Anomalous Mean-Squared Displacement in Quantum Active Matter from a Wigner Phase-Space Framework

Alexander P. Antonov; Benno Liebchen; Giovanna Morigi; Hartmut L\"owen; Michael te Vrugt; Sangyun Lee; Yehor Tuchkov

arxiv: 2604.22600 · v1 · submitted 2026-04-24 · ❄️ cond-mat.soft · quant-ph

Anomalous Mean-Squared Displacement in Quantum Active Matter from a Wigner Phase-Space Framework

Sangyun Lee , Yehor Tuchkov , Alexander P. Antonov , Benno Liebchen , Hartmut L\"owen , Giovanna Morigi , Michael te Vrugt This is my paper

Pith reviewed 2026-05-08 09:17 UTC · model grok-4.3

classification ❄️ cond-mat.soft quant-ph

keywords active matterquantum mechanicsWigner functionmean squared displacementanomalous diffusionphase spacemaster equationquantum fluctuations

0 comments

The pith

Quantum active matter described by a Wigner phase-space model exhibits mean-squared displacement scaling up to t^7.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a quantum theory of active matter by embedding classical self-propulsion into a Wigner phase-space description of quantum degrees of freedom. It then calculates the mean-squared displacement of the particles' positions over time. Analytical results show that the displacement typically grows as the sixth power of time. In some cases the growth becomes even steeper, reaching the seventh power of time. These power laws survive the addition of quantum noise in the initial conditions.

Core claim

By means of a hybrid Wigner master equation that treats active motion classically while keeping quantum evolution for other variables, the time-dependent mean-squared displacement is derived in closed form. The calculation reveals regimes of MSD ∼ t^6 and, for tuned parameters and initial states, MSD ∼ t^7. Both scalings prove stable when quantum fluctuations are present at the start.

What carries the argument

Hybrid Wigner master equation combining classical active particle dynamics with quantum phase-space evolution to compute the mean-squared displacement.

If this is right

Analytic expressions for the MSD time dependence are obtained for quantum active systems.
Characteristic scaling MSD ∼ t^6 occurs for standard parameter choices.
Steep scaling MSD ∼ t^7 is possible under specific conditions.
The anomalous behavior remains intact in the presence of initial-state quantum fluctuations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Classical active-matter results for MSD appear as the limit when quantum effects vanish.
The phase-space approach can be extended to predict transport in other hybrid quantum-classical driven systems.
Robustness against initial fluctuations suggests the scalings are generic features rather than fragile artifacts.

Load-bearing premise

The hybrid Wigner master equation that incorporates classical active motion and quantum degrees of freedom accurately captures the physics of the system without missing essential quantum-classical coupling terms.

What would settle it

A direct measurement of the position variance in a realized quantum active system that fails to show at least t^6 growth at intermediate times would falsify the predicted scalings.

Figures

Figures reproduced from arXiv: 2604.22600 by Alexander P. Antonov, Benno Liebchen, Giovanna Morigi, Hartmut L\"owen, Michael te Vrugt, Sangyun Lee, Yehor Tuchkov.

**Figure 1.** Figure 1: FIG. 1. Conceptual figure to illustrate the quantum active matter model from Ref. [ view at source ↗

**Figure 2.** Figure 2: FIG. 2. Plot of MSD versus time of the quantum active particle with the initial conditions, Eq. ( view at source ↗

**Figure 3.** Figure 3: FIG. 3. Plot of MSD versus time of the quantum active particle with different view at source ↗

**Figure 4.** Figure 4: FIG. 4. Plot of MSD versus time of the quantum active particle with different persistent time and noise intensity for the view at source ↗

**Figure 5.** Figure 5: FIG. 5. Plot of MSD [Eq. ( view at source ↗

**Figure 6.** Figure 6: FIG. 6. Plot of MSD [Eq. ( view at source ↗

**Figure 7.** Figure 7: FIG. 7. Plot of the difference between two moments, view at source ↗

**Figure 8.** Figure 8: FIG. 8. Plot of MSD versus time of the quantum active particle calculated with three different methods. The blue line view at source ↗

**Figure 9.** Figure 9: FIG. 9. Plot of MSD versus time of the quantum active particle. view at source ↗

read the original abstract

Active matter is driven out of equilibrium by a local influx of energy. While classical active matter has been extensively studied, the extension of active matter concepts to quantum systems has been explored far less. In this work we develop a full quantum description based on the Wigner function. By introducing a hybrid Wigner master equation that incorporates classical active motion and quantum degrees of freedom, we compute the quantum mean-squared displacement (MSD) using established techniques from classical active matter. We analytically derive the time dependence of the MSD and clarify the conditions under which the characteristic scaling with time $\mathrm{MSD}\sim t^{6}$ emerges. We further show that, for certain parameter and initial conditions, the MSD can exhibit an even steeper scaling regime $\mathrm{MSD}\sim t^{7}$, and we examine the robustness of these behaviors against quantum fluctuations of the initial state.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives t^6 and t^7 MSD scalings for quantum active matter via a hybrid Wigner master equation, but that equation lacks a shown microscopic derivation so the exponents rest on an unverified assumption.

read the letter

The key point is that this work takes classical active-matter methods for mean-squared displacement and applies them inside a Wigner-function description of quantum active particles. It reports an analytic t^6 regime under standard conditions and a steeper t^7 regime for certain parameters and initial states, plus a check that the scalings survive some quantum noise in the starting conditions. That t^7 result is new and the overall framing is straightforward. The authors also keep the derivations parameter-light and tie the scalings back to the active-motion terms, which is useful for anyone trying to organize simulations or experiments in this still-small subfield. The hybrid master equation is the central construction, and the paper shows how to extract the long-time asymptotics from it using the same moment techniques that work classically. That part is clean and reproducible in principle. The main weakness is that the hybrid equation itself is introduced without an explicit derivation from a microscopic model. The stress-test note flags the risk that active propulsion on the quantum state could generate extra commutator or drift terms that are missing here; if those terms exist, they would alter the phase-space current and change the extracted powers. Without seeing the full steps that justify the hybrid form, it is hard to know whether the t^7 scaling is robust or an artifact of the truncation. The rest of the analysis follows logically once the equation is granted, and there is no sign of circular fitting or invented parameters. This is the kind of paper that belongs in a reading group on quantum non-equilibrium systems or active matter extensions. Readers who already work with Wigner functions or who need concrete scaling predictions will get something concrete out of it. It is worth sending to peer review because the claims are specific, the novelty is real, and the gap in the master-equation derivation is fixable with a few added pages rather than a fundamental rewrite. Ask the authors to supply the microscopic origin of the hybrid equation and to test the scalings against a fully quantum simulation for at least one parameter set.

Referee Report

1 major / 1 minor

Summary. The paper develops a quantum description of active matter via the Wigner function, proposing a hybrid Wigner master equation that incorporates classical active motion into quantum phase-space evolution. Using established classical active-matter techniques on this equation, the authors analytically derive the time dependence of the mean-squared displacement (MSD), identifying a characteristic MSD ∼ t^6 scaling and, for certain parameters and initial conditions, a steeper MSD ∼ t^7 regime, while testing robustness against quantum fluctuations of the initial state.

Significance. If the hybrid master equation is shown to be complete, the analytical MSD derivations would provide concrete, falsifiable predictions for anomalous diffusion in quantum active systems, extending classical active-matter methods to the quantum regime in a phase-space setting. This could guide future work on quantum active particles or swimmers.

major comments (1)

[Section introducing the hybrid Wigner master equation] The hybrid Wigner master equation is introduced without an explicit microscopic derivation from an underlying Hamiltonian or Lindblad quantum master equation for the active system (see the section presenting the equation and the subsequent MSD calculation). Any omitted quantum-classical coupling terms—such as additional commutators or position-dependent drifts arising when the active force is Wigner-transformed—would alter the phase-space current and change the extracted long-time exponents. Since the t^6 and t^7 scalings are obtained by solving this equation, a detailed derivation or justification is required to establish that no essential terms are missing.

minor comments (1)

[Abstract] The abstract states that the MSD scalings are derived using 'established techniques from classical active matter' but does not cite the specific methods or references; adding these would improve clarity for readers unfamiliar with the classical literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion to strengthen the presentation of the hybrid Wigner master equation. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The hybrid Wigner master equation is introduced without an explicit microscopic derivation from an underlying Hamiltonian or Lindblad quantum master equation for the active system (see the section presenting the equation and the subsequent MSD calculation). Any omitted quantum-classical coupling terms—such as additional commutators or position-dependent drifts arising when the active force is Wigner-transformed—would alter the phase-space current and change the extracted long-time exponents. Since the t^6 and t^7 scalings are obtained by solving this equation, a detailed derivation or justification is required to establish that no essential terms are missing.

Authors: We agree that an explicit derivation from an underlying quantum master equation would improve the manuscript. In the revised version we will add a dedicated section (or appendix) that starts from the quantum Liouville–von Neumann equation for the particle density operator, augments the Hamiltonian with a classical self-propulsion term that depends on the particle orientation, and performs the Wigner transform. The active force enters the resulting phase-space equation solely as a deterministic drift term −F·∇_p, exactly as external forces appear in the standard Wigner formalism. Because the propulsion is treated as a classical drive rather than a quantum operator, no additional commutators or position-dependent quantum corrections arise beyond those already contained in the kinetic and potential terms. We will explicitly verify that the phase-space current used in the subsequent MSD calculation is unchanged by this construction, thereby confirming that the t^6 and t^7 scalings remain valid under the stated assumptions. We will also state the regime of validity of the hybrid approach (semiclassical limit or slow active motion relative to decoherence). revision: yes

Circularity Check

0 steps flagged

No circularity: MSD scalings derived from solving hybrid equation via established methods

full rationale

The paper introduces a hybrid Wigner master equation that adds classical active drift to quantum Wigner evolution, then applies established classical active-matter techniques to solve for the MSD analytically. The t^6 and t^7 regimes are obtained as solutions under specified parameter/initial conditions, with no reduction to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain is self-contained against the new equation and external classical methods; no step equates the output to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of a newly introduced hybrid Wigner master equation whose form is not shown; it combines standard quantum phase-space methods with classical active driving whose parameters are not specified here.

free parameters (1)

active motion parameters
Strength and form of classical self-propulsion incorporated into the master equation; required to produce the reported scalings.

axioms (2)

standard math Wigner function provides a valid phase-space representation for quantum states
Standard tool in quantum mechanics invoked to bridge to classical active motion.
domain assumption Classical active driving can be added to quantum evolution via a hybrid master equation without additional consistency conditions
Core modeling choice of the work.

pith-pipeline@v0.9.0 · 5476 in / 1319 out tokens · 52458 ms · 2026-05-08T09:17:36.167056+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

44 extracted references · 3 canonical work pages

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Ramaswamy, The mechanics and statistics of active matter, Annu

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2010

[2] [2]

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys.85, 1143 (2013)

2013

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Gompper, R

G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. L¨ owen, R. Golestanian, U. B. Kaupp, L. Alvarez,et al., The 2020 motile active matter roadmap, J. Phys.: Condens. Matter32, 193001 (2020)

2020

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Bechinger, R

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2024

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2025

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A. P. Antonov, S. Lee, B. Liebchen, H. L¨ owen, J. Melles, G. Morigi, Y. Tuchkov, and M. te Vrugt, Modeling dissipation in quantum active matter, arXiv:2511.21502 (2025)

work page arXiv 2025

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T. Nadolny, C. Bruder, and M. Brunelli, Nonreciprocal synchronization of active quantum spins, Phys. Rev. X15, 011010 (2025)

2025

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2023

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S. Lee, M. Ha, and H. Jeong, Quantumness and thermodynamic uncertainty relation of the finite-time otto cycle, Phys. Rev. E103, 022136 (2021)

2021

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