Active Systems Linked to New Symmetry Principle

Scientists have long sought to understand non-equilibrium phenomena, from the complex firing of neurons to the seemingly simple overflow of a fountain, often modelling these using nonreciprocal theories. Savdeep Sethi and Gabriel Artur Weiderpass, both from the University of Chicago, demonstrate a surprising connection between these stochastic theories describing nonreciprocal interactions and the mathematical framework of supersymmetry, revealing a mapping onto field theories possessing a single supercharge. This work significantly extends the earlier reciprocal theories pioneered by Parisi and Sourlas, which relied on interactions derived from potentials, and offers a novel approach to analysing a broad class of active matter systems characterised by non-Hermitian behaviour.

Can we understand complex systems, from living cells to flowing fluids, using the same mathematical tools. It appears we can, as a connection has been established between systems lacking balance and a powerful, elegant framework from theoretical physics. This mapping offers a fresh approach to modelling diverse natural phenomena with surprising simplicity.

Scientists routinely employ nonreciprocal theories to model diverse natural phenomena, extending from the complex activity of neuronal networks to the behaviour of overflowing fountains. These theories are particularly useful when describing active matter systems, where components exhibit self-propelled motion and internal forces. Scientists have demonstrated a surprising connection between these stochastic theories, those incorporating random fluctuations. The mathematical framework of supersymmetry, a concept originating in particle physics.

Specifically, they reveal that nonreciprocal interactions can be mapped onto quantum field theories possessing a single accelerate, a key element in supersymmetric systems. Unlike previous work focusing on reciprocal interactions derived from potential energies, these newly explored theories are generally non-Hermitian, meaning they do not adhere to the standard symmetry properties of many physical systems.

This effort builds upon earlier investigations, extending a reciprocal framework to encompass nonreciprocal forces. When considering two interacting components, a force exerted by one on the other is not necessarily met with an equal and opposite reaction. Envision two springs coupled in a way that violates typical energy gradients, creating a system where forces are not conservative.

Through modelling such systems requires a stochastic approach, accounting for inherent fluctuations. Even without introducing random noise, these deterministic systems can exhibit complex behaviour, observed in areas like machine learning and neuroscience. By mapping these stochastic frameworks into quantum field theories, scientists gain access to a powerful set of tools for analysis.

At the heart of this mapping lies a supersymmetric action, an equation exhibiting symmetry between bosons and fermions, fundamental particle types. The discovery of a manifest N = 1 supersymmetry, even with nonreciprocal interactions, is a significant step forward. Here, scientists have constructed an explicit supersymmetric action, mirroring previous approaches.

In turn, this action, though generally non-Hermitian, opens avenues for exploring a wider range of physical phenomena, including exceptional points where standard physical rules break down. Beyond simple ordinary differential equations, this framework extends to stochastic partial differential equations, offering insights into systems like the stochastic heat equation and the Ising model, potentially impacting our understanding of soft matter, active matter, and condensed matter physics.

Langevin dynamics and stochastic integral regularisation techniques

Initially, a set of variables, denoted as φi(t) where i ranges from 1 to n — define a dynamical system with an n-dimensional target space M. A vector field, fi(φ), residing on M. Dictates the system’s flow and forms the basis for a deterministic dynamical platform described by φi = fi, and this deterministic framework is extended into a stochastic ordinary differential equation, specifically the Langevin equation, given by φi = fi + ξi. Where ξi represents Gaussian white noise.

Meanwhile, the statistical properties of this noise are defined by a partition function, Z = ∫ Dξi exp(− ∫ dt ξiξi / 2σ). Establishing the foundation for probabilistic behaviour. Defining the stochastic theory necessitates a regularization of the noise term within the Langevin equation. Two common approaches exist, Ito and Stratonovich regularization, each offering a distinct method for handling the stochastic integral.

For physical applications, the Stratonovich method is favoured as it aligns with conventional calculus rules. At the same time, the project team adopted a path-integral formulation, defining the regularized stochastic system through a mapping into a path-integral description. Such an approach allows for the computation of correlation functions by averaging over the noise. Expressed as ⟨φi1 ξ(t1) … φin ξ(tn)⟩ = ∫ Dξi(t’) φi1 ξ(t1) … φin ξ(tn) exp(− ∫ dt’ ξi(t’)2 / 2σ).

In turn, the Martin-Siggia-Rose (MSR) procedure was implemented. The method, analogous to the Faddeev-Popov trick used in gauge theory, introduces a response field φi and utilizes a Faddeev-Popov identity — 1 = ∫ Dφi δEi −ξi Jac δEi δφj, within the path integral. By integrating over the noise and the response field, the team arrived at a functional integral, and z = ∫ Dφi det’ δEi δφj exp(− ∫ dt (φi −fi)2 / 2σ).

Instead of directly solving the stochastic differential equation, The project focuses on formulating the problem as a quantum mechanical one — the noise strength, σ, is treated as analogous to Planck’s constant, ħ, in the quantum theory. For simplification, σ was set to 1. At the same time, the Jacobian determinant, det’ δEi δφj, requires careful consideration of potential zero modes, and this must be addressed separately to ensure a well-defined path integral.

Supersymmetric field theory construction from nonreciprocal stochastic dynamics

The project details a mapping between stochastic theories describing nonreciprocal interactions and field theories defined by a supersymmetric action possessing a single accelerate. Calculations reveal a Hamiltonian density, H = (pi −iAi)2 2 + (δiS + Ai)2 2 −δjδiV + ∂jAi 2 [ψj. Ψi] + δjAi −δiAj 4 ψjψi, which arises from this mapping. This Hamiltonian incorporates terms related to momenta, a magnetic field-like coupling, and potential interactions between the fields.

Manipulation of the fermionic fields yields H = (pi −iAi)2 2 + (δiS + Ai)2 2 −δjδiV + ∂jAi 2 [ψj, ψi] + δjAi −δiAj 4 ψjψi. Another example involves coupling two kinetic Ising models nonreciprocally, leading to a supersymmetric action, S = ψ1 ∂t −∇2 ψ1 + m2φ1 + λφ3 1 ψ1ψ1 + ψ2 ∂t −∇2 ψ2 + m2φ2 + λφ3 2 ψ2ψ2 + K ψ1∇2ψ2 − ψ2∇2ψ1 + K 2 ψ2∇2ψ1 −ψ1∇2ψ2.

This action represents two copies of a base action with a nonreciprocal extension, maintaining N = 1 supersymmetry through the fermionic interactions. For stochastic differential equations, regularization is essential due to the distribution-valued nature of the noise term. A common approach involves defining the α stochastic integral as Z t 0 Ga(x(t)) ◦α dW a(t) ≡ms-lim N→∞ N X n=1 Ga (xα,n) ∆W a n, where xi α. N = αxi n+1+(1−α)xi n represents the α midpoint.

The mean square limit ensures the convergence of the stochastic integral. Also, the diffusion matrix, Dij(x) = n X a=1 σi a(x)σj a(x), is derived from the vector fields σi a(x) and can be interpreted as a metric on the manifold M.

Supersymmetry unlocks analytical tools for active matter physics

Scientists have long sought a unifying language to describe systems far from equilibrium, those messy, active realms where things happen irreversibly. For decades, modelling these active matter systems, from swarming bacteria to the complex firing of neurons, proved difficult because traditional physics tools, built for stable, predictable states, simply failed to capture their behaviour.

A new theoretical framework demonstrates a surprising connection between these systems and the seemingly distant world of supersymmetry, a concept originating in high-energy particle physics. Rather than being a mere mathematical curiosity, this mapping offers a powerful new way to analyse non-reciprocal interactions, those where cause and effect aren’t neatly symmetrical.

The implications extend beyond simply describing existing phenomena. Supersymmetryprovides established mathematical techniques for tackling complex calculations. By applying these to stochastic systems could unlock a deeper understanding of collective behaviour, potentially aiding the design of new materials with tailored properties or improving our ability to model biological processes.

The current work relies on specific mathematical mappings. The extent to which this framework can be generalised to all non-equilibrium systems remains an open question. This development represents a shift in perspective, moving away from trying to force these systems into conventional moulds. By revealing an underlying symmetry previously hidden, researchers open avenues for exploring topological properties and developing coarse-grained models that simplify complex interactions.

Translating these theoretical advances into practical applications requires bridging the gap between mathematical elegance and real-world complexity. Beyond this specific research, we might anticipate a broader adoption of supersymmetric techniques across diverse fields, offering a fresh approach to understanding the ubiquitous, yet often elusive, dynamics of non-equilibrium systems.