Autonomous ratcheting by stochastic resetting

The notion of stochastic resetting (SR) is attracting growing attention (see Ref. 1 for a recent review). This term refers to the sudden interruption of a stochastic process after random time intervals, followed by its starting anew, possibly after a further latency time, with the same initial conditions. Diffusion under SR is a non-equilibrium stationary process that has found applications in search contexts,² optimization of randomized computer algorithms,³ and many biophysical problems.^4,5 Surprisingly, under SR, the otherwise infinite mean first passage time of a freely diffusing Brownian particle⁶ from an injection point to an assigned target point becomes finite and, most notably, can be minimized for an optimal choice of the resetting time, τ.^7,8 Many analytical methods earlier developed in the theory of homogeneous stochastic processes^6,9 can be generalized to study diffusion under SR, for instance, to calculate the mean-first-exit time (MFET) of a reset particle out of a one-dimensional (1D) domain¹⁰ or potential well.¹¹ In general, SR speeds up (slows down) diffusive processes characterized by random escape times with a standard deviation larger (smaller) than the respective averages.⁵ 

In this Communication, we propose an SR mechanism with a degenerate resetting point. Let us consider an overdamped Brownian particle of coordinate x, diffusing in a 1D periodic potential, V(x), of period L. We assume for simplicity that the potential unit cells have one minimum each at x_n = x₀ + nL, with n = 0, ±1, …. Upon resetting, the particle stops diffusing and falls instantaneously at the bottom of the potential well it was in; it will resume diffusing after a latency time τ₀ ≥ 0; see Fig. 1(a). By using this mechanism of autonomous SR, we intend to model the dynamics of small motile tracers (such as bacteria or micro-robots¹²) capable of switching their internal engine on and off. In the case of undirected motility, the tracer would perform an unbiased Brownian motion. Let us further assume that the barriers separating two adjacent potential minima are asymmetric under mirror reflection, i.e., V(x − x₀) ≠ V(−x + x₀) (ratchet potential¹³). Extensive numerical simulations show that (i) SR rectifies diffusion at a ratchet potential. The particle’s net drift speed, ⟨v⟩, reaches a maximum for an optimal value of the resetting time, τ, which strongly depends on the potential profile; see Fig. 1(b); (ii) SR suppresses spatial diffusion. For large observation times, the particle’s mean-square displacement (MSD) turns proportional to time (normal diffusion); the relevant diffusion constant increases sharply with the resetting time in correspondence with the maximum of the drift speed; see Fig. 1(c).

While this variant of the SR mechanism may be reminiscent of a flashing ratchet with pulsated temperature,¹⁴ here the diffusing tracer exploits the substrate spatial asymmetry to autonomously rectify its random motion in the absence of external time-dependent fields of force or gradients,^13,15,16 simply by time-operating its internal engine to adjust to the substrate itself.

The key features of the resulting SR ratchet are shown in the bottom panels of Fig. 1: the particle motion gets rectified with the net speed ⟨v(τ)⟩ [Fig. 1(b)] and the asymptotic diffusion constant, D(τ), a monotonically increasing function of the SR time [Fig. 1(c)]. Rectification is maximum in an optimal τ range, as D approaches a stationary value (the same as in the absence of SR).

To explain ratcheting under SR, we anticipate two properties of the statistics of particle escape out of a potential well, shown in Fig. 3. In the absence of resetting, i.e., for asymptotically large τ, the probability current density of the process is zero, which rules out rectification,¹³ ⟨v(∞)⟩ = 0. Things change upon decreasing the SR time, as proven by the τ-dependence of the splitting probabilities, π_R,L(τ), for the particle to exit a potential well through the right/left barrier. Upon lowering τ, the asymmetry ratio π_R/π_L in Fig. 3(b) grows monotonically, the effect being more apparent at low noise, D₀ ≪ ΔV, so that ⟨v(τ)⟩ > 0. We qualitatively explain this property with the increased asymmetry of the probability density⁶ of the reset particle around the potential minima [Fig. 2(a)]. On the other hand, the data in Fig. 3(b) clearly show that in the limit τ → 0, ⟨T_R(τ)⟩ diverges exponentially, so that we anticipate ⟨v(τ → 0)⟩ = 0+. The combination of these two opposite effects determines the typical resonant profile of the ⟨v(τ)⟩ curves.

In more detail, the data in Fig. 2(b) suggest that ⟨v(τ)⟩ decays asymptotically like τ⁻¹. This behavior can be easily explained in the strong noise regime with D₀ ≫ ΔV and ⟨T_R,L(τ)⟩ ≪ τ. Under this condition, the particle executes many barrier crossings before being reset at the bottom of a V(x) well. At resetting, it is caught in average to the left of the well bottom; hence, at each resetting, the particle jumps to the right an average distance, $δx=x̄−x0>0$⁠, $x̄$ being the center of mass of the (periodic) particle’s stationary probability density function, p(x; τ, D₀), in the potential well with bottom at x = x₀. Accordingly, the particle gets rectified with a positive net drift speed ⟨v(τ)⟩ = δx/τ. In the strong noise regime, p(x; τ, D₀) approaches a uniform distribution; hence, δx = (L_L − L_R)/2, in good agreement with the numerical data in Fig. 2(a).

Upon decreasing the noise strength, δx diminishes for two reasons, as shown in Fig. 2(a). First, in the absence of SR, i.e., for τ → ∞, the probability density, p(x; τ, D₀), approaches its thermal equilibrium form, $p(x;∞,D0)=Nexp(−V(x)/D0)$⁠, with $N$ an appropriate normalization constant. For D₀ ≪ ΔV, p(x; ∞, D₀) shrinks around x₀, that is, δx diminishes. Second, by lowering D₀ in the presence of SR, i.e., for finite τ, ⟨T(τ)⟩ grows comparable with τ. Accordingly, barrier escape and resetting events grow correlated, which invalidates the above estimate of the particle’s drift speed. However, numerical data confirm that ⟨v(τ)⟩, though strongly suppressed, keeps decaying asymptotically like 1/τ, even at a low noise.

The plots of p(x; τ, D₀) for the lowest τ values in Fig. 2(a) consist of a central peak tapering off with asymmetric slow-decaying tails on both sides. In the limit τ → 0, (i) the peak gets sharper and more symmetric while remaining centered at the resetting point, x₀. Its square half-width can be easily calculated for D₀ ≪ ΔV by approximating $V(x)≃ω02(x−x0)2/2$ and averaging over the SR time, that is, $⟨(x−x0)2⟩≃2D0τ/(1+2ω02τ)$⁠; (ii) the tails get thinner but more asymmetric. This behavior is consistent with the τ-dependence of the escape asymmetry ratio, π_R/π_L, shown in Fig. 3(b).⁶ 

The sharp decay of ⟨v(τ)⟩ for τ → 0 proves that fast SR eventually suppresses interwell particle diffusion. In such a limit, as shown in Fig. 3, the particle tends to jump to the right with π_R(τ) ≫ π_L(τ); therefore, ⟨T(τ)⟩ ≃ ⟨T_R(τ)⟩ with ⟨T(τ)⟩ ≫ τ. Under these conditions, the resulting drift speed can be easily estimated under the renewal theory approximation,²¹ which is ⟨v(τ⟩ = L/⟨T_R(τ)⟩.

Regarding the intrawell diffusion, we remind that in the absence of SR, the MFET from x₀ to x₀ ± L amounts to the standard Kramers’ time⁹  $TK=(2π/ω02)exp(−ΔV/D0)$⁠. By the same token, one concludes that for τ → ∞, ⟨T_R⟩ ≃ ⟨T_L⟩, with both MFET’s tending to T_K for D₀/ΔV → 0, and their ratio, ⟨T_R⟩/⟨T_L⟩, approaching (1 + L/L_L)/(1 + L/L_R) ≃ 0.72 in the opposite limit, D₀/ΔV → ∞. On the other hand, for large τ, the splitting probabilities can be easily computed assuming no SR (see Sec. 5.2.7 of Ref. 9); their limits for D₀/ΔV → 0 (and →∞) are, respectively, π_R,L(∞) = 1/2 (and L_L,R/L), as shown in Fig. 3(b).

These remarks are useful to interpret the MSD datasets in Fig. 1(c). Numerical simulation indicates that diffusion at large times, t ≫ ⟨T(τ)⟩, is normal, as anticipated by the fitting law of Eq. (3). At small τ, a transient plateau for t ≲ ⟨T(τ)⟩, ⟨Δx²⟩ ≃ 2D₀τ, marks the particle relaxation inside a single potential well [with ⟨Δx²⟩ of the order of the square half-width of the p(x; τ, D₀) peak estimated above]. The τ-dependence of the asymptotic diffusion constants, D, is reported in Fig. 2(c). For large τ, the D(τ) curves approach the horizontal asymptotes,⁹  D = L²/2T_K, as to be expected in the absence of SR. Vice versa, for very short SR times, the diffusion constant is well approximated by D(τ) = L²/2⟨T_R(τ)⟩, as predicted by the renewal theory for a process with an average escape time constant ⟨T_R(τ)⟩.²¹ In both τ limits, our phenomenological arguments are supported by numerical simulation.

This instance of an SR ratchet lends itself to a simple laboratory demonstration. We start again from the LE (1) with the potential of Eq. (2), but instead of implementing the SR protocol with a latency time τ₀, we now assume a dichotomic noise strength, D₀(t), with D₀ = 0 for fixed time intervals, τ₀, and D₀(t) = D₀ for random time intervals exponentially distributed with average τ. The resulting LE describes a rectifier, which could be classified as a special case of a flashing ratchet.¹⁴ In one regard, the two rectification mechanisms are apparently similar: in both cases, the particle rests at the bottom of a potential well for the time interval, τ₀, before resuming Brownian diffusion because it is either reset that way (SR ratchet) or given enough time to relax there (flashing ratchet with $ω02τ0≫1$⁠). As shown in the inset of Fig. 4, for the same choice of the tunable parameters, D₀, τ, and τ₀, the rectification power of the two ratchets is almost identical. Therefore, one can utilize a ratchet with dichotomic noise strength to experimentally demonstrate the rectification properties of the proposed SR ratchet. However, an important difference between these two ratchets is also noteworthy. The flashing ratchet is fueled by an external source capable of “heating and cooling” the particle or its substrate.^22,23 SR ratcheting with a finite latency time, instead, can be controlled by the particle itself, by autonomously regulating its own internal motility mechanism for maximum efficiency.

In summary, we have proposed a new protocol of stochastic resetting, whereby a particle diffusing on a one-dimensional substrate gets reset not at a fixed point but rather at one of the degenerate minima of the substrate. We investigated, both numerically and analytically, the diffusion properties of the reset particle and showed that for spatially asymmetric substrates, the particle gets rectified in the direction determined by the substrate profile, with an optimal speed depending on the resetting time. We argue that, thanks to such a mechanism, a motile system (biological and synthetic) can exploit the substrate asymmetry to autonomously direct its motion, for instance, by randomly switching on and off its propulsion engine at an appropriate rate.

Y.L. was supported by the NSF China under Grant Nos. 11875201 and 11935010. P.K.G. was supported by SERB Core Research Grant No. CRG/2021/007394.

The authors have no conflicts to disclose.

Pulak K. Ghosh: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal). Shubhadip Nayak: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal). Jianli Liu: Data curation (equal); Formal analysis (equal); Methodology (equal). Yunyun Li: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Writing – review & editing (equal). Fabio Marchesoni: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.