🔬 condensed matter

State convertibility and fluctuation theorems from a dynamical reference: majorization meets martingales

This paper unifies and extends state convertibility criteria to arbitrary time-dependent reference distributions by introducing $g(t)$ -majorization and a martingale-based dual picture, which further enables the derivation of an exact fluctuation theorem that serves as a model-independent diagnostic for certifying reference evolution errors via $\chi^2$ -divergence bounds.

Original authors: Davide Cugini, Giacomo Guarnieri

Published 2026-08-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Davide Cugini, Giacomo Guarnieri

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

Thermodynamics is the science of how energy moves and changes form, a field built on the idea that the universe has a preferred direction, always flowing from order to disorder. For over a century, physicists have relied on a specific, stable state of equilibrium to measure this flow, treating it as a fixed backdrop against which all physical changes are judged. This backdrop acts like a ruler; if a system changes in a way that respects the ruler's markings, the change is allowed. If it tries to go against the grain, nature forbids it. However, this approach assumes the ruler itself never moves. In the real world, from the beating of a heart to the operation of a quantum computer, the conditions defining what is "normal" are often shifting, stretching, and evolving in time. When the reference point itself is in motion, the old rules for what is possible and what is not become difficult to apply, leaving scientists without a clear way to measure the cost of change or to know if a process is truly reversible.

A new study by Davide Cugini and Giacomo Guarnieri tackles this problem by reimagining how we measure the possibility of change. Instead of asking if a system can move from one state to another in isolation, the researchers ask if that move is compatible with a reference that is itself changing. They developed a method to track how a system's population of particles or energy levels deviates from a moving target. By converting the complex, multi-dimensional problem of comparing two changing states into a simpler, one-dimensional comparison of probabilities, they found a hidden mathematical structure that governs all such transitions. They discovered that for a change to be physically possible, the distribution of these deviations must behave like a specific type of random walk known as a martingale. In this framework, a transition is allowed only if the "spread" of the system's deviations tends to narrow or stay the same over time, never spreading out in a way that would violate the underlying rules of the moving reference.

This insight unifies several previously separate ideas in physics. It connects the concept of majorization, which determines what states can be reached using specific resources, with the theory of martingales, which describes how random variables evolve without a net drift. The authors showed that whether a system can transform from one configuration to another depends entirely on whether the probability distribution of its relative deviations can be connected by this martingale relationship. This means that the complex task of drawing and comparing intricate curves to check for feasibility can be replaced by checking a single, fundamental property of the probability distribution. If the distribution of deviations at the start can be linked to the distribution at the end through this specific probabilistic bridge, the transition is allowed. If not, the universe says no.

Beyond simply determining what is possible, the researchers used this framework to derive a precise rule for how much "entropy," or disorder, is produced during a process. They defined a new measure of entropy production that is relative to the chosen, moving reference. They proved that for any process that respects the rules of the reference, the average of a specific exponential quantity related to this entropy production always equals one. This is a fluctuation theorem, a powerful statement that holds true even when the system is far from equilibrium and the reference is changing rapidly. It provides a strict accounting of the cost of irreversibility, ensuring that on average, the system cannot violate the second law of thermodynamics relative to its own dynamic backdrop.

The study also addresses a very practical problem: what happens when scientists assume a reference state that is not quite right? In many experiments, researchers assume a system is following a smooth, ideal path, but in reality, the path might be jagged or lag behind due to noise or imperfect control. The authors showed that if the assumed reference is wrong, the fluctuation theorem they derived will appear to break down. The degree of this breakdown is not just a sign of error; it is a precise, measurable quantity that sets a lower limit on how far the assumed reference is from the true one. This allows scientists to quantify the mismatch between their model and reality without needing to know the true, complex dynamics of the system in advance. It turns a violation of a physical law into a diagnostic tool, certifying exactly how much the model has drifted from the truth.

To demonstrate this, the team applied their theory to a simple two-level system, like a single atom that can exist in a ground state or an excited state, driven by a changing energy gap. When they used the true, evolving state of the system as the reference, the laws held perfectly. However, when they used a simplified, instantaneous snapshot of the state as the reference, the system appeared to violate the second law of thermodynamics, showing a negative entropy production. This apparent violation was not a paradox but a clear signal. The size of the violation directly measured the "lag" caused by the system's inability to keep up with the rapid changes, effectively quantifying the non-adiabatic nature of the driving force. This result shows that what looks like a failure of physics is actually a precise measurement of how far a system has strayed from its ideal path.

The work offers a new way to think about the relationship between a system and its environment. By treating the reference not as a static wall but as a dynamic, evolving landscape, the researchers have provided a toolkit for understanding transitions in complex, time-dependent settings. Their findings suggest that the rules governing the flow of energy and information are more flexible and interconnected than previously thought, unified by a single mathematical structure that applies whether the reference is a simple equilibrium or a rapidly shifting drive. This approach allows for a more accurate diagnosis of errors in experimental setups and provides a rigorous foundation for studying systems that are constantly being pushed and pulled out of balance, from biological cells to quantum processors.

Technical Summary: State Convertibility and Fluctuation Theorems from a Dynamical Reference

Problem Statement
The paper addresses the fragmentation of statistical mechanics, where distinct frameworks—large deviation theory, information theory, resource theories of thermodynamics, and stochastic fluctuation relations—treat the distribution of system populations relative to a reference state in isolation. Historically, state convertibility (determining if a transformation $k(t_1) \to k(t_2)$ is possible) has been governed by majorization (for uniform references) or thermo-majorization (for Gibbs references). These criteria rely on constructing and comparing state-dependent Lorenz curves, a procedure tied to fixed, equilibrium references and operationally demanding. Furthermore, existing fluctuation theorems typically assume a known, static, or steady-state reference (e.g., Gibbs), leaving open the question of how to handle arbitrary, time-dependent reference distributions $g(t)$ and how to quantify errors when the assumed reference diverges from the true physical evolution.

Methodology
The authors unify these disparate concepts by introducing a "parent distribution" formalism. Instead of comparing population vectors directly, they map the system state $k(t)$ and the reference state $g(t)$ to a real-valued distribution of relative populations:
$\kappa_j(t) := \frac{k_j(t)}{g_j(t)}$
This relative population defines a parent distribution $p_\kappa(x; t) = \sum_j g_j(t) \delta(x - \kappa_j(t))$ , which represents the probability of drawing a specific deviation from the reference when sampling from the reference ensemble $g(t)$ .

Using this mapping, the authors reformulate the problem of state convertibility as a one-dimensional convex-order problem. They leverage martingale theory to establish a bijection between the existence of a column-stochastic transition matrix $T$ that preserves the reference dynamics ( $Tg(t_1) = g(t_2)$ ) and the existence of a martingale coupling between the parent distributions at times $t_1$ and $t_2$ .

Key Contributions and Results

$g(t)$ -Majorization and Martingale Coupling (Theorem 1):
The paper establishes that a transition $k(t_1) \to k(t_2)$ compatible with a reference dynamics $g(t)$ exists if and only if the associated parent distributions are connected by a martingale.
- This condition is equivalent to the convex order relation $p_\kappa(\cdot; t_1) \succeq_{cx} p_\kappa(\cdot; t_2)$ .
- This generalizes majorization and thermo-majorization to arbitrary, time-dependent references, replacing the sorting-based comparison of Lorenz curves with a one-dimensional convex-order comparison decidable by martingale existence. Physically, this implies that admissible transitions correspond to the concentration (narrowing) of the relative population distribution over time.
Reference-Generalized Fluctuation Theorem (Theorem 2):
By defining a reference-relative surprisal $s_j(t) = -\ln(k_j(t)/g_j(t))$ and the associated entropy production $\Delta s = s_{j_2}(t_2) - s_{j_1}(t_1)$ , the authors derive an exact integral fluctuation theorem for any transition matrix $T$ compatible with the reference dynamics:
$\mathbb{E}_\Pi [e^{-\Delta s}] = 1$
where the expectation is taken over the physical joint law of the process. This recovers the standard Second Law inequality $\mathbb{E}_\Pi[\Delta s] \geq 0$ via Jensen's inequality. This result extends integral fluctuation theorems from stochastic thermodynamics to arbitrary dynamical references.
Certifying Reference Mismatch (Theorem 3):
The authors address the practical scenario where the assumed reference $g(t)$ differs from the true reference evolution $g_{\text{true}}(t)$ . They show that the observed violation of the fluctuation theorem, $\mathbb{E}_\Pi [e^{-\Delta s}] - 1$ , provides a certified, model-independent lower bound on the $\chi^2$ -divergence between the true and assumed references:
$D_{\chi^2}(g_{\text{true}}(t_2) \parallel g(t_2)) \geq \frac{(\mathbb{E}_\Pi [e^{-\Delta s}] - 1)^2}{\sum_j k_j^2(t_2)/g_j(t_2)}$
This bound requires no independent characterization of the true dynamics $g_{\text{true}}$ . The denominator acts as a tunable diagnostic; concentrating the probe state $k(t_2)$ on microstates suspected of error sharpens the bound, whereas uniform spreading dilutes it. In the small-deviation limit, this bound certifies a lower bound on the Rao–Fisher distance between the assumed and true trajectories.

Illustrative Example
The framework is demonstrated on a driven two-level system with a time-dependent energy gap.

When the reference is the true propagated state, the fluctuation theorem holds exactly.
When the reference is the instantaneous Gibbs state (an approximation valid only in the adiabatic limit), the fluctuation theorem is violated for finite driving speeds.
The magnitude of this violation serves as a quantitative measure of the non-adiabatic lag, certifying the mismatch between the assumed instantaneous equilibrium and the true non-equilibrium dynamics without needing to know the true dynamics a priori.

Significance and Claims
The paper claims to provide a "common structure" that allows a single, arbitrary, and possibly time-dependent reference $g(t)$ to simultaneously unify:

The rate of fluctuations (Large Deviation Theory).
Informational meaning (Information Theory).
State convertibility (Resource Theories).
Trajectory statistics (Stochastic Fluctuation Relations).

By collapsing these questions into the statistics of relative populations, the work transforms state convertibility into a martingale problem and turns the breakdown of fluctuation relations into a diagnostic tool. The authors emphasize that this approach offers a model-independent method to quantify reference errors in complex systems (e.g., driven mesoscopic devices or quantum processors) where preparing or characterizing the true reference state is costly or impossible. The work does not propose new experimental setups but rather provides a theoretical framework to interpret existing measurements of non-equilibrium processes.

Technical Summary: State Convertibility and Fluctuation Theorems from a Dynamical Reference

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