Hamiltonian description of nonreciprocal interactions

TL;DR

This paper introduces a Hamiltonian framework for nonreciprocal interactions, enabling the use of classical statistical mechanics tools in systems lacking potential-based interactions.

Contribution

The authors construct a Hamiltonian with auxiliary degrees of freedom that reproduces nonreciprocal dynamics, extending Hamiltonian methods to nonpotential systems.

Findings

Glauber dynamics on the constrained Hamiltonian replicate original Langevin states.

The framework allows tuning interactions via periodic drives, changing lattice geometries.

It opens new avenues for applying Hamiltonian mechanics to nonreciprocal systems.

Abstract

In a vast class of systems, which includes members as diverse as sedimenting particles and bird flocks, interactions do not stem from a potential, and are in general nonreciprocal. Thus, it is not possible to define a conventional energy function, nor to use analytical or numerical tools that rely on it. Here, we overcome these limitations by constructing a Hamiltonian that includes auxiliary degrees of freedom; when subject to a constraint, this Hamiltonian yields the original nonreciprocal dynamics. We show that Glauber dynamics based on the constrained Hamiltonian reproduce both stationary and nonstationary states of the original Langevin dynamics, as we explicitly illustrate for dissipative XY spins with vision-cone interactions. Further, the symplectic structure inherent to our construction enables us to apply the well-developed notions of Hamiltonian engineering, which we demonstrate by varying the amplitude of a periodic drive to tune the spin interactions between those of a square and a chain lattice geometry. Overall, our framework for generic nonreciprocal pairwise interactions paves the way for bringing to bear the full conceptual and methodological power of conventional statistical mechanics and Hamiltonian dynamics to nonreciprocal systems.