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Temperature Beyond Equilibrium in Isolated Quantum Many-Body Systems and Their Subsystems

This paper proposes a generalized definition of temperature for isolated quantum many-body systems out of equilibrium by locating non-stationary states within a family of regular states compatible with their energy-coherence structure, thereby replacing the principle of maximum entropy with a principle of minimum discrimination information and extending the framework to subsystems via induced local thermodynamic structures.

Original authors: Maurizio Fagotti

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Maurizio Fagotti

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 have a giant, invisible dance floor made of quantum particles. In the old days, physicists thought that if you wanted to know how "hot" this dance floor was, you just had to wait until everyone stopped dancing wildly and settled into a calm, steady rhythm. That calm state is called equilibrium, and in that state, temperature is a well-behaved, easy-to-measure number.

But what happens when the music changes suddenly? What if you hit the dance floor with a quantum "quench," sending the particles into a chaotic, swirling frenzy that never quite settles down? For nearly two centuries, scientists have struggled to answer a simple question: What is the temperature of a system that is out of equilibrium?

In this paper, Maurizio Fagotti proposes a new way to look at the problem. He suggests that we stop trying to force these chaotic systems into the old "calm down" box. Instead, we should treat temperature as a way to locate a specific state within a vast, organized library of possibilities, even while the system is still dancing.

The Two Kinds of Energy Jitters

To understand the new idea, imagine the energy in the system has two different types of "wiggles":

The Populations: This is like the number of dancers on the floor. If you have more dancers, you have more energy. This is the "classical" part, the kind of ignorance we usually have about how much energy is in a system.
The Coherence: This is the tricky part. It's like the dancers holding hands in a specific, synchronized pattern. Even if the total number of dancers stays the same, the way they are connected (their relative phases) creates a special kind of energy uncertainty that has no classical equivalent. This coherence is what drives the system to change over time.

The paper argues that to define temperature out of equilibrium, we must separate these two wiggles. We need to find the temperature that fits the "population" part, while keeping the "coherence" part exactly as it is.

The Library of Leaves

Fagotti uses a clever metaphor called a foliation. Imagine the entire state of the quantum system is a giant book.

The Old Way (Commutant Foliation): In the past, scientists organized this book by grouping pages that shared the same "basis" (like grouping books by the author's name). But this only worked well when the system was calm (equilibrium). Once the system gets chaotic, this organization falls apart because the "author" (the pre-quench temperature) loses its meaning.
The New Way (Minimum-Variance Foliation): The author suggests a new way to organize the book. Imagine the pages are grouped into "leaves" based on how much coherent energy they contain. All states on the same "leaf" share the exact same pattern of quantum connections (coherence), but they differ in how the energy is distributed among the dancers (populations).

Think of a leaf as a specific "vibe" or "mood" of the system. Once you know the mood (the coherence), you can slide along the leaf to find the temperature. The temperature is just the coordinate that tells you where you are on that specific leaf.

The New Rulebook: Minimum Discrimination, Not Maximum

Here is where the paper makes a bold move. In standard thermodynamics, we usually find the state of a system by assuming it maximizes entropy (disorder). It's like saying, "The system will do whatever creates the most chaos."

However, this paper argues that out of equilibrium, the "maximum entropy" rule is wrong. Instead, the system follows a principle of minimum discrimination information.

Imagine you are trying to guess a secret code. The old rule said, "Guess the code that is most random." The new rule says, "Guess the code that is closest to what you already know, without changing the parts you are sure about (the coherence)." The paper shows through simulations that if you use the old "maximum entropy" rule on these chaotic leaves, you get the wrong temperature. The new rule, which minimizes the difference from a reference state, gives the correct answer.

What About the Subsystems? (The Partial View)

What if you only look at a small corner of the dance floor? Can you measure the temperature there?
The paper says no, not exactly. You cannot just look at the reduced state of a small piece and say, "Ah, this piece has a temperature of 5."

Why? Because the temperature of a small piece depends on how it is connected to the rest of the dance floor. The "temperature" of a subsystem is determined by the entire trajectory of the system's evolution. It's like trying to guess the speed of a car by looking only at the rearview mirror; you need to know the whole road ahead to understand the speed.

However, the paper suggests that if the subsystem is large enough, the "boundary" effects (the edges where the subsystem meets the rest of the world) become small. In the limit of a very large subsystem, we can define a local temperature, but there is always a tiny bit of uncertainty (proportional to the inverse of the subsystem's size) because of those boundary interactions.

What the Paper Rules Out

It rules out the idea that temperature is just a leftover from equilibrium. The paper argues that temperature is an intrinsic property of the state and its dynamics, not just a remnant of a calm past.
It rules out the idea that the principle of maximum entropy works for these non-equilibrium states. The authors show that using maximum entropy leads to biased, incorrect temperatures.
It rules out the idea that you can define a subsystem's temperature solely by looking at that subsystem's reduced state. The history and the global connection matter.

How Sure Are We?

The paper presents a theoretical framework that is mathematically rigorous for a specific class of systems (quantum spin chains with short-range interactions).

The core definition of the "leaf" and the "canonical flow" is derived from established quantum information theory (specifically the minimum-variance foliation).
The claim that "maximum entropy fails" and "minimum discrimination works" is supported by numerical simulations (Section 6 of the paper). The authors ran simulations showing that the old method gives a biased result, while the new method aligns with the expected physical behavior.
The paper suggests that this framework applies to the thermodynamic limit (infinite systems), but it acknowledges that some parts (like the existence of phase transitions on these "leaves") are still open questions for future research.

The Takeaway

Temperature isn't just a number you measure when things are calm. It's a coordinate on a map of possibilities. Even when a quantum system is in a chaotic, non-equilibrium frenzy, it still has a temperature—but to find it, you have to stop looking at the system as a static object and start looking at the "leaf" of possibilities it lives on, keeping its quantum connections (coherence) fixed while sliding along the path of least resistance. It's a new way of seeing the heat in the chaos.

Technical Summary: Temperature Beyond Equilibrium in Isolated Quantum Many-Body Systems and Their Subsystems

Problem Statement
The paper addresses the fundamental difficulty of defining temperature in isolated quantum many-body systems far from equilibrium. While temperature is a central concept in equilibrium thermodynamics, its extension to non-equilibrium regimes lacks a universally accepted definition. Existing approaches—ranging from fluctuation-dissipation relations to weak probe thermometers—often rely on reference to equilibrium or specific measurement protocols, lacking the conceptual solidity of equilibrium temperature.

The core challenge identified is that isolated quantum systems evolve unitarily, retaining energy coherence (information stored in relative phases between energy sectors) that has no classical thermodynamic analogue. Standard equilibrium concepts, which assume stationarity and the absence of such coherence, fail to capture the dynamics of these systems. The paper argues that treating temperature merely as a remnant of equilibrium is insufficient; instead, a unifying framework must treat temperature as an intrinsic property of a state and its dynamics, where equilibrium emerges as a special case.

Methodology
The authors propose a geometric and algebraic framework based on the minimum-variance foliation of the state space.

Foliation of State Space:
- The authors reinterpret stationarity not as a dichotomy but as the absence of coherent energy fluctuations.
- They introduce a minimum-variance foliation, which groups states into "leaves" based on their energy-coherence structure. States on the same leaf share the same optimal pure-state decomposition that minimizes the average energy variance (related to the Quantum Fisher Information).
- Unlike the "commutant foliation" (where leaves are sets of states commuting with a Hamiltonian), the minimum-variance foliation separates the "population-like" contributions to energy uncertainty from the "coherent" contributions.
Thermodynamic Limit and Pseudolocal Flows:
- Recognizing that the finite-dimensional construction of the foliation does not directly translate to infinite systems, the authors reformulate the theory in the thermodynamic limit using pseudolocal flows.
- A leaf is characterized by a symmetric logarithmic time derivative (SLtD), denoted as $\tilde{H}[\omega]$ , which acts as a "harmonic conjugate" to the Hamiltonian.
- States on the same leaf are those that share the same SLtD (or equivalently, satisfy specific algebraic conditions involving the dressed Hamiltonian).
Principle of Minimum Discrimination Information:
- To define temperature, the authors move away from the Principle of Maximum Entropy (which fails to extend naturally to non-commuting leaves).
- Instead, they employ the Principle of Minimum Discrimination Information (minimizing relative entropy) to define a canonical flow on each leaf.
- This flow connects a given state to a reference state (analogous to the infinite-temperature state) along a path parameterized by an inverse temperature $\beta$ .
Subsystems:
- For subsystems, the authors argue that temperature cannot be inferred from the reduced state alone. Instead, it is determined by the induced local thermodynamic structure resulting from the global canonical flow.
- They introduce the concept of a weak canonical flow on subsystems, acknowledging that boundary terms introduce an intrinsic uncertainty of order $O(|A|^{-1})$ in the definition of subsystem temperature.

Key Contributions and Results

Definition of Nonequilibrium Temperature: The paper defines a nonequilibrium temperature for regular states by locating the state on a canonical curve within its leaf of the minimum-variance foliation. The inverse temperature $\beta$ is the coordinate along this curve, defined relative to a reference state (zero inverse temperature).
Canonical Pseudolocal Flow: The authors construct a canonical flow generated by a "canonical charge" $\hat{H}_\omega$ . This flow preserves the local regularity properties (e.g., exponential clustering) of the state. The generator is given by:
$\hat{H}_\omega(O) = \langle\langle H, O \rangle\rangle^c_\omega + \frac{i}{2}\omega([ \tilde{H}[\omega], O ])$
where $\langle\langle \cdot, \cdot \rangle\rangle^c$ is the symmetrized connected correlation.
Failure of Maximum Entropy: The paper demonstrates that the principle of maximum entropy does not generally hold for non-equilibrium states on non-commuting leaves. The inverse temperature derived from the canonical flow is generally not the derivative of thermodynamic entropy with respect to energy. This is supported by numerical evidence showing a bias in the "bare" temperature estimator.
Generic vs. Non-Generic Leaves:
- Generic Leaves: Support a single thermodynamic direction (the canonical flow). These are the nonequilibrium counterparts of generic equilibrium dynamics.
- Non-Generic Leaves: May support additional pseudolocal flows (associated with extra conservation laws or integrability) or exhibit thermodynamic fragmentation (disconnected sectors).
Subsystem Temperature: The authors establish that subsystem temperature is a dynamical quantity determined by the global trajectory. They derive an equation for the time derivative of the subsystem inverse temperature ( $\delta \beta_A$ ), which depends on the variation of thermodynamic entropy and a generalized heat capacity. The definition is unique only up to boundary ambiguities that vanish in the thermodynamic limit.

Significance
The paper claims to provide a natural definition of temperature for regular nonequilibrium states that does not rely on reference to equilibrium. By shifting the perspective from "temperature as a parameter of an equilibrium ensemble" to "temperature as a coordinate on a leaf of the state space defined by coherence," the authors offer a framework where:

Equilibrium thermodynamics emerges as a special case (the commuting leaf).
The role of energy coherence is explicitly accounted for in the thermodynamic structure.
The principle of maximum entropy is replaced by a more general principle of minimum discrimination information.

The work suggests that the "minimum-variance foliation" provides a natural framework for a quantum-coherent thermodynamics, offering a way to organize the complex state space of isolated quantum systems beyond the dichotomy of stationary vs. non-stationary states. The authors emphasize that this approach is particularly suited for systems with short-range interactions and regular local properties, such as quantum spin chains.