NON-EQUILIBRIUM STATIONARY STATES IN 1D RUN & TUMBLE

Choose potential: harmonic well asymmetric barrier
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PERSISTENCE LENGTH

SPEED

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:
[1] Phys. Rev. E, 48, 2553 (1993).
[2] Phys. Rev. Lett. 104, 258101 (2010).