Understanding the thermodynamic properties of insect swarms

Abstract

Sinhuber et al. (Sci Rep 11:3773, 2021) formulated an equation of state for laboratory swarms of the non-biting midge Chironomus riparius that holds true when the swarms are driven through thermodynamic cycles by the application external perturbations. The findings are significant because they demonstrate the surprising efficacy of classical equilibrium thermodynamics for quantitatively characterizing and predicting collective behaviour in biology. Nonetheless, the equation of state obtained by Sinhuber et al. (2021) is anomalous, lacking a physical analogue, making its' interpretation problematic. Moreover, the dynamical processes underlying the thermodynamic cycling were not identified. Here I show that insect swarms are equally well represented as van der Waals gases and I attribute the possibility of thermodynamic cycling to insect swarms consisting of several overlapping sublayers. This brings about a profound change in the understanding of laboratory swarms which until now have been regarded as consisting of non-interacting individuals and lacking any internal structure. I show how the effective interactions can be attributed to the swarms' internal structure, the external perturbations and to the presence of intrinsic noise. I thereby show that intrinsic noise which is known to be crucial for the emergence of the macroscopic mechanical properties of insect swarms is also crucial for the emergence of their thermodynamic properties as encapsulated by their equation of state.

Figure 1

(a) Vertical density profiles of laboratory swarms are accurately represented by superpositions of Gaussians. (b) Constituent Gaussians in the quad-Gaussian representation with centres at •. In accordance with observations, the best superposition “contains around 3 horizontal layers, with the lowest layer being flattered than the others” [van der Vaart, Private Communication]. The Akaike weights for the single-, bi-, tri- and quad-Gaussian fits are 0.00, 0.00, 0.08 and 0.92 indicating strong support for the quad-Gaussian representation (albeit with one weak ‘embryonic’ slab). Data are taken from Sinhuber et al.: swarm Ob1 which contains on average 94 individuals. The swarm is centred on z = 0.

Figure 2

(a). Vertical density profiles of laboratory swarms are accurately represented by superpositions of Gaussians. (b) Constituent Gaussians in the tri-Gaussian representation with centres at •. In accordance with observations, the best superpositions “contains around 3 horizontal layers, with the lowest layer being flattered than the others” [van der Vaart, Private Communication]. The Akaike weights for the single-, bi-, tri- and quad-Gaussian fits are 0.00, 0.78, 0.21 and 0.01 indicating strong support for the bi and tri-Gaussian representations. Data are taken from Sinhuber et al.: swarm Ob5 which contains on average 22 individuals. The swarm is centred on z = 0.

Figure 3

Average number of Gaussian slabs $N={\sum }_{N=1}^4{w}_{N}N$ tends to increase with increasing average population size, n. w_N are the Akaike weights for best-fit profile containing N slabs. The least-squares linear regression (solid line) is added to guide the eye (R² = 0.50). Data are taken from Sinhuber et al.. All 17 daytime swarms.

Figure 4

Mean accelerations are consistent with slabs being coupled. Data (•) are shown (a) for swarm (Ob1) which contain on average 94 individuals and (b) for a swarm (Ob5) which contains on average 22 individuals. Shown for comparison are model predictions for coupled slabs, Eq. (2) (red lines) and for decoupled slabs, Eq. (3) (blue lines). Data are taken from Sinhuber et al..

Figure 5

(a). Distributions of vertical accelerations are accurately represented by superpositions of Gaussians. (b) Constituent Gaussians in the tri-Gaussian representation with centres at •. The Akaike weights for the single-, bi-, tri- and quad-Gaussian fits are 0.00, 0.03, 0.89 and 0.09 indicating strong support for the tri-Gaussian representation. Data are taken from Sinhuber et al.: swarm Ob1 which contains on average 94 individuals.

Figure 6

(a). Distributions of vertical accelerations are accurately represented by superpositions of Gaussians. (b) Constituent Gaussians in the tri-Gaussian representation with centres at •. The Akaike weights for the single-, bi-, tri- and quad-Gaussian fits are 0.05, 0.79, 0.14 and 0.02 indicating strong support for the bi-Gaussian representation. Data are taken from Sinhuber et al.: swarm Ob5 which contains on average 22 individuals.

Figure 7

Intrinsic noise causes compression and de-population of lower sublayers. In the absence of intrinsic noise, distributions of simulated individuals (•) match theoretical expectations (solid line) for noiseless swarms. The presence of noise in the lower sub-layers reduces their size and causes partially de-populate. Predictions are shown for the stochastic model, Eq. (4) (a) and its generalization to 3 sublayers (b).

Figure 8

Swarms are predicted to compress in response to external dynamic perturbations. Predictions were obtained using the stochastic model, Eq. (4), for the bipartite swarm shown in Fig. 7a.

Figure 9

Simulated and predicted thermodynamic cycling. Simulation data were obtained using the simple stochastic model, Eq. (4), for the bipartite swarm shown in Fig. 7a. Cycling the strengths of noises (strong multiplicative noise, weak additive noise; strong multiplicative noise, strong additive noise; weak multiplicative noise, strong additive noise; weak multiplicative noise, weak additive noise) drives the simulated swarm through a ‘thermodynamic’ cycle that resembles observations. (a) Predictions were obtained using the equation of state, Eq. (5), with c₁ = 0.001, c₂ = 1.59, c₃ = 2.69 and (b) with the virial-like expansion. (c) The second-order virial coefficient was obtained from fits to the simulation data. The second-order virial coefficient is well represented by $B=a-\frac{b}{{\sigma }_w^2}$ (dashed-line) which corresponds to a van der Waals gas. Each phase has duration 0.25 (a.u.) and F₁ = 1 throughout. Phase I F₂ = 2, F₃ = 1; Phase II F₂ = 2, F₃ = 2; Phase III F₂ = 0, F₃ = 2; Phase IV F₂ = 0, F₃ = 1.

Figure 10

Second virial coefficient for the largest laboratory swarm (•) mirrors theoretical expectations for a swarm with slabs and corresponds to a van der Waals gas (dashed line). Data are taken from Sinhuber et al.; swarm Ob1 which contains on average 94 individuals.