Canonical active Brownian motion

Active Brownian motion is the complex motion of active Brownian particles. They are “active” in the sense that they can transform their internal energy into energy of motion and thus create complex motion patterns. Theories of active Brownian motion so far imposed couplings between the internal energy and the kinetic energy of the system. We investigate how this idea can be naturally taken further to include also couplings to the potential energy, which finally leads to a general theory of canonical dissipative systems. Explicit analytical and numerical studies are done for the motion of one particle in harmonic external potentials. Apart from stationary solutions, we study nonequilibrium dynamics and show the existence of various bifurcation phenomena.

Figure 1

Two-dimensional motion of an active particle under the influence of harmonic forces in the $q_1$, $q_2$ plane, subject to Eqs. (9, 10, 11) with all parameters chosen to be equal to 1 except $d_1=5$ and $s^{1∕2}=0.3$. The radius of the circle can be calculated from Eq. (23): $r_0=2$. Initial conditions are $q_1\left(0\right)=0=q_2\left(0\right)$, $p_1\left(0\right)=0=p_2\left(0\right)$, $e\left(0\right)=0$.

Figure 2

Internal energy evolution in time corresponding to the parameter setup given in Fig. 1. The stationary value of $e$ can be calculated from Eq. (24): $e_0=1∕5$.

Figure 3

Trajectory to equilibrium (e1), parameters satisfying the corresponding stability conditions ${\gamma }_0=1$, $c_1=1$, $c_2=2$, $c_3=2$, $c_4=2$, $d_1=1$, $d_2=1$. (The internal energy is plotted with a factor of 2.)

Figure 4

Limit cycle appearing after the Hopf-bifurcation point. The parameter values are ${\gamma }_0=1$, $c_1=1$, $c_2=2$, $c_3=2=c_4$, $d_1=1$, $d_2=5$. Initial conditions were chosen to be $q\left(0\right)=0$, $p\left(0\right)=-1$, and $e\left(0\right)=0$. (The internal energy is plotted with a factor of 10.)

Figure 5

Trajectory in four-dimensional phase space to equilibrium (e3) for the following choice of parameters: ${\gamma }_0=11$, $c_1=4$, $c_2=0.01$, $c_3=129$, $c_4=31$, $d_1=0.01$, $d_2=48$ and initial conditions $S\left(0\right)=0.1$, $U\left(0\right)=0.1$, $K\left(0\right)=0.1$, $e\left(0\right)=0.2$. (The kinetic energy $K$ is plotted with a factor of 3.)

Figure 6

Limit cycle in the phase space of variables $(K,U,S,e)$ and its corresponding quasiperiodic motion in configuration space for $n=2$ after the transient phase. Parameter values are ${\gamma }_0=1$, $c_1=1$, $c_2=2$, $c_3=4$, $c_4=4$, $d_1=49$, $d_2=50$ and initial conditions $S\left(0\right)=0$, $U\left(0\right)=0$, $K\left(0\right)=1$, $e\left(0\right)=0$. Corresponding initial conditions in $(q_i,p_i,e)$: $q_1\left(0\right)=0=q_2\left(0\right)$, $p_1\left(0\right)=-1=p_2\left(0\right)$, $e\left(0\right)=0$. (The kinetic energy $K$ is plotted with a factor of 3.)