Hidden nonreciprocity as a stabilizing effective potential in active matter

TL;DR

The paper investigates how nonreciprocal interactions implemented as a transverse force affect the stationary states of active matter modeled by active Ornstein–Uhlenbeck particles. By decomposing forces into a conservative part and a transverse nonreciprocal part, and by analyzing persistent noise with finite $\tau$, it derives an effective potential $V_{eff}$ that stabilizes configurations near energy minima and certain nonequilibrium states, diverging from the Boltzmann form $\exp(-V/T)$ seen in the thermal limit. The authors prove and illustrate this stabilization across several models, including nonreciprocal harmonic oscillators (via $K_{eff}$), a two-species spherical model (magnetization enhancements), a nonreciprocal spherical Hopfield network (improved pattern retrieval), and nonreciprocal active swimmers (enhanced MIPS). The results imply that energy consumption and nonreciprocal couplings shape the steady states of active matter, with potential applications in inferring reciprocity and engineering desired collective behaviors.

Abstract

Nonreciprocal interactions are known to produce distinctive dynamics in active matter. To shed light on how the stationary state of such systems is affected by breaking reciprocity, we consider active Ornstein-Uhlenbeck particles coupled nonreciprocally by a transverse force, which is perpendicular to the gradient of the interaction energy. Focusing on the steady-state distribution of positions, we show that the nonreciprocal coupling helps keep the system at its stable configurations, including not only energy minima but also nonequilibrium configurations stabilized by the persistent noise which propels the particles. In contrast, the transverse force would not change the stationary distribution at all if the noise were thermal. For a variety of active systems, we demonstrate the stabilizing role of the nonreciprocity, finding that it stiffens springs, aligns spins, improves associative memory, and enhances motility-induced phase separation.

Figures (5)

• Figure 1: Illustration of the steady-state distribution $P(\mathbf{r})$ of active Ornstein-Uhlenbeck particles. The persistence in the noise $\bm\eta$ (black jagged arrows) captures the propulsion of the particles. The potential force $-\nabla V$ (blue arrow; the blue paraboloid shows the potential $V$) models reciprocal interactions, while the transverse force $\mathbf{F}_\perp \perp -\nabla V$ (red arrow) models nonreciprocal interactions. (a) In the limit of vanishing persistence, the noise is thermal, and $P(\mathbf{r})$ is the Boltzmann distribution determined by $V$ and independently of $\mathbf{F}_\perp$. (b) With persistence in the noise, $\mathbf{F}_\perp$ appears in $P(\mathbf{r})$ as an effective energy correction that helps keep the system at its stable configurations. This includes configurations that, due to particle propulsion, are not at the minima of $V$.
• Figure 2: Nonreciprocal spring-mass chain with periodic boundary conditions. (a) Schematic diagram. For each pair of neighboring masses, the left (right) mass acts on the right (left) mass with force constant $k+\alpha$ ($k-\alpha$). (b) Steady-state distribution $P(\Delta l)$ of spring displacements $\Delta l$ for various noise types and $\alpha$. Simulations (circles) match theoretical predictions (lines) for different conditions (color code). (c) Eigenvalues $\lambda_q$ for each wavenumber $q$, of the effective force constant matrix $\mathbf{K}_\text{eff}$ governing $P(\mathbf{r})$ for various noise types and values of $\alpha$.
• Figure 3: Nonreciprocal spherical model with two species of spins. (a) Schematic diagram. Spins of type $A$ and $B$ are reciprocally coupled to neighboring spins of the same type with strength $J$. In contrast, $A$ spins align with $B$ spins with strength $\alpha$, whereas $B$ spins (anti-)align with $A$ spins with strength $-\alpha$. (b) Steady-state distribution of potential energy, $P(V)$, for various noise types and values of $\alpha$. (c-f) Steady-state distribution $P(m_A,m_B)$ of the magnetizations $m_A$ and $m_B$ of $A$ and $B$ spins, respectively, under thermal (c, d) and persistent (e, f) noise and for $\alpha = 0$ (c, e) and $\alpha\neq 0$ (d, f).
• Figure 4: Nonreciprocal spherical Hopfield model of associative memory. (a, b) Schematic diagram. (a) Associative memory. Stored patterns can be retrieved from corrupted versions. (b) In the Hopfield model, the coupling among $N$ spins encodes $p$ random patterns. For small pattern loading $p/N$, the patterns are stored as minima of the potential $V$ (purple curve) and can be retrieved by gradient descent (purple arrow). We introduce nonreciprocity via a transverse force, $\mathbf{A}(-\nabla V)$ (magenta arrow), with $A_{ij}=-A_{ji} \sim \mathcal{N}(0, \alpha^2 / N)$ a random antisymmetric matrix. (c, d) We compare various quantities for thermal versus persistent noise and different values of $\alpha$. (c) Phase diagram of pattern retrieval in simulations, as a function of $T$ and $p/N$. Points of the same color form a transition line, below which the stored patterns can be retrieved and above which they cannot. (d) MSD from the energy minimum corresponding to the target pattern. Circles show values from retrieval simulations after an initial relaxation of the system, while lines are calculated from a linear stability analysis based on Eq. \ref{['eq:k-eff']}.
• Figure 5: Nonreciprocal active swimmers. (a) Schematic diagram of the interparticle forces. In addition to short-range repulsion (purple arrows) mediated by the WCA potential ($V$), the particles experience nonreciprocal interactions with strength $\alpha$. Illustrated is the case of $\alpha > 0$, in which the nonreciprocity drives particle $i$ to follow the motion of particle $i+1$ (blue arrow) and particle $i+1$ to do the opposite of particle $i$ (red arrow). (b) Steady-state distribution of potential energy, $P(V)$. (c) Steady-state pair correlation function, $g(r)$, zoomed in around the cutoff distance beyond which the pairwise repulsion vanishes ($r_\text{cut}$, dashed line). In (b) and (c), results are shown for various noise types and values of the nonreciprocity parameter $\alpha$. (d-g) Steady-state properties related to density (subpanels i-iii) for persistence times of $\tau = 0$ (thermal noise) (d), $\tau = 25$ (e), $\tau = 100$ (f), and $\tau = 400$ (g). (i, ii) Representative particle configurations for $\alpha = 0$ (i) and $\alpha \neq 0$ (ii). The colors indicate the local density ($\rho$) in each region formed by discretizing the simulation box into a $10 \times 10$ square grid. (iii) Distribution of local density, $P(\rho)$. The total number density ($\rho_0$) is indicated by the dashed line. In (b), (c), and panels (iii), error bars (vertical solid lines) are shown but, for most of the data, are too small to be seen.