Thermodynamics of collective motion

Research summary

One paper, summarised for reference. Thermodynamics and computation during collective motion near criticality — Emanuele Crosato, Richard E. Spinney, Ramil Nigmatullin, Joseph T. Lizier & Mikhail Prokopenko, Phys. Rev. E (2018) 97, 012120 (arXiv:1708.04004). doi:10.1103/PhysRevE.97.012120.

The thesis

Treat the self-organization of collective motion as a thermodynamic phenomenon and track it through the first law: as a flock orders, how do generalized internal energy and work trade off against configuration entropy? Driving a self-propelled-particle model across its kinetic (disorder→coherence) transition, the paper finds that at criticality the entropy drops, the rates of change of work and energy fall while their curvatures diverge, and — the headline — the entropy reduction achieved per unit of work (a thermodynamic efficiency of computation) peaks at the critical point. This is the concrete flocking demonstration behind the “super-efficiency at criticality” idea.

The setup

The Grégoire–Chaté model ($N=512$ self-propelled particles: alignment + cohesion + perturbation), with two control parameters — the alignment strength $J = v_0 a/n_c$ and the number of nearest neighbours $n_c$ (topological). A quasi-static protocol varies them infinitesimally slowly so the system stays in equilibrium, and the velocity distribution (relative to each particle’s neighbourhood) is estimated from simulation — connecting to Bialek et al.’s maximum-entropy model $p(v_i\mid J,n_c)\propto\exp(\tfrac{J}{2}\sum v_i\!\cdot\!v_j)$.

Fisher information ↔ thermodynamics

The engine is the identity that Fisher information is the curvature of the free entropy $\psi=\ln Z$ (the thermodynamic metric), which in the quasi-static limit equals minus the curvature of work: $F(\theta) = -\,d^2\langle\beta W_{\text{gen}}\rangle/d\theta^2$. Combined with the first law $d\langle\beta U_{\text{gen}}\rangle/d\theta = dS/d\theta + d\langle\beta W_{\text{gen}}\rangle/d\theta$, this gives the information-geometric decomposition

$$\frac{d^2\langle\beta U_{\text{gen}}\rangle}{d\theta^2} = \frac{d^2 S}{d\theta^2} - F(\theta),$$

i.e. the internal-energy curvature is a computational balance between the system’s sensitivity (Fisher information) and its uncertainty (configuration-entropy curvature) — a balance that is broken at criticality.

Results

• Fisher information diverges at criticality ($J\approx0.075$ for $n_c=20$), letting the critical point be localized without any order parameter, and used to draw a phase diagram over $(J, n_c)$ (critical curve $J = v_0 a/n_c$) — useful where a good order parameter is hard to define.
• Configuration entropy drops sharply across the transition (its curvature is discontinuous), and the internal-energy curvature diverges with it.
• The thermodynamic efficiency of computation peaks at criticality: $$\eta = \frac{-\,dS/d\theta}{d\langle\beta W_{\text{gen}}\rangle/d\theta} = \frac{-\,dS/d\theta}{\int_\theta^{\theta^*} F(\theta')\,d\theta'},$$ the order generated per unit work. Because the denominator (cumulative Fisher information to the perfectly-ordered “zero-response” point $\theta^*$) acts like a distance to perfect order, one bit of uncertainty reduction near order counts for more than one bit in disorder — so the reduction of uncertainty is most significant at the critical point.

Why it’s collected here

This is the foundational, self-propelled-particle result underlying Chen & Prokopenko’s Principle of Super-efficiency — the same lab, the same $\eta$, shown in an actual flocking model rather than an Ising lattice. Two methods it contributes are directly usable: Fisher information as an order-parameter-free phase detector (a cleaner criticality probe than a hand-picked susceptibility), and the sensitivity-vs-uncertainty decomposition of internal energy, which is a principled way to read what diverges — and what balance breaks — at a collective transition.

References

Crosato et al. (2018), Phys. Rev. E 97, 012120 · Grégoire & Chaté (2004), Phys. Rev. Lett. 92, 025702 · Vicsek et al. (1995), Phys. Rev. Lett. 75, 1226 · Bialek et al. (2012), PNAS 109, 4786 · Prokopenko, Lizier, Obst & Wang (2011), Phys. Rev. E 84, 041116 · Crooks (2007), Phys. Rev. Lett. 99, 100602 · Nigmatullin & Prokopenko (2021), Entropy 23, 757 · Mora & Bialek (2011), J. Stat. Phys. 144, 268 · Cavagna et al. (2010), PNAS 107, 11865.