Active Hamiltonian Systems
Jürgen Horbach (Prof. Dr.)

We consider a class of non-standard, two-dimensional (2D) Hamiltonian
models that may show features of active particle dynamics, and
therefore, we refer to these models as active Hamiltonian (AH) systems.
The idea is to consider a spin fluid where -- on top of spin-spin and
particle-particle interactions -- spins are coupled to the particle's
velocities via a vector potential. Continuous spin variables interact
with each other as in a standard $XY$ model. Typically, the AH models
exhibit non-standard thermodynamic properties (e.g.~for temperature and
pressure) and equations of motion with non-standard forces. This implies
that the derivation of symplectic algorithms to numerically solve
Hamilton's equations of motion, as well as the thermostatting for these
systems, is not straightforward. However, one can make use of the fact
that for Hamiltonian systems the equipartition theorem holds, providing
a clear definition of temperature (note, however, that the temperature
is not given by the average kinetic energy in this case) [1]. We derive
a symplectic integration scheme and propose a Nos\'e-Poincar\'e
thermostat, providing a correct sampling in the canonical ensemble [2].
Results for two different AH models are presented: (i) A model proposed
by Casiulis et al. [3] shows transition from a fluid at high temperature
to a cluster phase at low temperature where, due to the coupling of
velocities and spins, a center-of-mass motion of the cluster occurs. The
claim in Ref. [3] that this cluster motion is reminiscent of real flocks
of birds has been challenged by Cavagna et al. [4]. (ii) We propose an
AH model where spins and velocities are coupled such that as a result
particles feel a generalized Lorentz force. We show that our model leads
to a collective motion of particle clusters that is closer to the
behavior of flocks of birds.

[1] K. Huang, Statistical Mechanics (John Wiley \& Sons, New York,
1987).
[2] A. Bhattacharya, J. Horbach, and S. Karmakar, arXiv:2409.14864
(2024).
[3] M. Casiulis, M. Tarzia, L. F. Cugliandolo, and O. Dauchot, Phys.
Rev. Lett. {\bf 124}, 198001 (2020).
[4] A. Cavagna, I. Giardina, and M. Viale, arXiv:1912.07056 (2019).