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Generalized fluctuation-dissipation theorem and Einstein relation in rotating equilibrium

This paper derives a generalized fluctuation-dissipation theorem and a model-independent Einstein relation for vector fields in rotating thermal equilibrium, revealing rotation-induced tensorial corrections and a new mechanism for heavy quarkonium orbital polarization linked to spectral zero modes.

Original authors: Shuo Fang, Shi Pu

Published 2026-10-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shuo Fang, Shi Pu

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 quiet, invisible world of quantum particles, there is a fundamental rule that governs how things move and settle down. When a system is in thermal equilibrium, meaning it has settled into a steady temperature and is not changing overall, the random jiggling of its parts is inextricably linked to how it resists being pushed. This connection is known as the fluctuation-dissipation theorem. It tells us that the same microscopic chaos that causes a particle to wander randomly is also responsible for the friction that slows it down. For decades, physicists have relied on this rule to understand everything from the diffusion of smoke in the air to the behavior of electrons in a wire. However, this rule was derived for systems that are sitting still or moving in a straight line. It did not account for what happens when the entire environment is spinning.

Rotation is a common feature of the universe, from the spin of a neutron star to the swirling vortex of a fluid in a laboratory. When a system rotates, it introduces a new kind of force and a new way for particles to organize themselves. In recent years, experiments with heavy-ion collisions have created tiny, super-hot droplets of matter that spin faster than any other fluid known to science. These droplets, called quark-gluon plasma, offer a unique laboratory to test how rotation changes the basic laws of physics. The question remained: does the familiar link between random motion and resistance hold true when the entire universe of the particle is turning?

A team of researchers has now answered this question by deriving a new, more general version of the fluctuation-dissipation theorem that works specifically for rotating systems. They started by looking at the mathematical foundations of thermal equilibrium, which usually rely on a concept called the Kubo-Martin-Schwinger relation. This relation connects the way particles interact with the flow of time. The researchers realized that in a rotating system, time does not flow in a simple, straight line; it is twisted by the rotation. By carefully rewriting this fundamental relation to include the effects of rotation and acceleration, they uncovered a new layer of physics that was previously hidden.

The result is a generalized rule that shows rotation adds a new, directional component to how particles fluctuate and dissipate energy. In a non-rotating world, the resistance a particle feels is the same in every direction. But in a spinning environment, the researchers found that the resistance becomes dependent on the direction of motion relative to the spin. This means that the random jiggling of a particle is no longer just a simple blur; it carries a specific signature of the rotation. The team showed that this new rule applies without needing to know the specific details of how the particles interact with each other, making it a universal law for any rotating quantum matter.

One of the most striking findings is the discovery of a new type of connection between the random motion of particles and the drag they experience. In standard physics, the rate at which a particle slows down is directly tied to how much it jiggles. The researchers found that in a rotating system, this relationship is modified by a new term that depends on the speed of the rotation. More surprisingly, they identified a specific "zero mode" in the system's behavior—a state where the particle's motion is perfectly balanced by the rotation. This zero mode creates a new kind of link between the particle's orbital motion and the drag it feels, even when the drag itself is not changing.

This discovery has immediate implications for understanding heavy particles, such as heavy quarkonium, moving through the spinning quark-gluon plasma created in particle accelerators. The researchers showed that the rotation of the plasma can cause these heavy particles to align their orbits in a specific direction, a phenomenon known as orbital polarization. This alignment is not caused by the spin of the particles themselves, but by the way their orbital motion interacts with the spinning fluid. This provides a new way to measure the properties of the plasma and understand how angular momentum is transferred in the most extreme environments in the universe.

The team also applied their new rule to the process of detailed balance, which describes how likely a particle is to jump from one energy state to another compared to the reverse process. In a stationary system, this probability depends only on the temperature and the energy difference. In a rotating system, the researchers found that the probability is also influenced by the direction of the jump relative to the spin. This means that the rotation can subtly bias the way particles transition between states, favoring some directions over others. This effect is governed by the angular momentum of the transition itself, offering a new way to control and predict the behavior of quantum systems in rotating environments.

These findings establish a set of microscopic constraints that any theory of rotating quantum matter must obey. They provide a rigorous framework for studying the quark-gluon plasma and other rotating fluids without needing to solve the incredibly complex equations of every single interaction. By revealing how rotation reshapes the fundamental link between fluctuation and dissipation, the researchers have opened a new window into the behavior of matter under extreme conditions. Their work suggests that the spin of the universe leaves a permanent mark on the way particles move, even in the smallest, most fleeting moments of existence.

Technical Summary: Generalized Fluctuation-Dissipation Theorem and Einstein Relation in Rotating Equilibrium

Problem Statement
Thermal equilibrium imposes fundamental constraints on correlations and responses in many-body systems, traditionally encapsulated by the Kubo–Martin–Schwinger (KMS) relation and the fluctuation-dissipation theorem (FDT). In non-rotating, isotropic media, these relations lead to the Einstein relation, which links momentum diffusion to dissipative drag. However, physical systems ranging from relativistic heavy-ion collisions to rotating Bose–Einstein condensates often exist in states with non-zero total angular momentum. While the KMS relation has been studied separately in rotating and accelerating systems, a unified formulation incorporating total angular momentum conservation for vector fields in phase space has been lacking. Specifically, it remains unclear how the conservation of angular momentum modifies the Einstein relation itself, particularly whether momentum diffusion depends on low-frequency spectral information beyond the standard dissipative drag in the presence of thermal vorticity.

Methodology
The authors derive an exact phase-space KMS relation for bosonic vector fields in global thermal equilibrium, subject to the conservation of energy, particle number, and total angular momentum.

Generalized KMS Relation: Starting from the density operator $\varrho_J = Z_J^{-1} \exp[-b_\mu P^\mu + \alpha N + \varpi_{\mu\nu} J^{\mu\nu}]$ , where $\varpi_{\mu\nu}$ is the thermal vorticity tensor, the authors utilize the cyclicity of the trace to establish a generalized KMS relation. Unlike the conventional imaginary-time translation, this relation incorporates spacetime translations and Lorentz transformations (boosts and rotations) via the operator $\tilde{A}(x; b, \varpi) = e^{-C(P,J)} A(x) e^{+C(P,J)}$ .
Phase-Space Formulation: To handle the lack of translational invariance in rotating systems, the authors employ the Wigner transform of Wightman functions. This yields an exact phase-space KMS relation (Eq. 5) that explicitly separates orbital effects (via the orbital Lorentz generator $L_{\rho\sigma}$ ) and intrinsic spin effects (via the spin generator $R$ ).
Expansion and Transport Limit: The authors expand the resulting relations to second order in thermal vorticity $\varpi_{\mu\nu}$ . They define a generalized distribution function and derive the generalized FDT in phase space. Subsequently, they apply this to the local transport limit for a heavy nonrelativistic particle described by Langevin dynamics, analyzing the low-frequency ( $E \to 0$ ) behavior of spectral and statistical functions.

Key Contributions and Results

Generalized Fluctuation-Dissipation Theorem: The authors derive a model-independent generalized FDT (Eq. 6) for vector fields in rotating equilibrium. This relation connects the statistical function $G_F$ to the spectral function $\rho$ through a scalar thermal factor and an additional rotation-induced tensorial contribution involving the symmetrized product of vorticity and the spectral function. This result is non-perturbative in interaction strength.
Generalized Einstein Relation: In the transport limit, the authors derive a generalized Einstein relation (Eq. 8) linking the symmetric momentum diffusion tensor $\kappa_{ij}$ to the drag tensor $\eta_{D}^{ij}$ . Crucially, this relation includes rotation-induced corrections up to second order in vorticity. These corrections couple symmetric momentum diffusion to low-frequency spectral coefficients ( $\gamma_{2}^{ij}$ and $\gamma_{3}^{ij}$ ) that are distinct from the drag coefficient, indicating that diffusion depends on spectral information beyond the dissipative drag.
Spectral Zero Modes and New Einstein-Type Relation: The analysis reveals the existence of rotation-induced zero modes ( $i\gamma_{0}^{ij}$ ) in the spectral function, which are odd in vorticity. By requiring the finiteness of the diffusion tensor, the authors derive a new Einstein-type relation (Eq. 9): $\gamma_{0}^{ij} = 4MT\eta_D \varpi^{ij}$ . This links the spectral zero mode directly to the non-rotating drag coefficient $\eta_D$ in a non-perturbative manner.
Orbital Polarization of Heavy Quarkonium: Applying the zero-mode constraint to heavy quarkonium dynamics in a rotating medium, the authors show that spectral zero modes drive the evolution of orbital angular momentum. This leads to a mechanism for orbital polarization (population imbalance between orbital magnetic sublevels) distinct from spin polarization. This effect manifests in helicity-dependent photon angular distributions in radiative decays (e.g., $h_c \to \eta_c + \gamma$ ) and daughter polarization in hadronic decays (e.g., $\chi_{c1} \to \Lambda\bar{\Lambda}$ ).
Modified Detailed Balance: The generalized FDT yields a modified detailed-balance relation (Eq. 11) for transition rates in a rotating fluid. To first order in vorticity, the correction is governed by the transition polarization (angular-momentum polarization of the transition matrix element) and is independent of the specific microscopic bath model.

Significance
The paper establishes rigorous microscopic equilibrium constraints for rotating quantum matter. By deriving the generalized KMS relation and FDT without relying on weak-coupling expansions or specific interaction models, the results provide a foundational framework for studying spin and orbital transport in strongly coupled systems, such as the quark-gluon plasma. The identification of the new Einstein-type relation for spectral zero modes offers a novel mechanism for orbital polarization, potentially explaining experimental observations of spin polarization in heavy-ion collisions. Furthermore, the modified detailed-balance relation provides a model-independent tool for analyzing transition rates in vortical fluids. These findings bridge the gap between angular-momentum conservation constraints and observable transport phenomena in rotating quantum systems.

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