Session M29: Non-Reciprocity in Soft and Active Matter I

Generalized time-reversal symmetry and effective theories for nonequilibrium matter

The past decade has witnessed the development of systematic effective theories for dissipative thermal systems. In this talk, I will describe an analogous effective theory framework that applies to the classical stochastic dynamics of nonequilibrium systems. This approach applies to a range of examples, including nonreciprocal (predator-prey) dynamics, dissipative and driven rigid-body motion, and active chiral fluids and solids. Many of these systems exhibit a generalized time-reversal symmetry, which plays a crucial role within our formalism, and which in many cases can be implemented within the Martin-Siggia-Rose path integral. This effective theory formalism yields generalizations of the fluctuation-dissipation theorem and the second law of thermodynamics valid out of equilibrium. By stipulating a stationary distribution and a set of symmetries—rather than postulating the stochastic equations of motion directly—the effective theory framework provides an alternative route to building phenomenological models of driven and active matter.   

Effective field theory approach to odd elasticity

We have developed an effective field theory (EFT) framework applicable to active, non-equilibrium systems. Our approach to applying the EFT paradigm in the context of active matter departs from traditional methods that start with equations of motion. Instead, we postulate the form of the non-equilibrium steady state, which captures equal-time correlations. This approach offers an alternative pathway for constructing phenomenological models for driven and active matter. In this work, we apply our EFT formalism to problems in active solids—solids with active, non-conservative microscopic interactions that enable the material to perform work on its environment through quasi-static cycles of deformations. In the realm of odd elasticity, these active components contribute to the odd component of the static elastic modulus tensor. Specifically, we apply our formalism to a paper by Scheibner et al. [1], which established the equations of motion for odd elastic solids by postulating the most general stress tensor consistent with symmetry principles. By assuming there must be some notion of stationarity in this system, we discover additional constraints on certain active moduli. Our approach brings to light subtleties related to the nature of noise and time reversal symmetry breaking, leading to nontrivial results that have previously been overlooked.

[1] Scheibner, C., Souslov, A., Banerjee, D., Surówka, P., Irvine, W. T., & Vitelli, V. (2020). Odd elasticity. Nature Physics, 16(4), 475–480.    

Optimal design of odd elastic metamaterials

Non-reciprocal interactions in active solids yield elastic moduli forbidden in equilibrium. These odd moduli offer a bottom-up approach to designing autonomous materials that spontaneously crawl, roll or swim. However, current schemes for designing odd materials are typically overactuated, using excessively many active elements. This overactuation limits the feasibility of large-scale experimental realisations, and invites the challenge of rational design: how can microscopic non-reciprocity be optimally converted into macroscale odd response?

 

Here we show that odd moduli emerge in a broad range of lattices made of non-reciprocal springs. However, the strength of odd response strongly depends on the precise lattice geometry. Hyperstatic lattices are needlessly hard to actuate, leading to sub-optimal odd response. By contrast, we find that in overly floppy lattices, zero modes couple to microscopic non-reciprocity, destroying odd moduli entirely. By avoiding these pitfalls, we identify optimal design principles for building odd lattices.

 

Our results offer a blueprint for efficient engineering of odd lattices, opening the door to experimental realisations that truly probe the continuum limit of these robotic materials.   

Non-reciprocity permits novel dynamics in living and active topological systems

Living and active systems exhibit various emergent dynamics during system regulation, growth, and motility. However, how robust dynamics arises from stochastic components remains unclear. Towards understanding this, I develop topological theories that support robust edge currents, effectively reducing the system dynamics to a lower-dimensional subspace. I will introduce stochastic networks in molecular configuration space that model different systems from a circadian clock to ring attractors. The edge localization results in new properties, e.g., the clock demonstrates increased precision with simultaneously decreased cost. Crucially, we find that unlike in quantum systems, non-reciprocity is strictly necessary for edge states and strong localization in stochastic topological systems. Our work indicates new pathways for the design and control of active systems and their dynamics.   

Tuning non-reciprocal pattern

Reciprocity is a fundamental principle guiding the behavior of the majority of physical systems. However, when these systems are out of equilibrium, a lack of reciprocity becomes more common. In recent years, researchers have extensively explored non-reciprocity across various systems. Nevertheless, the vast majority of tend to converge towards a single specific state. Here, we demonstrate the potential of multiple states in non-reciprocity active metamaterials, both theoretically and experimentally. By manipulating the initial states, we have the capability to specifically target a desired final state. Furthermore, when the system is initiated from random noise, countless mixed states, combinations of the basic states over time or space, can also emerge. We foresee that exploring multiple non-reciprocal states will pave the way for a deeper understanding of active matter and living matter, and open avenues in the fields of wave propagations, versatile soft robots and metamaterials.

Lorentz Reciprocal Theorem in Fluids with Odd Viscosity

The Lorentz reciprocal theorem---that is used to study various transport phenomena in hydrodynamics---is violated in chiral active fluids that feature odd viscosity with broken time-reversal and parity symmetries. Here we show that the theorem can be generalized to fluids with odd viscosity by choosing an auxiliary problem with the opposite sign of the odd viscosity [1]. To demonstrate its applicability, we use the thorem to determine the swimming velocity of two categories of microswimmers in a Stokes fluid with odd viscosity: first to swimmers with an imposed slip velocity on their surface and then to swimmers with a prescribed tangential force density. We solve both problems for spherical swimmers in a 3D fluid as well as disk-shaped swimmers in 2D. In both cases, we show that odd viscosity does not affect the motion of swimmers with prescribed surface velocity, but it does affect those with prescribed propulsive forces. A particularly interesting case is a "twister" which propels itself by exerting only a torque dipole on the fluid.

[1] Y. Hosaka, R. Golestanian, and A. Vilfan, Phys. Rev. Lett. 131, 178303 (2023).   

Non-reciprocal stochastic topological networks show new properties as compared to quantum counterparts

Stochastic topological systems draw from topological invariants first developed for quantum systems. While stochastic and quantum systems can share the same invariant in the bulk, their spectra differ under open boundary conditions, leading to new properties. We systematically investigate how spectra in both systems differ. Solving the spectrum analytically using Chebyshev polynomials, we find that in a 1D uniform chain and SSH model with even sites, the stochastic spectrum is given by a one-site smaller quantum system. The spectral differences are most prominent in small system sizes and large non-reciprocity, where quantum states converge while the gap increases between the steady-state and the slowest decaying state in stochastic systems. In the 2D SSH model, we find that exceptional points emerge in different areas of the spectrum in the topological phase. More broadly, this work characterizes unique physical properties that emerge from identical networks described by Laplacian and adjacency matrices respectively.   

Active Boltzmann equation for self-propelled particles with non-reciprocal interactions

We study models of self-propelled particles with alignment interactions by means of an active Boltzmann equation. To this end, we evaluate the collision integral directly based on the microscopic equations of motion. This kinetic theory is founded on the assumption of one-sided molecular chaos, a refinement to the molecular chaos assumption underlying commonly employed mean-field approaches. We have introduced the methods for Vicsek-type active particles with anti-alignment that allows for an asymptotically exact analytical solution. One direct application is the extraction of the self-diffusion coefficient which cannot be done to mean-field order. This exact solution has then been used to study the behaviour of binary mixtures with non-reciprocal and chiral interactions. In addition to analytical approaches, the methodology allows for numerical calculation of the collision integral in systems that cannot be treated analytically which we use to evaluate the beyond mean-field corrections to the flocking transition under noise.   

Multicomponent active diffusion

Nonreciprocal interactions are ubiquitous in the natural and living world and can endow systems with properties that have no analog in passive materials. A familiar example of a nonreciprocal interaction is the so-called "predator-prey" interaction whereby one entity (the predator) feels an attractive force towards the other while the other (the prey) is repelled. These effective nonreciprocal interactions can emerge from a host of complex factors and can have far-reaching implications on collective phenomena, phase transitions, and pattern formation. In this talk, we present a dynamical framework for determining the stability of multicomponent active systems. We present phenomenological linear transport relations and derive Green-Kubo relations for the linear transport coefficients that govern the stability of these nonequilibrium systems. For a large class of systems, we can further relate these transport coefficients to mechanical variables, providing a simple criterion for the emergence of traveling phases. Our perspective thus provides a multiscale framework for understanding nonreciprocal phase transitions with multiple conserved order parameters. We demonstrate the utility of this perspective by applying it to two model active systems and predicting the emergence of both stationary and traveling phase transitions.   

Odd diffusivity and odd mobility in a system with equilibrium structure

Odd transport coefficients may generally appear in systems with broken time-reversal and parity symmetry [1,2]. Typical examples are systems that are driven at the level of individual constituents, and thus have stationary state distributions that are not known explicitly. This makes the analysis of these systems challenging. Here we consider a system of  interacting Brownian particles evolving under the influence of forces that have components transverse to energy gradients [3]. The evolution breaks the time-reversal and parity symmetry but its stationary distribution coincides with the equilibrium distribution. We analyze the tracer dynamics and derive expressions for the self-diffusion and mobility coefficients, including their odd components. A mode-coupling approximation predicts that the ratio of the odd diffusion and odd mobility diverges at the dynamic glass transition, which implies an extreme violation of the Einstein relation.

[1] C. Hargus, J.M. Epstein and K.K. Mandadapu, PRL 127, 178001 (2021).

[2] A.R. Poggioli and D.T. Limmer, PRL 130, 158201 (2023).

[3] F. Ghimenti, L. Berthier, G. Szamel and F. van Wijland, arXiv:2307.02840.   

Space-time symmetry and non-reciprocal transport in dynamic mechanical systems

Active mechanical structures whose parameters are modulated in space and time harbor wave phenomena, such as amplification and non-reciprocal transport, that are forbidden in passive materials. One such phenomenon is parametric resonance, which generates exponentially growing or decaying oscillations when parameters are modulated periodically in time. While parametric resonance of individual oscillators is well understood, identifying the conditions for parametric resonance in systems of coupled oscillators remains challenging. In this talk, we will identify and use the non-Hermitian internal symmetries arising from the real-valued and symplectic nature of classical mechanics to determine these parametric resonance conditions. Upon including external symmetries, we find the conditions for modes to be protected from resonating at certain modulation frequencies where they were expected to resonate in the absence of the external symmetry. In particular, we analyze systems with space-time symmetry where the system remains invariant after a combination of discrete translation in both space and time. For such systems, we identify a combined space-time translation operator that provides more information about the system than the Floquet operator does, and use it to derive conditions for non-reciprocal wave amplification in mechanical metamaterials. Our results establish an exact theoretical framework based on symmetries to engineer non-reciprocal transport in out-of-equilibrium systems.   

Molecular modelling of odd viscoelastic fluids

In materials that break chiral symmetry, the stress response to applied strain can contain unusual coefficients dubbed odd viscosity (in fluids) or odd elasticity (in solids), both of which can be seen as special cases of a general phenomenon of odd viscoelasticity. In this talk, I will describe and analyze the first known microscopic model that produces an odd viscoelastic fluid. The model consists of a collection of dumbbells suspended in a fluid, with the beads of the dumbbells featuring an active driving system that exerts torque on the dumbbells. After coarse-graining the model we analytically calculate the odd viscoelastic coefficients and corroborate the findings using molecular dynamics simulations.   

Non-reciprocal alignment can induce asymmetric clustering in active repulsive mixtures

It is now well established that non-reciprocal systems exhibit intriguing, novel dynamical phases, the characteristics of which are shaped by the type and degree of non-reciprocity [1-3]. Here, we study a paradigmatic model of non-reciprocal active matter, namely a binary mixture of motile particles with completely symmetric repulsive interactions and non-reciprocal alignment couplings [3]. Using a combination of hydrodynamic theory, linear stability analysis, and particle-based simulations, we find dynamical, asymmetrical clustering situations, in which weakly polarized clusters form out of only one of the two species. Importantly, these asymmetric clusters emerge even though the isotropic repulsive interactions do not distinguish one species. Instead, the clustering is driven solely by non-reciprocal orientational couplings. For systems with antagonistic (anti-)alignment couplings, the resulting single-species clusters move and chase more dilute accumulations of the other species. We present a full non-equilibrium phase diagram in the parameter space of inter-species coupling strengths and compare with particle-based simulations, highlighting the impact of non-reciprocity on various scales.

[1] Z. You et al., PNAS 117, 19767 (2020).

[2] M. Fruchart et al., Nature 592, 363 (2021).

[3] K. L. Kreienkamp and S. H. L. Klapp, New J. Phys. 24, 123009 (2022).