🔬 condensed matter

Memory with Onsager-Casimir symmetry: Rotating particle in a viscoelastic fluid

This paper combines experiments and theory to demonstrate that a rotating Brownian particle in a viscoelastic fluid exhibits enhanced diffusivity and time-antisymmetric cross-correlations, which are explained by a minimal model featuring a non-reciprocal memory kernel that satisfies Onsager-Casimir symmetry and establishes a novel geometric fluctuation-response relation linking cross-correlations to transverse response.

Original authors: Debankur Das, Niloyendu Roy, Niklas Windbacher, Clemens Bechinger, Matthias Krüger

Published 2026-08-04

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CC0 1.0

Original authors: Debankur Das, Niloyendu Roy, Niklas Windbacher, Clemens Bechinger, Matthias Krüger

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 a world where the rules of how things move aren't just about pushing and pulling, but also about how the past whispers to the present. This is the realm of statistical physics, the branch of science that studies how tiny particles dance in a chaotic crowd. For over a century, scientists have relied on a simple idea: when you push a particle, it moves immediately, and when you stop pushing, it stops immediately. It's like sliding a book across a smooth table; the friction is instant, and the past doesn't matter. But in many real-world environments, like the thick goo inside a cell or a complex gel, this simple picture breaks down. These fluids have "memory." If you push a particle, the fluid remembers the shove for a while, creating a drag that lingers. This is called non-Markovian behavior.

Now, add a twist: what happens if that particle is also spinning? In our everyday world, spinning things can do weird things, like a spinning top that wobbles or a curveball in baseball that dips unexpectedly. Scientists call this the Magnus effect. But when you combine this spinning with a fluid that has a memory, the physics gets even stranger. The big question is: how does a spinning particle behave when it's just floating around on its own, without anyone pushing it? Does the memory of the fluid change how it wiggles? Does the spin create a hidden connection between moving left and moving up? Understanding this isn't just about spinning toys; it helps us figure out how to move tiny machines inside our bodies or how to sort microscopic particles in complex liquids.

This paper takes a deep dive into that exact mystery. The researchers, a team of physicists from Germany and Switzerland, set up a clever experiment where they watched a tiny magnetic bead, about 4.5 micrometers wide (roughly the width of a human hair), spin inside a special, gooey fluid made of worm-like micelles. They spun the bead using a magnetic field and watched how it jittered around. They found two surprising things. First, the spinning made the particle wander much faster over long periods than it would if it were just sitting still. Second, and even more strangely, the particle started developing a secret link between its movements in different directions. If it moved a bit to the right, it was statistically likely to have moved up or down in a specific, time-reversed pattern. It was as if the particle was drawing a spiral in the air, connecting its past and future movements in a way that defies the usual rules of symmetry.

To explain this, the team built a simple mathematical model. They imagined the spinning bead (the "tracer") was attached by a spring to a ghostly "bath particle" representing the fluid's memory. To mimic the spinning, they added a special rule that made the spring act differently depending on the direction, kind of like a one-way street for forces. This model predicted that the "memory" of the fluid wouldn't just be a simple drag; it would rotate over time, forming a shape called a logarithmic spiral. This rotating memory creates a non-reciprocal relationship: the fluid pushes back on the particle in a way that isn't the same if you reverse time.

The paper confirms that this spinning, memory-filled system follows a specific, elegant rule known as Onsager-Casimir symmetry. In plain English, this means that if you were to reverse the direction of the spin, the way the particle responds would flip in a predictable, mirror-image way. The researchers showed that the strange cross-correlations they saw in the experiment (the particle moving sideways when it shouldn't) are directly linked to the Magnus effect (the sideways deflection when pushed). They even derived a new geometric trick: by looking at how the particle's position correlations spiral in a graph, you can actually calculate the angle of the Magnus deflection without ever applying a force to the particle.

While the theory and the experiment matched up beautifully in their general shape and behavior, the numbers weren't a perfect 100% match. The authors suggest this is likely due to the complexity of the real fluid or statistical noise in the measurements, but the qualitative agreement is strong. They didn't just observe a weird wobble; they established that rotating colloids in viscoelastic fluids are a real-world example of these deep symmetry principles in action. This work suggests that by understanding how memory and rotation mix, we might be able to design better ways to steer tiny particles through complex environments, turning the chaotic dance of the microscopic world into something we can predict and control.

Technical Summary: Memory with Onsager-Casimir Symmetry: Rotating Particle in a Viscoelastic Fluid

Problem Statement
The paper addresses the stochastic dynamics of a Brownian particle in a non-Markovian fluid (a fluid with memory) when time-reversal symmetry is broken by rotation. While memory kernels in time-reversal symmetric systems obey Onsager reciprocity, the behavior of memory kernels in systems with broken time-reversal symmetry—specifically those involving a pseudovector like a rotation vector—is less understood. The authors investigate how rotation and memory interact to produce non-reciprocal responses and how these manifest in the fluctuations of a tracer particle in the absence of external driving forces. Specifically, they seek to understand the nature of fluctuations arising from time-reversal symmetry breaking and whether the Magnus effect (a transverse deflection under force) is encoded in the unperturbed cross-correlations of the particle's position.

Methodology
The study employs a combined theoretical and experimental approach:

Experimental Setup: The authors use a superparamagnetic colloidal particle ( $\sigma \sim 4.5$ µm) rotating around the $x_3$ -axis with frequency $\Omega$ in a viscoelastic solution of worm-like micelles. This fluid exhibits structural relaxation times on the order of seconds, inducing non-Markovian dynamics. They measure:
- The Magnus effect (transverse deflection) under an external force.
- The Mean Squared Displacement (MSD) to determine diffusivity.
- The two-time displacement correlation matrix $C(t, t')$ to analyze cross-correlations between orthogonal displacement components ( $x_1, x_2$ ).
- Control experiments were performed in purely viscous fluids and semi-dilute polymer solutions to isolate the effects of memory and rotation.
Theoretical Model: A minimal linear model is introduced, consisting of a tracer particle coupled via a harmonic spring to a "bath" particle.
- The bath particle is subject to an advective coupling term $\Omega \times (y - x)$ , mimicking the rotation of the surrounding fluid. This term breaks time-reversal symmetry.
- By integrating out the bath variable, the authors derive a Generalized Langevin Equation (GLE) for the tracer.
- The resulting memory kernel is analyzed for its symmetry properties and its relation to the fluctuation-dissipation theorem (FDT).

Key Contributions and Results

Experimental Observations:
- Enhanced Diffusivity: Rotation enhances the long-time diffusivity of the particle in the plane perpendicular to the rotation axis, a phenomenon not observed in purely viscous fluids.
- Time-Antisymmetric Cross-Correlations: In the viscoelastic fluid, the off-diagonal components of the displacement correlation matrix ( $C_{12}$ and $C_{21}$ ) are non-zero and exhibit time-antisymmetry: $C_{ij}(t, t') = -C_{ij}(t', t)$ . This implies that cross-correlations vanish at equal times ( $t=t'$ ).
- Magnus Effect: The system exhibits a Magnus deflection when driven by an external force, confirming the breaking of time-reversal symmetry.
Theoretical Findings:
- Non-Reciprocal Memory Kernel: The derived memory kernel $L(t)$ is non-symmetric (non-reciprocal). In the complex plane, the kernel traces a logarithmic spiral.
- Onsager-Casimir Symmetry: The memory kernel satisfies the Onsager-Casimir symmetry relation $L^T(\Omega) = L(-\Omega)$ . This symmetry holds for the fluctuations of a rotating sphere in a viscoelastic medium, analogous to systems with magnetic fields.
- Fluctuation-Response Relation: The authors derive a generalized fluctuation-dissipation relation. Unlike the standard Einstein relation which links the symmetric part of the response to the MSD, this work establishes a link between the antisymmetric sector of the response (the Magnus effect) and the time-antisymmetric cross-fluctuations.
- Geometric Construction: A geometric method is proposed to extract the Magnus angle (the ratio of transverse to longitudinal response) directly from the unperturbed correlation data. Specifically, the slope of the correlation trajectory in the $(x_1, x_2)$ plane near the "kink" at $t=t'$ relates to the steady-state Magnus angle.
Comparison:
- The theoretical model qualitatively reproduces the experimental findings, including the enhanced diffusivity, the antisymmetric cross-correlations, and the spiral structure of the correlations in the $(x_1, x_2)$ plane.
- Quantitative deviations exist between the model and experiment, which the authors attribute to the simplicity of the phenomenological model (e.g., neglecting specific flow field details) and statistical limitations in the experimental data.

Significance and Claims
The paper claims to establish rotating colloids in viscoelastic fluids as a minimal realization of Onsager-Casimir symmetry in time-nonlocal stochastic dynamics. The primary significance lies in:

Demonstrating that the Magnus effect is encoded in the unperturbed fluctuations of the system, providing a new way to probe non-reciprocal responses without external driving.
Deriving a specific relation (Eq. 25) that connects off-diagonal response functions to time-antisymmetric cross-correlations, distinct from standard Einstein relations.
Showing that the interplay of memory and rotation leads to a memory kernel that rotates in time (forming a logarithmic spiral) while strictly obeying Onsager-Casimir symmetry.

The authors maintain a modest tone regarding the quantitative agreement, noting that while the qualitative behavior is robust, the specific values of parameters (like the Magnus angle extracted from correlations) show systematic overestimation compared to direct force measurements, suggesting the need for further investigation into which classes of models strictly obey the derived relations in complex media.

Technical Summary: Memory with Onsager-Casimir Symmetry: Rotating Particle in a Viscoelastic Fluid

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