Hamiltonian description of non-reciprocal interactions

Area of Science:

• Statistical mechanics
• Non-reciprocal systems
• Hamiltonian dynamics

Background:

• Many physical systems, such as sedimenting particles and bird flocks, exhibit non-reciprocal interactions.
• These interactions do not stem from a potential and violate the action-reaction principle, precluding conventional energy function definitions.
• This limitation hinders the application of standard analytical and numerical tools in these systems.

Purpose of the Study:

• To overcome the limitations of analyzing non-reciprocal systems.
• To develop a method for defining Hamiltonian dynamics and statistical mechanics for systems without conventional energy functions.
• To enable the simulation and analysis of complex non-reciprocal phenomena.

Main Methods:

• Constructed a Hamiltonian incorporating auxiliary degrees of freedom.
• Applied a constraint to this Hamiltonian to generate the original non-reciprocal dynamics.
• Utilized Monte Carlo simulations based on the constrained Hamiltonian.
• Employed Hamiltonian engineering techniques, including periodic (Floquet) drives.

Main Results:

• The constrained Hamiltonian successfully reproduced both stationary and non-stationary states of the original Langevin dynamics.
• Demonstrated the method's efficacy using dissipative XY spins with vision-cone interactions.
• Showcased the ability to tune spin interactions between different lattice geometries (square and chain) by varying the amplitude of a Floquet drive.
• Validated the extension of statistical mechanics and Hamiltonian dynamics to non-reciprocal systems.

Conclusions:

• The proposed Hamiltonian construction provides a viable framework for studying non-reciprocal systems.
• This method extends the applicability of Hamiltonian dynamics and statistical mechanics beyond potential-based interactions.
• The approach opens new avenues for research in diverse fields involving non-reciprocal phenomena.