NON-EQUILIBRIUM STATIONARY STATES IN 1D RUN & TUMBLE
Interactive simulation of 5000 (only 50 are shown)
active particles moving in 1D with a "run and tumble" stochastic dynamics [1]:
\begin{equation}
\dot x(t) = -\mu \frac{dU(x)}{dx}+\xi(t)
\end{equation}
with $\mu$ the particle mobility, \(U(x)\) an external potential (solid black line)
and \(\xi(t)\) is the self-propulsion speed switching between the two
values \(\pm v_0\) with a constant probability per unit time \(\lambda\).
Free particles (\(U(x)=\) const) perform an unbiased
random walk with persistence length \(\ell=v_0/\lambda\) and a long time
diffusivity \(D=v_0\ell\).
In the Brownian limit \(\ell\rightarrow 0\) the particles distribute with a
Boltzmann probability density
\begin{equation}
\rho(x)\propto\exp[-\frac{\mu U(x)}{D}]
\end{equation}
which is plotted as a blue dashed line.
Increasing the persistence length \(\ell\) strong deviations from the Boltzmann
distribution are observed, characterized by an accumulation of particles
over the repulsive boundaries of the confining potential well [2,3].
Switching the potential to the asymmetric barrier we can observe again an equilibrium
(Boltzmann) like behavior for small persistence lengths resulting in an equal probability
density for each side of the barrier (same energy). Increasing \(\ell\) (e.g. \(\ell\sim0.3, v_0\sim1.1\))
again a deviation from equilibrium is observed resulting in the possibility of accumulating
particles on the right side of the barrier which faces the higher slope [4].
This non-equilibrium property can be exploited for the targeted delivery of colloidal particles
driven by an active bath of swimming bacteria [5,6].
References:
