  We study self-organisation of collective motion as a thermodynamic
phenomenon, in the context of the first law of thermodynamics. It is expected
that the coherent ordered motion typically self-organises in the presence of
changes in the (generalised) internal energy and of (generalised) work done on,
or extracted from, the system. We aim to explicitly quantify changes in these
two quantities in a system of simulated self-propelled particles, and contrast
them with changes in the system's configuration entropy. In doing so, we adapt
a thermodynamic formulation of the curvatures of the internal energy and the
work, with respect to two parameters that control the particles' alignment.
This allows us to systematically investigate the behaviour of the system by
varying the two control parameters to drive the system across a kinetic phase
transition. Our results identify critical regimes and show that during the
phase transition, where the configuration entropy of the system decreases, the
rates of change of the work and of the internal energy also decrease, while
their curvatures diverge. Importantly, the reduction of entropy achieved
through expenditure of work is shown to peak at criticality. We relate this
both to a thermodynamic efficiency and the significance of the increased order
with respect to a computational path. Additionally, this study provides an
information-geometric interpretation of the curvature of the internal energy as
the difference between two curvatures: the curvature of the free entropy,
captured by the Fisher information, and the curvature of the configuration
entropy.
