) \leq 0$

for a selected convex entropy functional $\eta$ (Ranocha et al., 2019, Kang et al., 2021, Doehring et al., 7 Jul 2025). The existence and uniqueness of $\gamma_n$ is guaranteed by convexity, and the entropy condition holds provided the semi-discretization is entropy-conservative or entropy-stable in the sense of Tadmor.

The entropy relaxation approach is readily extended to partitioned and multirate integration, allowing physically principled, robust time stepping in systems with strong scale separation or stiffness. For quadratic entropy functions (e.g., $S(u) = \frac{1}{2}\|u\|^2$), the relaxation parameter admits a closed-form expression; for more general entropies, a scalar root-finder with an initial guess at $\gamma_n=1$ suffices.

5. Relative Entropy Methods for Relaxation Limits in PDEs

In mathematical analysis of kinetic and hydrodynamic models, entropy relaxation is formalized via the method of relative entropy (sometimes termed modulated energy). This framework quantifies the distance between a non-equilibrium solution and an equilibrium or reduced (e.g., diffusive or overdamped) limit.

Given a convex entropy $\mathcal{H}(f)$ for a kinetic system and a target limit equilibrium $g$, the relative entropy is

$$\mathcal{H}(f|g) = \mathcal{H}(f) - \mathcal{H}(g) - \langle \mathcal{H}'(g), f-g \rangle$$

By exploiting the entropy structure, one can derive a differential inequality with a strictly non-negative dissipation term, which directly yields stability and convergence results: as the relaxation parameter $\varepsilon \to 0$, solutions converge to the reduced model at a quantifiable rate (Bianchini, 2019, Lattanzio et al., 2012, Tzavaras, 2014, Carrillo et al., 2019, Chen et al., 2023). Applications encompass the rigorous derivation of porous media and aggregation-diffusion equations from kinetic or hydrodynamic models with friction, relaxation of stress models in polyconvex elastodynamics, and the design of entropy dissipative schemes for nonlinear viscous conservation laws.

6. Mesoscopic and Quantum Extensions of Entropy Relaxation

At the mesoscopic level, entropy relaxation principles are embedded in Fokker–Planck models for relaxation of vectorial or orientational degrees of freedom, such as dielectric polarization. Using the Gibbs entropy postulate and mesoscopic chemical potential, one derives nonlinear Fokker–Planck equations, where entropy production underpins the Onsager structure and determines relaxation times under external fields (Méndez-Bermúdez et al., 2010). In homogeneous and inhomogeneous contexts, these approaches quantitatively capture kinetic relaxation times and multi-modal loss features observed in experiments.

In closed quantum systems, recent approaches use the local relative entropy $S(\rho_A(t)\|\rho_A(\infty))$ as a diagnostic for relaxation, showing it can be reliably approximated by the difference in von Neumann entropies, $S(\rho_A(\infty)) - S(\rho_A(t))$ (Ares et al., 8 Jul 2025). This provides both theoretical insight (relating relaxation to entanglement build-up and symmetry restoration) and numerical advantage in simulations of the quantum Mpemba effect and related non-equilibrium phenomena.

7. Insights, Limitations, and Practical Implications

Entropy relaxation, in its various guises, provides a unifying language and method for describing the loss of information and the approach to equilibrium (or maximal disorder) across physical, mathematical, and data-driven settings.

• In quantum and classical many-body systems, it supplies direct measures of dynamical mixing and the emergence of highly correlated, dephased states.
• In thermodynamic and stochastic processes, it grounds the Second Law, detailed fluctuation theorems, and bounds on irreversibility as strict statements about information contraction.
• In materials science, it enables predictive modeling of dynamic responses based on static structure, with clear empirical scaling laws.
• In computational mathematics, entropy relaxation is crucial for constructing stable, high-order numerical integrators consistent with the underlying physics even for stiff and multiscale problems.
• In PDE and kinetic theory, relative entropy serves as a Lyapunov function for stability analysis and provides explicit convergence rates for singular and asymptotic limits.

Limitations of entropy relaxation arise when the underlying entropy structure is not convex (e.g., polyconvexity only), when regularity conditions required for the proofs are violated (e.g., in the presence of shocks or singularities), or when additional constraints (e.g., strong multibody, topological, or quantum coherence effects) invalidate the closure or dissipation properties assumed.

In sum, entropy relaxation forms a cornerstone of modern analysis and simulation in both equilibrium and non-equilibrium sciences, enabling rigorous derivations, robust numerics, and physical understanding of relaxation processes driven by the universal principle of entropy increase or information loss.