Revisiting the Ratchet Principle: When Hidden Symmetries Prevent Steady Currents

Abstract

The "ratchet principle", which states that non-equilibrium systems violating parity symmetry generically exhibit steady-state currents, is one of the few generic results outside thermal equilibrium. We study exceptions to this principle observed in active and passive systems with spatially varying fluctuations sources. For dilute systems, we show that a hidden time-reversal symmetry prevents the emergence of ratchet currents. At higher densities, pairwise forces break this symmetry but an emergent conservation law for the momentum field may nevertheless prevent steady currents. We show how the presence of this conservation law can be tested analytically and characterize the onset of ratchet currents in its absence. Our results show that the ratchet principle should be amended to preclude parity symmetry, time-reversal symmetry, and bulk momentum conservation.

Figures (5)

• Figure 1: (a): Light-activated particles, like bacteria whose flagella rotors are controlled by proteorhodospin arlt_painting_2018frangipane_dynamic_2018, accumulate in low-speed regions. (b): Speed and temperature fields. (c): Steady-state density of ABPs and OBPs interacting via soft repulsive forces with the fluctuation sources shown in (b), compared to the non-interacting case. (d): Interactions induce a steady directed current for ABPs, but not for OBPs. All numerics are detailed in End Matter.
• Figure 2: Entropy production rate and stress measured in simulations of OBPs in the temperature landscape shown in panel (a). (b) Heat map of the entropy production rate density field $\sigma( {\bf r})=\langle \hat{\sigma}( {\bf r})\rangle$. Inset: Net entropy production up to time $t$. (c) Plot of the $(x,x)$ component of the Irving-Kirkwood stress tensor and of the ideal gas pressure. Their sum is uniform, which prevents the emergence of directed currents.
• Figure 3: Simulations of interacting ABPs in the activity landscape shown in panel (a). (b) As in the passive case (Fig. \ref{['fig:EOSpassive']}), the contributions to the total stress---${\boldsymbol{\sigma}}_{{ \rm A}}$ and ${\boldsymbol{\sigma}}_{{ \rm IK}}$---vary throughout space. Here, however, the total stress ${\boldsymbol{\sigma}}_{\rm tot}$ also varies due to the momentum source $\delta F_{ \rm A}$. The exchanged momentum $\Delta p_1$ in region $\mathcal{R}_1$ and $\Delta p_2$ in $\mathcal{R}_2$ do not cancel out, leading to the directed current $\langle J\rangle$ shown in panel (c). Outside $\mathcal{R}_1$ and $\mathcal{R}_2$, the current is supported by non-uniform density profiles, which lead to a non-uniform stress. (c) The momentum source $\delta F_A$ and the variation of the total stress add up to produce a uniform current, $J_{\rm stress} = \partial_x \sigma_{\rm tot}^{xx} + \delta F_{ \rm A}^x$. It matches the current $J_{\rm sim}$ measured directly by tracking particle displacements.
• Figure 4: Non-interacting 1d RTPs in the potential landscape $U(x)$ and activity landscape $v(x)$ depicted in (a). (b) Pseudo potential $\Phi(x)$ in the presence (solid line) or in the absence (dashed line) of $U(x)$. An aperiodic $\Phi$ leads to a steady current $J$. (c) Comparison between the steady-state current measured in particle simulations (symbols) and its predictions from Eq. \ref{['eq:exactRTP']} (lines) for various tumbling rates $\tau^{-1}$.
• Figure 5: (a) At the mean-field level, an RTP (bright red) moving in an activity landscape $v(x)$ (green) experiences an effective external potential $V_\mathrm{eff} = U_{\text{int}}\ast \rho$ induced by its neighbors (dim red). (b) Comparison between the current measured in simulations (circles) and its prediction by the perturbative solution of Eqs. \ref{['eq:CWr']}-\ref{['eq:CWm']}, as the scaled interaction strength $N\varepsilon$ is varied. The first order, $J_1$ (gray line), captures the slope of $J$ at $\varepsilon=0$ while the sixth order (black line) captures the non-monotonicity of $J$ as $N \varepsilon$ increases. Note that the radius of convergence of the expansion is limited by a jamming transition that occurs at larger values of $N\varepsilon$.