Exceptions to the Ratchet Principle in active and passive stochastic dynamics

TL;DR

The paper analyzes when non-equilibrium fluctuations can rectify into steady currents, revealing that breaking time-reversal symmetry and parity is not always sufficient to generate transport. A central finding is the existence of effective momentum conservation in several stochastic systems, which can suppress interaction-induced currents even under TRS violation; momentum sources become the decisive factor for currents to appear. Across UBPs, OBPs, ABPs, RTPs, and AOUPs, the authors develop a combination of path-integral and operator methods, perturbation theory, and mean-field arguments to establish when currents can arise and when they are inherently blocked. The work highlights the nuanced role of discretization, interactions, and fluctuation landscapes (including blowtorch configurations) in shaping non-equilibrium transport, and it identifies momentum conservation as a critical dividing line for the ratchet principle in active and passive stochastic dynamics.

Abstract

The "ratchet principle" asserts that non-equilibrium systems which violate parity symmetry generically exhibit steady-state currents. As recently shown, there are exceptions to this principle, due to the existence of hidden time-reversal symmetry or bulk momentum conservation. For underdamped and overdamped Brownian dynamics, we show how thermal fluctuations cannot power the momentum sources required to sustain steady ratchet currents, even when time-reversal symmetry is broken due to an inhomogeneous temperature field. While Active Brownian and Run-and-Tumble particles display interaction-induced ratchet currents in asymmetric activity landscapes, we show that this is not the case for Active Ornstein-Uhlenbeck particles: not all inhomogeneous active fluctuations lead to net momentum sources. For each of the systems considered in this article, we numerically test for the emergence of interaction-induced ratchet currents. We then characterize time-reversal (a)symmetry in position space using a combination of path-integral and operator methods. When the existence of effective momentum conservation is ruled out, we develop perturbation theories to characterize the onset of interaction-induced currents.

Figures (10)

• Figure 1: Spatially-inhomogeneous fluctuations cause interaction-induced currents in ABPs [ (i-l)], but not in UBPs [ (a-d)], OBPs [ (e-h)], or AOUPs [ (m-p)]. In panels (a), (e), (i), and (m), we show the fluctuation landscape used in each system. In panels (b), (f), (j), and (n), we show the steady-state density $\rho$ (heatmap) and current field (arrows) for the systems in the presence of interactions. In panels (c), (g), (k), and (o), we do the same for the non-interacting systems. In panels (d), (h), (l), and (p), we show the average displacement per particle both in the presence (purple) and absence (pink) of interactions. Simulation details are provided in Appendix \ref{['appendix:sim-examp']}.
• Figure 2: Directed currents are prevented in a two-dimensional system of UBPs by an emergent bulk momentum conservation.(a) The temperature landscape $T(x)$ used in simulations. (b) The steady-state density distribution of UBPs both without (gold) and with (dark red) interactions. The density in the $\gamma\rightarrow \infty$ limit, $\propto 1/T(x)$, is shown for reference (thin tan line). (c) Directly measuring the net particle displacement over time, both without and with interactions, reveals the absence of directed current. (d) In the steady state, variations of the momentum flux (blue) are balanced by those of the Irving-Kirkwood stress (green), resulting in a homogeneous pressure $\sigma_{\rm tot}^{xx}$ and preventing the emergence of a steady current. (e) The $(y,y)$ component of the stress tensor, $\sigma_{\rm tot}^{yy}$, varies along $x$, unlike $\sigma_{\rm tot}^{xx}$. It is, however, constant along $y$ by symmetry, thus generating no current. Simulation details are provided in Appendix \ref{['appendix:sim-ud-pbp']}.
• Figure 3: Interaction-induced current for OBPs with different time discretizations. (a) Particles interacting via a soft repulsive harmonic potential, Eq. \ref{['eq:appendix-Uint']}, are placed in a temperature landscape $T(x)$. (b) Steady-state particle density as the time-discretization parameter $\alpha$ is varied. (c) The net integrated displacement of the particles show the emergence of nonzero currents for discretizations $0<\alpha<1$. (d) Steady-state particle current as $\alpha$ is varied. Momentum conservation prevents a directed current in the Itō-discretized ($\alpha$=0) system. The Hänggi-discretized ($\alpha$=1) system is equivalent to a time-reversed Itō dynamics and thus also current free. Finally, no such conservation exists in other discretizations which thus lead to directed currents. The mean-field prediction for the current $J$, to 1st-order in the interaction strength, Eq. \ref{['eq:od-pbp-alpha-J1']}, is plotted in black. Simulation details are provided in Appendix \ref{['appendix:sim-od-pbp']}.
• Figure 4: EPR measured in simulations of OBPs. (a) Temperature landscape. (b) Heat map of the average EPR density field $\langle \hat{\sigma}( {\bf r})\rangle$. (c) Net entropy production up to time $t$. Adapted from Fig. 2 of our companion letter metzger_exceptions_letter. Simulation details are provided in Appendix \ref{['appendix:sim-od-pbp']}.
• Figure 5: Interaction-induced current for ABPs and RTPs in $d=2$ spatial dimensions. We use a soft repulsive harmonic potential proportional to $\varepsilon$. (a) Self-propulsion landscape $v(x)$ used in all simulations. Panels (b-d) correspond to simulations with $N\varepsilon/L_y=7.5$. (b) Snapshot of particle locations and self-propulsion vectors for an ABP simulation. (c) Steady-state density distributions, compared to the non-interacting result $\rho_{\rm s}(x)\propto 1/v(x)$. (d) Net average displacement per particle over time for interacting (solid) and non-interacting (dashed) ABPs and RTPs. The shaded region indicates $\pm 3\sigma$, where $\sigma$ is the standard error of the mean. (e) Average current vs. interaction strength, along with the mean-field prediction to first order in $\varepsilon$. Simulation details are provided in Appendix \ref{['appendix:sim-abp-rtp']}.