Memory-induced active particle ratchets: Mean currents and large deviations

Abstract

We analyse a continuous-time random walk model with stochastic reversals of direction. There is no external potential but the reorientation mechanism generates a non-zero current from asymmetry in the forward and backward waiting-time distributions (even when they have the same mean); the system can therefore can be considered as a type of active particle ratchet. We derive an explicit expression for the mean ratchet current with exponentially distributed reorientation times and also develop a general renewal-theory framework to obtain the full large deviations, using this to comment on the possibility of dynamical phase transitions.

Figures (10)

• Figure 1: Two-channel continuous-time random walk on a ring. The particle transitions are restricted to the respective channels with rates $\beta^{+}(\tau)$ and $\beta^{-}(\tau)$ until a reorientation occurs with rate $r$ and the particle switches channels.
• Figure 2: HMM cell structure for a hypoexponential-exponential ratchet. The dashed box shows the hidden sites at one position on the ring.
• Figure 3: HMM cell structure for a hyperexponential-exponential ratchet. The dashed box shows the hidden sites at one position on the ring.
• Figure 4: Mean current versus reorientation rate $r$. Solid lines show the prediction of \ref{['mean_j']}. Dots show the result from equivalent HMMs. Dashed black line shows the large-$r$ limit. Dashed blue line shows approximation for small $r$, with the inset displaying the behavior close to zero. Top: Mean current in the hypoexponential-exponential ratchet. A two-parameter hypoexponential distribution was used in the forward channel with parameters $p_1=1, p_2=2$. Bottom: Mean current in hyperexponential-exponential ratchet. A two-parameter hyperexponential distribution was used in the forward channel with equiprobable branches and parameters $p_1=1, p_2=2$.
• Figure 5: Non-monotonic behavior of mean current with respect to reorientation rate $r$ as seen in a ratchet with gamma-distributed waiting times in both channels. Parameters of the two gamma distributions are $(a_1, p_1)=(2,1)$ and $(a_2,p_2)=(5,2.5)$. Solid line shows the analytical mean current from \ref{['non_monotonic_gamma']} while dots show results from an equivalent HMM.