Thermodynamics and computation during collective motion near criticality

We study self-organization 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 (generalized) internal energy and of (generalized) 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 behavior 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.

Figure 1

Two kinetic phases of the model of collective motion. Each arrow represents a particle with its position and velocity in the $(x,y,z)$ space. Figure 1 is taken from a simulation of the model in which $J=0.001$ and $n_c=20$, after the relaxation time, and shows the system in its disordered motion phase. Figure 1 is taken from a simulation in which $J=0.2$ and $n_c=20$, after the relaxation time, and shows the system in its coherent motion phase.

Figure 2

Average normalized velocity of the group over the alignment strength $J$. The horizontal axis represents $J$ from 0 and 0.2, at steps of 0.001, and the vertical axis represents the average normalized velocity of the group $v_a=\frac{1}{N{v}_0}\left|{\sum }_{i=1}^N{\mathbf{v}}_i\right|$ over the simulation time. The parameter $n_c$ is fixed at 20.

Figure 3

Probability distribution of particles' velocity for different values of $J$. Figure 3 illustrates how a particle's velocity, with respect to the average velocity of the particles in its neighborhood, is defined by two spherical coordinates. The orange arrow represents the average velocity of the neighboring particles. A coordinate system is created so that the average velocity of the neighborhood is the $z$ axis. The vector ${\mathbf{v}}_i$ (blue arrow) is then the velocity of a particle with respect to this coordinate system, which can be expressed by the spherical coordinates $(\rho ,{\alpha }_p,{\alpha }_a)$, where $\rho$ is the radial distance, ${\alpha }_p\in [-\pi ,\pi ]$ is the polar angle and ${\alpha }_a\in [-\pi /2,\pi /2]$ is the azimuthal angle. Since in the model that we consider the speed of the particles is constant, ${\alpha }_p$ and ${\alpha }_a$ are sufficient for specifying ${\mathbf{v}}_i$. Figures 3–3 show the probability distribution of the discretized cluster velocity $p\left({\mathbf{s}}_k\right|J,n_c)$, for increasing values of $J$ from 0.001 to 0.2 and with $n_c$ fixed at 20. The horizontal and vertical axis are used for representing ${\mathbf{s}}_k$ and indicate the azimuthal and polar angles, respectively, while $p\left({\mathbf{s}}_k\right|J,n_c)$ is represented using a color scale, which varies from dark blue for the lowest values to light yellow for highest values.

Figure 4

Fisher information over the parameter $J$. The horizontal axis represents $J$ from 0 to 0.2, at steps of 0.001, and the vertical axis represents the Fisher information $F(J,n_c)$, with the parameter $n_c$ is fixed at 20.

Figure 5

Phase diagram using maximum Fisher information. The horizontal axis represents the alignment strength $J$ between particles, while the vertical axis represents the number of nearest neighbors $n_c$. Red crosses indicate the values of $J$ that yield to the higher Fisher information for fixed values of $n_c$ from 10 to 30 at steps of 5. Analogously, blue circles indicate the values of $n_c$ that yield to the higher Fisher information, for fixed values of $J$ from 0.05 to 0.2 at steps of 0.25. The yellow dotted line is the function $J=v_{0}a/n_c$, with $a=30$ and $v_0=0.05$, which approximates the critical curve that separates the coherent and disordered motion phases.

Figure 6

Curvature of the configuration entropy of the system over $J$. The horizontal axis represents $J$ from 0 to 0.2, at steps of 0.001, and the vertical axis represents the curvature of the configuration entropy of the system $S(J,n_c)$, with the parameter $n_c$ is fixed at 20.

Figure 7

Second derivative of the generalized internal energy with respect to $J$ ($\beta =1$). The horizontal axis represents $J$ from 0 to 0.2, at steps of 0.001, and the vertical axis represents the second derivative of the generalized internal energy with respect to $J$. The parameter $n_c$ is fixed at 20.

Figure 8

Aggregated curvature $d^2{\left(\mathbb{S}\right)}^{+}/d{J}^2$. The horizontal axis represents $J$ from 0 to 0.2, at steps of 0.001, and the vertical axis represents the aggregated curvature. The parameter $n_c$ is fixed at 20.

Figure 9

Rates of change of work, internal energy, and configuration entropy with respect to $J$, with $n_c$ is fixed at 20. In both graphs, the horizontal axis represents $J$ from 0 to 0.2, at steps of 0.001. The green crosses in Fig. 9 represent the rate of change with respect to $J$ of the generalized work ($\beta =1$), the red dots represent the rate of change with respect to $J$ of the generalized internal energy ($\beta =1$), and the blue asterisks represent the rate of change with respect to $J$ of the configuration entropy. Figure 9 shows the thermodynamic efficiency of computation $\eta$ over $J$.

Figure 10

Entropy of the system over $J$. The horizontal axis represents $J$ from 0 to 0.2, at steps of 0.001, and the vertical axis represents the entropy of the system $S(J,n_c)$, with the parameter $n_c$ is fixed at 20.

Figure 11

Fisher information over $n_c$. The horizontal axis represents $n_c$ from 1 to 30, and the vertical axis represents the Fisher information $F_{n_c}(J,n_c)$, with the parameter $J$ is fixed at 0.1.

Figure 12

Rates of change of work, internal energy, and configuration entropy with respect to $n_c$ with $J$ is fixed at 0.1. The horizontal axis represents $n_c$ from 1 to 30. The green crosses represent the rate of change with respect to $n_c$ of the generalized work ($\beta =1$), the red dots represent the rate of change with respect to $n_c$ of the generalized internal energy ($\beta =1$), and the blue asterisks represent the rate of change with respect to $n_c$ of the configuration entropy.