🔬 condensed matter

Interplay of activity and non-reciprocity in tracer dynamics: From non-equilibrium fluctuation-dissipation to giant diffusion

This paper derives a generalized Langevin equation for a tracer in a non-reciprocal active or passive bath, revealing that non-reciprocal interactions induce a resonance-driven "giant diffusivity" regime accompanied by significant heat dissipation, thereby establishing a direct thermodynamic cost for enhanced transport.

Original authors: Subhajit Paul, Manish Patel, Debasish Chaudhuri

Published 2026-09-02

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CC BY 4.0

Original authors: Subhajit Paul, Manish Patel, Debasish Chaudhuri

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 microscopic world of fluids and gels, scientists often use tiny probe particles, known as tracers, to understand how a material behaves. By watching how these particles wander, researchers can measure the material's thickness, its ability to flow, and how it responds to forces. In a calm, balanced environment, this wandering follows predictable rules where the energy of the particles and the resistance they face are perfectly linked. However, life and many modern materials are rarely calm. They are often driven by a constant input of energy, such as bacteria swimming or synthetic particles that move on their own. In these active, out-of-balance systems, the usual rules break down, and particles can move in surprising, sometimes chaotic ways. Understanding how these particles move is crucial for everything from designing better drugs to understanding how cells function, but a complete picture of how they move when the forces between them are uneven has remained elusive.

A team of researchers has now mapped out exactly what happens when a single tracer particle moves through a crowd of other particles that push and pull on it in an uneven way. They focused on a specific type of interaction called non-reciprocity. In the everyday world, forces usually come in equal and opposite pairs; if you push a wall, the wall pushes back with the same strength. But in many complex systems, this balance is broken. One particle might push another hard, while the second particle barely pushes back. The researchers built a mathematical model to track a tracer particle coupled to a bath of other particles, allowing them to control exactly how much this "push-back" symmetry was broken. They treated the particles as either passive, drifting with the flow, or active, moving with their own internal energy, and studied how the tracer behaved in all possible combinations.

The study revealed a striking and counterintuitive result. As the researchers increased the degree of this uneven pushing, the tracer did not just move a little faster; it suddenly began to diffuse, or spread out, with extraordinary speed. This surge in movement happened at a very specific point where the imbalance between the particles reached a precise mathematical relationship determined by the number of particles in the crowd. The researchers call this phenomenon "giant diffusivity." It is as if the particles, by pushing against each other in a lopsided way, create a perfect storm that launches the tracer into a state of rapid, efficient exploration. This effect was not limited to just one type of particle or interaction; the researchers confirmed through computer simulations that this giant speed-up occurs whether the particles are moving on their own or being pushed by external energy, and whether they interact over long distances or only when they are very close.

However, the universe demands a price for such efficiency. The researchers found that this massive increase in movement comes with a significant thermodynamic cost. To maintain this state of giant diffusion, the system must dissipate a large amount of heat. The energy required to keep the particles moving so fast spikes at the exact same moment the diffusion peaks. This means that the enhanced transport is not free; it is fueled by a substantial expenditure of energy that is released as heat. The study shows that while non-reciprocal interactions can be used to control and amplify how particles move, doing so requires a continuous and heavy investment of energy.

The work provides a clear, general framework for understanding transport in systems where forces are not balanced. It suggests that by tuning the asymmetry of interactions, scientists could potentially design materials that switch between slow, controlled movement and rapid, giant diffusion. This has potential relevance for understanding biological processes, such as how cells navigate their environments or how predator and prey interact, as well as for engineering synthetic active matter. The findings confirm that the violation of action and reaction symmetry is a fundamental driver of nonequilibrium behavior, capable of generating dramatic changes in how matter moves, provided the system is willing to pay the energetic price.

Technical Summary: Interplay of Activity and Non-reciprocity in Tracer Dynamics

Problem Statement
Tracer particles serve as fundamental probes for understanding transport in complex media, ranging from colloidal suspensions to biological systems. While equilibrium tracer dynamics are well-described by Brownian motion and the fluctuation-dissipation relation (FDR), non-equilibrium environments—particularly active baths—exhibit anomalous transport, non-Gaussian fluctuations, and FDR violations. Existing theoretical frameworks for tracer-bath dynamics typically assume interaction reciprocity (action-reaction symmetry), introducing nonequilibrium effects solely through self-propulsion or colored noise. However, non-reciprocal interactions (where the force exerted by particle $A$ on $B$ differs from $B$ on $A$ ) are increasingly recognized as a generic feature of driven many-body systems, arising in hydrodynamic couplings, chemical fields, and biological chase-and-run dynamics. The specific role of non-reciprocity in modifying single-particle tracer transport, distinct from particle activity, remains underexplored.

Methodology
The authors develop a generalized system-bath framework where a tracer (system) is coupled to a bath of $N$ particles. The dynamics are governed by a linear stochastic model allowing for non-reciprocal coupling:
$\frac{dx_i}{dt} = -\lambda(x_i - X_s) + \xi_i$
$\frac{dX_s}{dt} = -\Lambda \sum_{i=1}^N (X_s - x_i) + \zeta$
Here, $\lambda$ and $\Lambda$ are independent coupling constants. Reciprocity corresponds to $\Lambda = \lambda$ , while the ratio $\chi = \Lambda/\lambda$ serves as the non-reciprocity parameter. The noise terms ( $\xi_i, \zeta$ ) are modeled as either white noise (passive) or exponentially correlated colored noise (active), allowing the study of all four combinations of active/passive tracers and baths.

The authors integrate out the bath degrees of freedom to derive an overdamped generalized Langevin equation (GLE) for the tracer. This equation features a non-Markovian memory kernel and an effective noise term composed of intrinsic tracer noise and bath-induced noise. Using this framework, they analytically compute:

The velocity and displacement response functions.
A generalized nonequilibrium fluctuation-dissipation relation.
The mean-squared displacement (MSD) and effective diffusivity.
The heat dissipation rate to quantify the thermodynamic cost.

These analytical predictions are validated against numerical simulations using both the harmonic interaction model (Eq. 1) and a short-range interaction model, integrated via the Euler–Maruyama scheme.

Key Contributions and Results

Generalized Langevin Framework: The study establishes a unified description for tracer dynamics in active and passive baths with non-reciprocal coupling, deriving an exact GLE with a memory kernel determined by the bath coupling strength $\lambda$ .
Resonant Giant Diffusivity: The most significant finding is the existence of a resonance in the effective diffusivity ( $D_{eff}$ ) as a function of the non-reciprocity parameter $\chi$ . The analytical expression for diffusivity is:
$D_{eff} = D_s + \frac{N(\Lambda/\lambda)^2 D_b}{(1 + N\Lambda/\lambda)^2}$
This expression exhibits a divergence (in the ideal linear model) at a critical non-reciprocity value:
$\chi^* = -1/N$
This "giant diffusivity" regime is robust; it depends solely on the number of bath particles ( $N$ ) and the degree of non-reciprocity, remaining insensitive to whether the particles are active or passive. Numerical simulations with short-range interactions confirm that this resonance persists beyond the idealized harmonic model, though physical nonlinearities regularize the divergence.
Violation of FDR: The authors formulate a generalized FDR relating velocity fluctuations to the dissipative part of the response. They demonstrate that the departure from equilibrium is strongly frequency-dependent, with low-frequency fluctuations significantly enhanced near the resonance $\chi^*$ .
Thermodynamic Cost: The enhanced transport is not free. The study derives the heat dissipation spectrum, showing that heat dissipation is maximized at the same non-reciprocal coupling ( $\chi^*$ ) where diffusivity peaks. This establishes a direct link between giant diffusivity and increased energetic dissipation, indicating that the enhancement is a thermodynamic trade-off rather than a violation of physical laws.

Significance
The paper identifies non-reciprocity as a fundamental control parameter for tracer transport, distinct from particle activity. It extends conventional system-bath descriptions to a broader class of active, biological, and information-driven systems. The predicted resonant enhancement of diffusivity provides a specific, experimentally accessible signature for detecting non-reciprocal interactions in nonequilibrium media. Furthermore, the work clarifies that while non-reciprocal interactions can generate exceptionally large diffusivity, this comes with a well-defined thermodynamic cost, offering a new perspective on energy dissipation in active matter and biological systems such as predator-prey dynamics. The results suggest that tuning coupling asymmetry and bath size could provide experimental routes to control exploration and transport in nonequilibrium environments.

Technical Summary: Interplay of Activity and Non-reciprocity in Tracer Dynamics

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