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Non-reciprocally interacting Ornstein-Uhlenbeck processes: Exceptional points, Anomalous relaxation, Pseudo-equilibrium and Boundary refrigeration

This paper investigates non-reciprocally interacting Ornstein-Uhlenbeck processes to demonstrate how exceptional points induce anomalous polynomial relaxation, how disorder creates non-self-averaging singularities, and how complete asymmetry leads to pseudo-equilibrium states and boundary refrigeration effects.

Original authors: Soumya Kanti Pal, Shamik Gupta

Published 2026-09-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Soumya Kanti Pal, Shamik Gupta

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, predictable world of classical physics, forces usually come in pairs. If you push a door, the door pushes back with equal strength; if one particle pulls another, the second pulls back just as hard. This principle of reciprocity is the bedrock of equilibrium, the state where systems settle down and stop changing. But the universe is full of systems that refuse to sit still. From the bustling activity of cells to the chaotic flow of traffic, many systems operate far from this calm balance. In these active worlds, interactions are often one-sided: one part influences another without receiving the same influence in return. This lack of reciprocity breaks the rules of equilibrium, driving systems into a constant state of flux. Scientists have long known that such non-reciprocal interactions create strange, collective behaviors, but the precise mathematical machinery behind these phenomena has remained elusive, particularly in how these systems relax toward a steady state.

A team of researchers at the Tata Institute of Fundamental Research in India has now built a set of simplified, yet powerful, models to explore this uncharted territory. They focused on a class of mathematical descriptions known as Ornstein-Uhlenbeck processes, which are essentially the standard way physicists describe particles jiggling in a fluid while being tugged by springs. By introducing a tunable knob that controls how strongly one particle influences its neighbor compared to how the neighbor influences it back, they created a hierarchy of models that range from simple pairs of particles to long chains of interacting bodies. Their work reveals that when this non-reciprocal influence is tuned to a very specific value, the system hits a mathematical singularity known as an exceptional point. At this precise moment, the usual rules of how the system relaxes break down, giving way to a new, slower form of decay that is dictated by the geometry of the system itself.

The researchers discovered that at these exceptional points, the system does not simply fade away into its final state as it normally would. Instead of a smooth, exponential decay where the activity drops off quickly and predictably, the relaxation becomes "anomalous." The decay is dressed with a polynomial factor, meaning the system lingers longer, and the rate at which it slows down depends on the size of the system and the specific arrangement of the particles. In their simplest models, consisting of pairs of particles, this effect is straightforward. However, when they scaled up to a long chain of particles interacting with their nearest neighbors, the behavior became even more intricate. They found that the mathematical structure of the chain, specifically how the particles are arranged in space, is directly encoded into the way the system relaxes over time. The further apart two particles are, the more complex the polynomial factor becomes, effectively imprinting the spatial layout of the chain onto the time it takes for the system to settle.

One of the most striking findings emerged when the researchers pushed the non-reciprocity to its extreme limit, creating a state of complete asymmetry. In this regime, the system enters a peculiar condition the authors call "pseudo-equilibrium." Here, the statistical distribution of the particles looks exactly like a system in thermal equilibrium, where everything is balanced and calm. Yet, despite this appearance of stillness, the system is actually churning with a steady, non-zero flow of probability current. It is a state that mimics the quiet order of equilibrium while secretly maintaining a constant, directed motion. Remarkably, in this extreme state, the complex interactions of the entire chain can be mathematically reduced to a set of independent, simpler processes, suggesting a hidden simplicity beneath the apparent complexity of the non-reciprocal drive.

The study also delved into the energetic cost of maintaining these non-equilibrium states. By calculating the heat dissipated by the system into its surrounding environment, the researchers uncovered a phenomenon they describe as boundary refrigeration. In a typical scenario, the boundaries of a system act as heat sources, dumping energy into the environment. However, by tuning the strength of the non-reciprocal interaction, the researchers found that the particles at the very ends of the chain could switch roles. Depending on the settings, one boundary particle could act as a hot reservoir, releasing heat, while the other acted as a cold reservoir, absorbing heat from the surroundings. This reversal happens without violating the fundamental laws of thermodynamics, as the total heat dissipated by the entire system remains positive. It is a counterintuitive result where the edges of the system can be cooled by the very mechanism that drives the whole system out of equilibrium.

The implications of these findings extend beyond the specific models studied. The researchers showed that their chain of interacting particles is mathematically identical to a famous model in non-Hermitian quantum physics known as the Hatano-Nelson model. This connection bridges the gap between classical stochastic systems, which describe random motion, and quantum systems, which describe the behavior of particles at the smallest scales. By demonstrating that simple, exactly solvable classical models can host these exotic exceptional points and exhibit such rich behavior, the work provides a new toolkit for understanding how non-reciprocity shapes the dynamics of active matter, biological networks, and disordered systems. The results suggest that the strange, singular behaviors once thought to be the exclusive domain of complex quantum mechanics are actually accessible in much simpler, classical settings, offering a clearer path to understanding the physics of systems that never truly rest.

Technical Summary: Non-reciprocally interacting Ornstein-Uhlenbeck processes: Exceptional points, Anomalous relaxation, Pseudo-equilibrium and Boundary refrigeration

Problem Statement
Reciprocal interactions, governed by action-reaction symmetry, form the basis of equilibrium statistical mechanics. However, many physical systems—including active matter, biological networks, and disordered systems—operate under non-reciprocal interactions where the influence of one constituent on another is not reciprocated. While non-reciprocity is known to break detailed balance and drive systems out of equilibrium, the specific role of exceptional points (EPs)—spectral singularities where eigenvalues and eigenvectors coalesce—in classical stochastic systems remains underexplored. Unlike non-Hermitian quantum systems, where EPs are well-studied, a systematic framework connecting non-reciprocal classical stochastic dynamics to EPs of various orders, and their consequences for relaxation and steady-state thermodynamics, has been lacking.

Methodology
The authors construct a hierarchy of exactly solvable Non-Reciprocally Interacting Ornstein-Uhlenbeck (NROU) models governed by a tunable non-reciprocity parameter $g$ . The study utilizes the multivariate OU process formalism, where the dynamics are described by a drift matrix $M$ . The authors analyze three specific models:

Model I: Non-interacting dimers in a uniform harmonic trapping potential with non-reciprocal coupling between dimer constituents.
Model II: Non-interacting dimers in a disordered harmonic trapping potential (quenched disorder in stiffness constants) with non-reciprocal coupling.
Model III: A many-body chain of particles in a uniform harmonic potential with nearest-neighbor non-reciprocal interactions and open boundary conditions. This model is shown to map exactly onto the Hatano-Nelson model, a paradigmatic non-Hermitian quantum system.

The analysis relies on linear algebra techniques, specifically the Jordan canonical form, to identify conditions where the drift matrix becomes non-diagonalizable. The authors calculate autocorrelation functions, covariance matrices, and steady-state probability currents. For thermodynamic analysis, they employ the Harada-Sasa relation to compute heat dissipation and quantify the violation of the fluctuation-dissipation theorem.

Key Contributions and Results

Realization of Exceptional Points (EPs): The authors demonstrate that tuning the non-reciprocity parameter $g$ to a critical value $g^*$ causes the drift matrix to become non-diagonalizable, realizing EPs of different orders.
- In Model I and II (dimers), EPs of order 2 occur.
- In Model III (many-body chain), an EP of order $N$ (where $N$ is the system size) occurs at $g^*=0$ under open boundary conditions. This is linked to the Jordan block structure $J_N(0)$ .
- The study highlights that EPs are strictly conditioned on non-Hermiticity; Hermitian matrices cannot host EPs.
Universality in Disordered Systems (Model II): For the disordered model, the location of the EP ( $g^*$ ) fluctuates across disorder realizations. The authors show that the distribution of $g^*$ develops a universal edge singularity precisely at the EP location of the corresponding clean system ( $g^*=0$ ), regardless of the underlying disorder distribution. Furthermore, this distribution is non-self-averaging, meaning the mean and the most probable value do not coincide.
Anomalous Relaxation: A central finding is the modification of relaxation dynamics at the EP.
- Autocorrelation: Away from the EP, relaxation is exponential. At the EP, the exponential decay is "dressed" by a polynomial-in-time prefactor. The degree of this polynomial is determined by the order of the EP.
- Spatial Encoding: In the many-body chain (Model III), the structure of the polynomial prefactor in the covariance (cross-correlation) function encodes the spatial architecture of the chain. Specifically, the degree of the polynomial depends on the spatial separation between particles, and the functional form reflects the particle locations.
- Phase Behavior: The relative autocorrelation function exhibits oscillatory behavior for $g < g^*$ , polynomial growth (anomalous relaxation) at $g = g^*$ , and exponential growth for $g > g^*$ .
Pseudo-Equilibrium at Complete Asymmetry: At the special point $g = -1$ in Model III, the system exhibits "pseudo-equilibrium."
- The steady-state distribution factorizes into independent, equilibrium-like single-particle measures (Gibbs form), resulting in zero equal-time cross-correlations.
- Despite this equilibrium-like static distribution, the system sustains a nonzero steady-state probability current, making it a genuine Non-Equilibrium Steady State (NESS).
- At this point, the $N$ -particle interacting dynamics can be exactly mapped to $N/2$ independent complex OU processes.
Heat Dissipation and Boundary Refrigeration: Using the Harada-Sasa relation, the authors derive a closed-form expression for total steady-state heat dissipation.
- While the total heat dissipation remains positive (consistent with the second law), individual particle analysis reveals distinct behaviors.
- Boundary Refrigeration: A transition is observed where boundary particles switch roles. As $g$ is tuned, one boundary particle can act as a heat source (positive dissipation) while the other acts as a heat sink (negative dissipation, absorbing heat from the bath). This switch occurs around the equilibrium point $g=1$ , realizing a "boundary refrigeration" effect.

Significance and Claims
The paper claims to bridge the gap between non-Hermitian spectral theory and nonequilibrium stochastic thermodynamics. By constructing simple, exactly solvable models, the authors demonstrate that:

Non-reciprocity naturally gives rise to EPs of various orders in classical stochastic systems.
These singularities fundamentally alter relaxation dynamics, introducing polynomial time-dependencies that encode spatial information.
Non-equilibrium systems can exhibit "pseudo-equilibrium" static properties while maintaining non-zero currents.
Tunable non-reciprocity can induce thermodynamic phenomena like boundary refrigeration, where boundary conditions effectively switch between hot and cold reservoirs.

The work suggests that these simple models provide a foundational framework for understanding complex phenomena in active matter, biological systems, and disordered materials where non-reciprocal interactions are ubiquitous.

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