🔬 condensed matter

Restoring detailed balance in non-Hermitian Markov processes

This paper introduces a general and computationally tractable method using Dyson maps to restore monotonic entropy growth in non-Hermitian Markov processes with broken detailed balance, thereby enabling the application of standard statistical physics tools and inference techniques to a broad class of non-equilibrium systems.

Original authors: Tim Van Wesemael, Gilberto Nakamura, Jan Baetens, Odemir M. Bruno, Alexandre S. Martinez, Christophe Deroulers

Published 2026-06-17

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CC BY 4.0

Original authors: Tim Van Wesemael, Gilberto Nakamura, Jan Baetens, Odemir M. Bruno, Alexandre S. Martinez, Christophe Deroulers

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

Imagine you are watching a chaotic game of "musical chairs" played by a crowd of people. Some people are standing up (infected), and some are sitting down (healthy). They move around, stand up, and sit down based on certain rules. In the world of science, this is called a stochastic process—a system driven by chance and uncertainty.

Usually, scientists try to understand these systems by looking at entropy. Think of entropy as a measure of "messiness" or "uncertainty." In a perfect, calm world, you'd expect the messiness to either stay the same or get messier over time, never getting tidier on its own. This is the "arrow of time."

However, the paper explains that in many real-world systems (like the spread of a disease or a rumor), the "messiness" doesn't behave nicely. It goes up, then down, then up again. It's non-monotonic. This makes it very hard for scientists to use their standard mathematical tools to predict what will happen next, because those tools rely on the idea that things always get messier (or stay the same) until they settle down.

The Problem: A Broken Mirror

The authors describe the system using a mathematical object called a matrix (let's call it H). In a "nice" system, this matrix is symmetric, like a perfect mirror. If you look at the reflection, it matches the object. But in these chaotic, out-of-balance systems, the matrix is non-Hermitian.

Imagine looking into a funhouse mirror that distorts your image. The reflection is there, but it's warped. Because of this warping, the "messiness" (entropy) of the system behaves strangely, sometimes decreasing when it should be increasing. This breaks the standard rules of physics that scientists use to make sense of the world.

The Solution: The "Dyson Map"

The authors introduce a clever mathematical trick called a Dyson map. Think of this as a special pair of glasses or a lens.

When you put these "glasses" on the distorted, funhouse-mirror system, something magical happens:

The distortion disappears: The warped matrix transforms into a perfect, symmetric mirror (a Hermitian matrix).
The rules are restored: Suddenly, the "messiness" (entropy) starts behaving normally. It grows steadily and smoothly until the system reaches a calm, stable state.
The view is still accurate: Even though the image looks different through the glasses, the underlying information is preserved. You can still calculate the exact same probabilities and outcomes as before, you just have to apply a small "correction factor" (called a metric) to your final answer.

The Analogy: Straightening a Wobbly Table

Imagine a wobbly table (the original system) that keeps tipping left and right. It's hard to balance a cup of coffee on it because the table's movement is unpredictable.

The Dyson map is like adding a set of adjustable legs to the table. You don't move the table itself; you change the perspective from which you view it. Suddenly, the table looks perfectly level. The coffee cup sits still. The "wobble" is gone.

However, the table is still sitting on the same floor. If you want to know exactly where the table is in the room, you have to remember that you added those legs. The paper provides the math to translate the "level table" view back into the "wobbly table" reality, ensuring no information is lost.

What They Tested: A Disease Model

To prove this works, the authors applied their method to a model of disease spreading (called the SIS model).

Before the fix: They watched the "uncertainty" of the disease spread. It went up as the disease spread, then went down as it settled, then went up again. It was a confusing, jagged line.
After the fix: Using their Dyson map, they watched the same process. The uncertainty went up smoothly and steadily, like a ball rolling down a hill, until it reached the bottom.

This smooth, predictable behavior allowed them to easily find the "tipping point" (phase transition) where the disease goes from dying out to becoming an epidemic. In the original, broken view, this was much harder to spot.

The Big Picture

The paper claims that by using this "lens" (the Dyson map), scientists can take messy, unpredictable systems that break the standard rules of thermodynamics and transform them into clean, orderly systems that follow the rules.

This doesn't change the physical reality of the system; it just changes the mathematical language used to describe it. It allows researchers to use powerful, standard tools to solve problems that were previously too chaotic to handle, such as predicting how a disease spreads or how a rumor travels through a crowd, by first "straightening out" the math.

In short: The paper offers a way to turn a chaotic, confusing math problem into a neat, orderly one, solve it easily, and then translate the answer back to the real world without losing any details.

Technical Summary: Restoring Detailed Balance in Non-Hermitian Markov Processes

Problem Statement
Stochastic processes out of equilibrium, particularly those modeled by Markov processes with discrete states, frequently involve asymmetric transition rates that break detailed balance. These non-Hermitian stochastic generators ( $H$ ) lead to non-monotonic entropy production, where entropy may decrease after transient increases. This behavior undermines standard thermodynamic interpretations and inference techniques that rely on entropy maximization or the assumption of ever-increasing uncertainty. Furthermore, the lack of time reversibility in these systems complicates the physical interpretation of equilibrium states, which may encode low-entropy configurations. While tensor network methods address the "curse of dimensionality" in high-dimensional interacting systems, they do not resolve the fundamental issue of non-Hermitian dynamics and the resulting non-monotonic entropy.

Methodology
The authors propose the use of Dyson maps ( $\eta$ ) to transform non-Hermitian stochastic generators into Hermitian operators while preserving the real eigenvalue spectrum. The core methodology involves:

Dyson Transformation: A reversible transformation is applied to the master equation $\partial_t |P(t)\rangle = -H|P(t)\rangle$ , yielding a transformed equation $\partial_t |\phi(t)\rangle = -\tilde{H}|\phi(t)\rangle$ , where $|\phi(t)\rangle = \eta|P(t)\rangle$ and $\tilde{H} = \eta H \eta^{-1}$ .
Metric Operator: The transformation introduces a metric operator $\Omega = \eta^\dagger \eta$ , which re-weights contributions in the original system. This operator connects measurements in the transformed coordinates back to the original probability vector, ensuring that statistical averages $\langle O \rangle$ can be recovered via $\langle \phi | \tilde{\Omega} \Xi' O' | \phi \rangle$ .
Entropy Restoration: In the transformed system, the Rényi entropy $S' = -\ln |\phi(t)|^2$ is proven to increase monotonically over time ( $\frac{1}{2}\frac{d}{dt}S'(t) \geq 0$ ). This contrasts with the original system, where the squared norm $|P(t)|^2$ (and thus the entropy) can exhibit local minima.
Computational Construction:
- Analytical Approach: The authors seek a Hermitian generator $\Lambda = \ln \eta$ within a closed operator algebra. By expanding the condition $H = H^\dagger$ in terms of commutators, they derive a set of algebraic equations (involving coefficients $G_{ijk}$ ) to solve for $\Lambda$ . This approach assumes a closed structure under the commutator, which may not hold for all non-trivial interacting problems.
- Numerical Approach: For general cases where analytical closure is difficult, an iterative numerical algorithm is proposed. This method treats $\Lambda$ as a rotation axis in the algebra, performing iterative rotations to suppress the anti-Hermitian component ( $\Delta H$ ) of the generator until a tolerance is met.

Key Results
The methodology was applied to a Susceptible-Infected-Susceptible (SIS) epidemic model on a static contact graph:

Monotonic Entropy Growth: In the transformed system, the Rényi entropy $S'$ increases monotonically until equilibrium is reached, whereas the original system's entropy exhibits a transient maximum followed by a decrease.
Statistical Equivalence: Despite the transformation, the statistical averages (mean and standard deviation of infectious vertices) computed in the transformed system match the original system's results with high precision (Kolmogorov-Smirnov distance $< 10^{-3}$ ), provided the metric operator is correctly applied.
Phase Transition Detection: The maximum of the Rényi entropy in both the original and transformed systems coincides at the same critical ratio of infection to recovery rates ( $\beta/\gamma$ ). This confirms that the transformed system, despite having restored time reversibility, retains the signature of the phase transition found in the original non-Hermitian system.
Structure Preservation: The steady-state distributions in the transformed system ( $|\phi\rangle$ ) preserve the structural indices of the original distribution ( $|P\rangle$ ), suggesting that diagonal contributions of the Dyson map govern the transformation while off-diagonal components serve to mix components to increase entropy.

Significance and Claims
The paper claims to provide a general and computationally tractable method to restore detailed balance in a broad class of Markovian systems. By converting non-Hermitian generators into Hermitian ones, the approach imbues these systems with a "familiar physical interpretation" characterized by strictly positive entropy production. This allows for the application of standard statistical physics tools, such as entropy maximization, to determine equilibrium states.

The authors highlight that this formalism establishes a connection between stochastic processes and quantum dynamics (via Wick rotation $t \to i\tau$ ) for systems with real, non-negative spectra. This analogy suggests that quantum simulators could potentially be utilized to solve difficult classical stochastic problems dominated by noise and fluctuations. However, the authors modestly note limitations: the current approach requires real spectra and cannot directly address systems with complex eigenvalues (oscillatory behavior), and the analytical construction relies on specific algebraic closures that may not exist for all interacting problems. The method effectively trades the complexity of non-monotonic entropy for the complexity of computing statistical averages via the metric operator $\tilde{\Omega}$ .