Large Fluctuations and Dynamic Phase Transition in a System of Self-Propelled Particles

We study the statistics, in stationary conditions, of the work $W_{\tau }$ done by the active force in different systems of self-propelled particles in a time $\tau$. We show the existence of a critical value $W_{\tau }^{†}$ such that fluctuations with $W_{\tau }>W_{\tau }^{†}$ correspond to configurations where interaction between particles plays a minor role whereas those with $W_{\tau }<W_{\tau }^{†}$ represent states with single particles dragged by clusters. This twofold behavior is fully mirrored by the probability distribution $P(W_{\tau })$ of the work, which does not obey the large-deviation principle for $W_{\tau }<W_{\tau }^{†}$. This pattern of behavior can be interpreted as due to a phase transition occurring at the level of fluctuating quantities and an order parameter is correspondingly identified.

Figure 1

A typical stationary configuration at $\varphi =0.1$ and $\mathrm{Pe}=200$, just inside the binodal line [22], (only a portion of the system of size $L=200$ is shown). For each dumbbell, the color represents the value of $W_{\tau }$ defined in Eq. (2) (with $\tau =10$) according to the color code in the bar on the right. The black arrow indicates the value of $W^{†}(\tau )$. The inset is a magnification of the highlighted box, where also the arrows representing the active force directions have been drawn [see movies M1–M5 in the SM [34] to visualize the evolution of the system (for dumbbells, colloids, tetrabells, and rods of different aspect ratio, respectively)].

Figure 2

Probability distributions (on log-scale) of $W_{\tau }$, with $\tau =20$, for $\mathrm{Pe}=1$, 50, 100, 200, 400, at $\varphi =0.1$ (main) and $\varphi =0.001$ (inset). All cases fall inside the homogeneous low-density phase, except those at $\varphi =0.1$ and $\mathrm{Pe}=200$, 400 that are just inside the coexistence region. The threshold $W_{\tau }^{†}$ is signaled by vertical arrows. The continuous black curve is the analytical result for a single dumbbell (see the SM [34]).

Figure 3

The quantity $-(1/\tau )\ln [P(W_{\tau })/A_{\tau }]$, is plotted vs $W_{\tau }$ for different values of $\tau$, $\mathrm{Pe}=200$, and $\varphi =0.1$.

Figure 4

The quantity $m(W_{\tau })$ is plotted vs $W_{\tau }$ for $\tau =30$ and $\tau =100$ (see key), with $\mathrm{Pe}=200$ and $\varphi =0.1$. In the two insets the histogram of the values of $\tilde{m}$ is shown (for a case with $W_{\tau }=0.215>W_{\tau }^{†}$ on the right and with $W_{\tau }=0.045<W_{\tau }^{†}$ on the left).