⚛️ high-energy theory

On the relaxation dynamics of non-equilibrium quantum systems

This paper investigates the relaxation dynamics of approximately conserved charges in interacting quantum systems near local equilibrium by demonstrating that Zubarev's non-equilibrium statistical operator approach and a simpler local-equilibrium construction both yield a leading relaxation rate determined by equilibrium correlation functions, which can be extended to diffusion-relaxation equations and validated against kinetic descriptions in models like electroweak B+L washout.

Original authors: Matthias Carosi, Björn Garbrecht, Silvia Pla, Nils Wagner, Edward Wang

Published 2026-09-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthias Carosi, Björn Garbrecht, Silvia Pla, Nils Wagner, Edward Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, humming machinery of the universe, from the seething plasma of the early cosmos to the quiet hum of a laboratory gas, there is a constant tension between order and chaos. At the smallest scales, particles interact in a frantic, complex dance, governed by strict laws that usually keep certain quantities, like electric charge or particle number, perfectly constant. However, nature is rarely perfect. Sometimes, these rules are slightly bent, allowing a conserved quantity to slowly leak away. When this happens, the system does not collapse instantly; instead, it drifts toward a new state of balance, a process known as relaxation. Understanding how fast this drift occurs is crucial for physicists. It tells us how long the universe held onto specific conditions after the Big Bang, or how long a quantum computer might keep its delicate information before the environment scrambles it. For decades, scientists have relied on two main ways to calculate this speed: one that tracks individual particle collisions like a traffic report, and another that looks at the collective behavior of the system as a whole, like watching the flow of a river.

A team of researchers has now bridged the gap between these two perspectives, showing that they are actually describing the same underlying reality. In a new study, they focused on a specific type of quantum system where a charge is almost, but not quite, conserved. They wanted to understand the precise mechanism by which this charge fades away over time. To do this, they revisited a sophisticated mathematical framework developed in the 1970s, known as the non-equilibrium statistical operator approach. This method is powerful but notoriously difficult to use, often requiring complex calculations that obscure the simple physical reasons why things relax. The researchers stripped away the unnecessary complexity, creating a clear, step-by-step guide that reveals exactly how the system forgets its past and settles into equilibrium. They demonstrated that the rate at which the charge disappears is determined entirely by how the system fluctuates when it is already in a state of balance. In essence, the speed of the decay is a direct reflection of the system's natural, random jiggling when it is at rest.

The team did not stop at theory. They tested their new, simplified method against two very different physical scenarios. First, they looked at a theoretical model involving charged particles that can decay into neutral ones. In this setting, they could calculate the relaxation rate using their new method and compare it directly with the traditional "traffic report" approach, known as the Boltzmann equation. The results matched perfectly, confirming that their streamlined approach captures the same physics as the more cumbersome, standard techniques. Second, they applied their findings to a profound cosmological problem: the disappearance of matter in the early universe. Specifically, they examined how the combined number of protons and neutrons (baryons) and their antimatter counterparts (leptons) was washed out by the extreme conditions of the infant cosmos. Their work showed that the rate at which this matter vanished is directly linked to how quickly topological defects in the fabric of space-time diffuse, a connection that had been suspected but not so clearly derived.

What makes this work particularly significant is not just that it confirms old results, but that it clarifies why those results are true. The researchers highlighted three key conditions that must be met for a system to relax in this predictable, smooth way. First, the violation of the conservation law must be weak; the charge must be able to survive for a long time. Second, there must be a clear separation between the fast, chaotic interactions of individual particles and the slow, steady drift of the overall charge. Third, the system must lose its memory of the specific details of its past very quickly. When these conditions are met, the complex history of the system becomes irrelevant, and the future evolution depends only on the current state. The researchers showed that whether one uses the heavy machinery of the non-equilibrium statistical operator or a simpler, more intuitive construction based on local equilibrium, the answer is the same.

By providing a pedagogical review and a simplified alternative derivation, the authors have made a powerful tool more accessible to the broader scientific community. They demonstrated that the relaxation rate is not a mysterious parameter to be guessed, but a quantity that can be calculated directly from the equilibrium properties of the system. This means that to predict how a quantum system will behave out of balance, one does not need to simulate the entire chaotic history of its evolution. Instead, one only needs to measure or calculate how the system fluctuates when it is calm. This insight simplifies the study of non-equilibrium dynamics, offering a direct path to understanding phenomena ranging from the behavior of exotic quantum gases to the evolution of the universe itself. The work stands as a reminder that even in the most complex quantum systems, the path to equilibrium is governed by simple, universal principles that can be uncovered with the right perspective.

Technical Summary: On the relaxation dynamics of non-equilibrium quantum systems

Problem Statement
The paper investigates the relaxation dynamics of an approximately conserved charge in interacting quantum systems close to local equilibrium. While interacting quantum systems possess complex microscopic degrees of freedom, their late-time behavior is often governed by a few slow variables, such as charges associated with weakly broken symmetries. The central challenge is to derive a local relaxation law for these charges, connecting the macroscopic decay rate to the microscopic properties of the underlying theory. Specifically, the authors aim to rigorously derive the relaxation rate $\Gamma_{\text{relax}}$ for a charge $Q$ that is weakly violated by a perturbation $\lambda H^{(\text{viol})}$ , where $\lambda \ll 1$ , and to establish its connection to equilibrium correlation functions and diffusion rates.

Methodology
The authors employ two complementary theoretical frameworks to derive the relaxation rate, ensuring consistency between a formal statistical operator approach and a more intuitive local-equilibrium construction.

Zubarev's Non-Equilibrium Statistical Operator (NSO) Approach:
- The authors provide a pedagogical review and detailed derivation using Zubarev's NSO method. They construct a non-equilibrium statistical operator $\rho^{(\text{NEQ})}(t)$ based on a set of relevant observables (total energy and the charge $Q$ ).
- The construction involves a "local equilibrium" (LEQ) operator $\rho^{(\text{LEQ})}(t)$ , which maximizes entropy locally in time given instantaneous averages, and a retarded averaging procedure over past times to enforce causality and the vanishing of initial correlations.
- Key assumptions include a separation of timescales between the microscopic equilibration time $\tau_{\text{micro}}$ and the macroscopic relaxation time $\tau_{\text{macro}}$ (where $\tau_{\text{micro}} \ll \tau_{\text{macro}} \sim \lambda^{-2}$ ).
- The derivation utilizes a Markovian approximation, justified by the rapid decay of microscopic correlations, to linearize the dynamics near equilibrium.
Simplified Local-Equilibrium Approach:
- As an alternative, the authors propose a direct derivation based on the evolution of a local-equilibrium state over a mesoscopic time interval $\Delta t$ ( $\tau_{\text{micro}} \ll \Delta t \ll \tau_{\text{macro}}$ ).
- This method avoids constructing the full non-equilibrium operator. Instead, it evolves the LEQ state forward in time using the interaction picture with respect to the charge-conserving Hamiltonian.
- The finite change in the charge over $\Delta t$ isolates the slow drift, yielding the relaxation rate directly from the microscopic correlators without requiring the self-consistency conditions inherent to the full NSO formalism.
Connection to Diffusion and Kinetic Theory:
- The authors relate the derived relaxation rate to the equilibrium diffusion rate of the same charge using the fluctuation-dissipation theorem.
- They generalize the formalism to inhomogeneous systems, deriving a diffusion-relaxation equation.
- To validate the formalism, they apply the results to two specific models: a perturbative charged scalar field and the electroweak $B+L$ (baryon plus lepton) anomaly. In the scalar model, they explicitly compare the correlator-based result with the linearized Boltzmann equation.

Key Contributions and Results

Derivation of the Relaxation Rate: The paper explicitly derives the leading-order relaxation rate $\Gamma_{\text{relax}}$ as an equilibrium correlation function of the charge-violating operator $J = i[H, Q]/\hbar$ . The result is given by the Green-Kubo type relation:
$\Gamma_{\text{relax}} = \frac{1}{\langle Q^2 \rangle_{\text{EQ}}} \int_0^\infty dt \, \langle J(t) J(0) \rangle_{\text{Bogoliubov}}$
where the Bogoliubov inner product accounts for thermal smearing. The authors show that under the relevant assumptions, this reduces to the standard equilibrium correlation function limit.
Clarification of Assumptions: The work explicitly highlights the physical assumptions required for a local relaxation law: weak charge violation, a clear separation of timescales between microscopic correlations and macroscopic relaxation, and the resulting loss of microscopic memory (Markovian limit).
Equivalence of Approaches: The paper demonstrates that the simplified local-equilibrium construction yields the same leading-order result as the more complex Zubarev NSO approach. This provides a direct, practical alternative for calculations while making the common physical ingredients (scale separation, memory loss) explicit.
Relation to Diffusion: A direct link is established between the relaxation rate and the diffusion rate $\gamma_{\text{diff}}$ via the Einstein relation:
$\Gamma_{\text{relax}} = \frac{V}{\langle Q^2 \rangle_{\text{EQ}}} \gamma_{\text{diff}}$
where $V$ is the volume. This connects the decay of a homogeneous charge to the spatial diffusion of charge fluctuations.
Validation via Examples:
- Scalar Model: For a charged scalar field weakly coupled to a neutral scalar, the correlator-derived relaxation rate is shown to match exactly the result obtained from the linearized Boltzmann equation. This establishes a direct connection between equilibrium-correlator and kinetic descriptions in a perturbative regime.
- Electroweak $B+L$ Washout: The formalism is applied to the early universe, relating the relaxation of the anomalous $B+L$ charge to the Chern-Simons diffusion rate (sphaleron rate), recovering known results in this context.

Significance and Claims
The authors claim that their work offers a clearer and more explicit derivation of the relaxation rate for weakly violated charges compared to existing literature, which often presents these derivations in a compressed form that obscures the role of timescale separation and memory suppression. By providing a simplified local-equilibrium alternative, the paper aims to facilitate practical calculations while maintaining rigorous physical grounding.

The paper modestly notes that while the NSO formalism provides a self-consistent scheme, it possesses an ambiguity at next-to-leading order (NLO) in the symmetry-breaking parameter $\lambda$ due to different possible definitions of the local equilibrium operator. However, the authors state that their analysis is restricted to leading-order effects, where this ambiguity does not affect the results. The work serves to unify the statistical operator approach with kinetic theory and diffusion concepts, providing a robust framework for studying non-equilibrium dynamics in systems with approximately conserved quantities.

More like this