Periodic potential can enormously boost free-particle transport induced by active fluctuations

Active fluctuations are detected in a growing number of systems due to self-propulsion mechanisms or collisions with an active environment. They drive the system far from equilibrium and can induce phenomena that are forbidden at equilibrium states by, e.g., fluctuation-dissipation relations and detailed balance symmetry. Understanding their role in living matter is emerging as a challenge for physics. Here we demonstrate a paradoxical effect in which a free-particle transport induced by active fluctuations can be boosted by many orders of magnitude when the particle is additionally subjected to a periodic potential. In contrast, within the realm of only thermal fluctuations, the velocity of a free particle exposed to a bias is reduced when the periodic potential is switched on. The presented mechanism is significant for understanding nonequilibrium environments such as living cells, where it can explain from a fundamental point of view why spatially periodic structures known as microtubules are necessary to generate impressively effective intracellular transport. Our findings can be readily corroborated experimentally, e.g., in a setup comprising a colloidal particle in an optically generated periodic potential.

Figure 1

The average velocity $\langle v\rangle$ of the Brownian particle dwelling in the symmetric potential $U\left(x\right)=sinx$ and driven by active fluctuations $\eta \left(t\right)$ vs the potential barrier height $\varepsilon$ for selected values of $\langle \eta \left(t\right)\rangle =v_0$. The parameters are as follows: the spiking rate $\lambda =30$, the variance and skewness of the amplitude distribution $\rho \left(z\right)$ are ${\sigma }^2=3.1$ and $\chi =0.99$, respectively, and the temperature is set to $D_T=0.01$.

Figure 2

The average velocity $\langle v\rangle$ vs the spiking rate $\lambda$ of active fluctuations $\eta \left(t\right)$ for selected values of the potential barrier height $\varepsilon$ and fixed $\langle \eta \left(t\right)\rangle =v_0=1$. Solid lines correspond to the asymmetry $\chi =0.99$, whereas dashed lines correspond to $\chi =-0.99$. Other parameters are the variance ${\sigma }^2=3.1$ and temperature $D_T=0.01$.

Figure 3

Exemplary Brownian particle trajectory is shown for the potential barrier $\varepsilon =40$. Other parameters are as follows: the spiking rate $\lambda =30$, the average bias $\langle \eta \left(t\right)\rangle =v_0=1$, the variance ${\sigma }^2=3.1$, and the asymmetry $\chi =0.99$ of the amplitude distribution $\rho \left(z\right)$. Temperature is fixed at $D_T=0.01$.

Figure 4

The characteristic time $1/{\lambda }_{\text{max}}\left(\varepsilon \right)$ between two successive $\delta$-kicks of active fluctuations $\eta \left(t\right)$ and the estimated median time ${\tau }_{\text{median}}\left(\varepsilon \right)$ of particle relaxation towards the potential $U\left(x\right)$ minimum both as a function of the barrier height $\varepsilon$. Other parameters are as follows: the average value of active fluctuations $\langle \eta \left(t\right)\rangle =v_0=1$, variance ${\sigma }^2=3.1$, skewness $\chi =0.99$ of the amplitude distribution $\rho \left(z\right)$, and temperature $D_T=0.01$.

Figure 5

Absolute value of the relaxation time $|{\tau }_{AB}|$ from the position $x_A$ to $x_B$ calculated according to Eq. (7).

Figure 6

Illustration of the role of potential symmetry on enhancement of free-particle transport. The potential barrier is $\varepsilon =40$. The average bias $\langle \eta \left(t\right)\rangle =v_0=1$. Solid lines correspond to the asymmetry $\chi =0.99$, whereas dashed lines correspond to $\chi =-0.99$. Other parameters are as follows: the variance ${\sigma }^2=3.1$, and temperature is fixed to $D_T=0.01$.

Figure 7

The average velocity $\langle v\rangle$ vs the spiking rate $\lambda$ of active fluctuations $\eta \left(t\right)$ depicted for three amplitude distributions and fixed $\langle \eta \left(t\right)\rangle =v_0=\lambda \zeta =1$, i.e., $\zeta =1/\lambda$. $\rho \left(z\right)$ corresponds to the skew-normal statistics with ${\sigma }^2=3.1$ and $\chi =0.99$; ${\rho }_G\left(z\right)$ indicates the Gaussian density with ${\sigma }^2=3.1$; ${\rho }_e\left(z\right)$ is the exponential distribution. The potential barrier height reads $\varepsilon =40$, and temperature is fixed at $D_T=0.01$. The upper panel corresponds to the symmetric potential $U\left(x\right)$, whereas in the bottom panel it is asymmetric, $U_{+}\left(x\right)$.

Figure 8

The probability distribution $\rho \left(z\right)$ of amplitudes $z_i$ of active nonequilibrium fluctuations $\eta \left(t\right)$ is presented in panel (a) for fixed mean $\zeta =1/30$, variance ${\sigma }^2=3.1$, and different values of skewness $\chi$, and in panel (b) for various mean $\zeta$ with skewness $\chi =0.99$.

Figure 9

Three exemplary realizations of active nonequilibrium noise $\eta \left(t\right)$ for different parameters: (a) $\lambda =30$, $\zeta =1/30$, ${\sigma }^2=3.1$, and $\chi =0.99$; (b) $\lambda =6$, $\zeta =1/6$, ${\sigma }^2=3.1$, and $\chi =0.99$; (c) $\lambda =30$, $\zeta =1/30$, ${\sigma }^2=3.1$, and $\chi =-0.99$. The solid red line corresponds to the average $\langle \eta \left(t\right)\rangle =1$.