††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Ratchet-mediated resetting: Current, efficiency, and exact solution

Connor Roberts  Department of Mathematics, Imperial College London, London SW7
2AZ, United Kingdom  Emir Sezik  Department of Mathematics, Imperial College
London, London SW7 2AZ, United Kingdom  Eloise Lardet  Department of
Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

###### Abstract

We model an overdamped Brownian particle that is subject to resetting
facilitated by a ratchet potential on a spatially periodic domain. This
asymmetric potential switches on with a constant rate, but switches off again
only upon the particle’s first passage to a resetting point at the minimum of
the potential. Repeating this cycle sustains a non-equilibrium steady-state,
as well as a directed steady-state current which can be harnessed to perform
useful work. We derive exact analytic expressions for the probability
densities of the free-diffusion and resetting phases, the associated currents
for each phase, and an efficiency parameter that quantifies the return in
current for given power input. These expressions allow us to fully
characterise the system and obtain experimentally relevant results such as the
optimal current and efficiency. Our results are corroborated by simulations,
and have implications for experimentally viable finite-time resetting
protocols.

††preprint: APS/123-QED

##  I Introduction

<EMAIL_ADDRESS>
The field of stochastic resetting has received much attention since its
inception over a decade ago [1, 2, 3]. In stochastic resetting, the recurrent
restarting of a process from its initial state establishes internal
probability currents and, remarkably, a non-equilibrium steady-state even if
the underlying dynamics otherwise satisfy detailed balance. One of the most
striking properties of stochastic resetting is that it can render the mean
first-passage time of a diffusive search process finite [4, 5, 6], thus
offering an attractive way of improving transport properties [7, 8]. On top of
its myriad variations and applications, such as to biology [9, 10], computer
science [11], and finance [12], resetting has most recently been shown to
improve the efficiency of Brownian heat engines [13].

The prototypical model of stochastic resetting involves a Brownian particle
subject to instantaneous resets to its initial position at a constant rate.
This is in stark contrast to experimental implementations of stochastic
resetting, where it takes a finite time to reset a particle back to its
initial position [14, 15, 16, 17]. With these recent experimental advances,
others have sought to devise theoretical models that emulate finite-time
resetting [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Unlike instantaneous
resetting, finite-time resetting can be mediated by a physical mechanism, such
as an external potential [29, 30, 31, 32]. Aside from being more physically
realistic, it was recently shown that potential-mediated resetting can
actually expedite diffusive search processes compared to instantaneous
resetting [31, 33].

The probability currents generated through stochastic resetting naturally lead
to questions about work extraction and the efficiency of resetting [34, 35,
36, 37]. For instance, these models could be modified to generate useful work
by introducing an underlying asymmetry into the resetting mechanism.
Harnessing the asymmetry of an autonomous engine to generate work is an idea
dating back to Feynman’s “ratchet and pawl” machine [38]. In a similar vein,
models of molecular motors provide a mechanism for Brownian particles to
exhibit a net drift [39, 40], akin to the movement of motor proteins along
filaments [41, 42]. This directed motion can be brought about by the asymmetry
of a ratchet potential, in combination with either potentials, external forces
or states that fluctuate in time [43, 44, 45]. Alternatively, the particles
themselves can self-propel, in which case their motion in an asymmetric
potential readily breaks time-reversal symmetry, as has been the study of so-
called “active ratchets” in recent times [46, 47, 48]. Active ratchets are
essentially autonomous engines that can be exploited to extract useful
thermodynamic work [49, 50]. Finding exact analytical solutions to these
minimal models of non-equilibrium systems is an intense area of ongoing
research [51, 52, 53, 54, 55, 56, 18, 50].

There has been a recent surge of numerical studies investigating resetting in
combination with asymmetric potentials [22, 57, 58]. However, analytical
studies are comparatively lacking, especially for experimentally relevant
models. In this paper, we study the motion of an overdamped Brownian particle
subject to finite-time resetting facilitated by a piecewise-linear ratchet
potential on a periodic domain. The potential switches on with a constant
rate, but switches off only upon the particle’s first passage to a resetting
point at the minimum of the potential, at which point the particle returns to
a freely diffusive motion and restarts the cycle. This is in contrast to other
models of fluctuating potential ratchets [44, 45, 40], which have previously
been used to model molecular motors [39]. In these models, a time-varying
ratchet potential switches between two or more states (such as “on” and
“off”). In non-resetting ratchet models, this is controlled by some other form
of systematic driving, e.g. switching between states with a constant rate. In
the present model, the switching off being determined by the particle reaching
the minimum of the potential means the particle can fully exploit the
potential’s height to further bias its motion through the ratchet’s asymmetry,
paving the way for an optimised current. This work brings together stochastic
resetting and fluctuating potentials to provide a minimal finite-time
resetting model that exhibits a net current. We derive the exact steady-state
solution, backed up by simulation results, in an experimentally motivated
stochastic resetting set-up.

The remainder of the paper is organised as follows. After introducing the
model in Section II, we derive the system’s steady-state probability density
in Section III. This is supplemented by a derivation of the time-dependent
probability densities in Appendix B. In Section IV, we turn our attention to
the current, which, to the best of our knowledge, constitutes the first
systematic analytic study of a steady-state net current generated through
finite-time resetting. The fact that we can generate a net current motivates
us to study the energetic cost of this resetting in Section V, and ultimately
devise a measure of the efficiency of the current generation for given power
input. Throughout the paper, our analytical results are supported by extensive
numerical simulations. Finally, we conclude and summarise our results in
Section VI.

##  II Model

Figure 1: Example simulated trajectory from numerically integrating Eqs. (1)
and (2) for parameters $L=1$, $a=0.2$, $h=0.1$, $D_{0}=0.1$, and $r=2$, see
Appendix A for simulation details. Here, the trajectory is plotted on an
extended domain for visualisation purposes; however, all results in the main
text are calculated for a position $x$ wrapped back onto a periodic domain
$x\in[0,L)$. The blue (respectively, red) sections of the line indicate when
the particle is in the diffusion (respectively, resetting) phase. The ratchet
potential $V(x)$, Eq. (3), is visualised alongside the particle trajectory.
The horizontal dashed lines indicate the resetting positions at $x=nL$ for
$n\in\mathbb{Z}$. Resetting phases can start at any position $x$, but must end
at these resetting points.

We consider an overdamped Brownian particle in one dimension which has two
phases to its motion. In the “diffusion phase”, the particle freely diffuses
and so its stochastic motion is governed by the following Langevin equation,

$\dot{x}_{D}(t)=\sqrt{2D_{0}}\xi(t),$ (1)

where $t$ is time, $D_{0}$ is the diffusion constant, $\xi(t)$ is a Gaussian
white noise with zero mean $\xi(t)=0$ and correlation
$\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})$, and the subscript
$D$ indicates that motion is in the diffusion phase. The friction coefficient
that would normally appear in Eq. (1) has been set to unity without loss of
generality.

While in the diffusion phase, a potential $V(x)$ with period $L$ is switched
on with a Poissonian rate $r$, upon which the particle enters the “resetting
phase”. The potential is switched off again only upon the particle reaching a
resetting point located at $x=nL$ for $n\in\mathbb{Z}$, which coincide with
the potential minima. To obtain a model with non-vanishing steady-state
density and current, we wrap the position $x$ onto a periodic domain, i.e.
identify positions $x+nL$ with $x$, thus constraining the particle’s motion to
the interval $x\in[0,L)$. Due to these periodic boundary conditions, any
reference henceforth to $x=0$ can be assumed to equally refer to $x=L$, and
vice versa.

By iterating the cycle of diffusion and resetting phases, a non-equilibrium
steady-state is sustained from the constant injection of energy at the
beginning of each resetting phase to increase the particle’s potential energy.
Denoting the period where the potential $V(x)$ is switched on with a subscript
$R$, the Langevin equation for the particle while in the resetting phase is

$\dot{x}_{R}(t)=-V^{\prime}\left(x_{R}(t)\right)+\sqrt{2D_{0}}\xi(t).$ (2)

We are interested in using the resetting potential $V(x)$ to rectify the
particle’s motion in order to generate a steady-state current, in which case
the potential needs to impart a bias on the particle’s motion. To render our
system exactly solvable, we choose $V(x)$ to be a piecewise-linear asymmetric
potential with an apex at $x=a$ [50, 44], i.e. a “ratchet potential” defined
by

$V(x)=\begin{cases}V^{[1]}(x)=\frac{h}{a}x,&0\leq x<a,\\\
V^{[2]}(x)=\frac{h}{L-a}(L-x),&a\leq x<L,\end{cases}$ (3)

where $h$ is the maximum height of the potential, occurring at $x=a$. Notably,
this setup generalises instantaneous stochastic resetting, since the limit
$h\to\infty$ corresponds to the particle being instantaneously reset to the
origin $x=0$ when the potential is switched on. In the instantaneous resetting
case, the parameter $a$ controls whether the particle instantaneously jumps a
distance $x$ to the left, if $0\leq x<a$, or a distance $L-x$ to the right, if
$a\leq x<L$. The full model for finite $h$ is illustrated by an example
stochastic trajectory in figure 1.

To demonstrate the rectification mechanism, consider the case where $a<L/2$.
When the potential is switched on, the particle is more likely to be found in
the larger $a\leq x<L$ region of the resetting potential, which imparts a
positive drift velocity $-V^{[2]^{\prime}}(x)=h/(L-a)$ on the particle towards
the origin $x=L^{-}$. Hence, on average, the particle experiences a positive
drift while in the resetting phase. Since motion in the diffusion phase is
symmetric, the particle’s overall motion is rectified in the positive
direction, corresponding to a positive net current. The equivalent negative
current is obtained for a ratchet potential where the position of the apex is
reflected across $x=L/2$, i.e. $a\to L-a$.

Using a periodic asymmetric potential in combination with resetting to
generate a net current was recently investigated in Ref. [22]. However, in
that work, the current was not induced by recurrent activation of the
potential, as in the present study. Instead, the authors of Ref. [22] obtained
numerical results for a system in which a Brownian particle moves in a static
asymmetric potential. The particle switches off its diffusion with a constant
rate, leading to a deterministic return to the potential minimum proceeded by
the particle restarting its diffusion. We argue that our present model is more
experimentally accessible than this “stop-and-start” model, as diffusion is
not typically an internal degree of freedom that can be autonomously switched
on or off, whereas an external potential can be readily modulated by an
external controller. Furthermore, we obtain exact results for our model that
can be used to analytically study experimentally relevant observables, such as
the probability densities and currents for each phase, as well as the
efficiency of the current generation. For completeness, we derive the analytic
solution for the stop-and-start model in Appendix C.

##  III Probability densities

The dynamics of the system can be equivalently encoded in the probability
densities $P_{D/R}(x,t)$ to observe the particle in the diffusion/resetting
phase at a position $x$ at time $t$. Since the ratchet potential, Eq. (3), is
piecewise linear, it is convenient to separate the probability density for the
resetting phase into separate components for each region. Here and throughout,
we distinguish these separate components with a superscript $[i]$, where
$i\in\\{1,2\\}$, i.e.

$P_{R}(x,t)=\begin{cases}P_{R}^{[1]}(x,t),&0\leq x<a\\\
P_{R}^{[2]}(x,t),&a\leq x<L\end{cases}.$ (4)

The probability densities obey the following coupled Fokker-Planck equations,

$\displaystyle\frac{\partial P_{D}(x,t)}{\partial t}$
$\displaystyle=-\frac{\partial J_{D}(x,t)}{\partial
x}-rP_{D}(x,t)+\left(J_{R}^{[2]}(L^{-},t)-J_{R}^{[1]}(0^{+},t)\right)\delta(x),$
(5a) $\displaystyle\frac{\partial P_{R}^{[1]}(x,t)}{\partial t}$
$\displaystyle=-\frac{\partial J_{R}^{[1]}(x,t)}{\partial
x}+rP_{D}(x,t)+J_{R}^{[1]}(0^{+},t)\delta(x),$ (5b)
$\displaystyle\frac{\partial P_{R}^{[2]}(x,t)}{\partial t}$
$\displaystyle=-\frac{\partial J_{R}^{[2]}(x,t)}{\partial
x}+rP_{D}(x,t)-J_{R}^{[2]}(L^{-},t)\delta(x),$ (5c)

where $L^{-}$ (respectively, $0^{+})$ is the limit of $x$ approaching $L$
(respectively, $0$) from below (respectively, above), and the currents for
each phase are given by

$\displaystyle J_{D}(x,t)$ $\displaystyle=-D_{0}\frac{\partial
P_{D}(x,t)}{\partial x},$ (6a) $\displaystyle J_{R}^{[i]}(x,t)$
$\displaystyle=-D_{0}\frac{\partial P_{R}^{[i]}(x,t)}{\partial
x}-V^{\prime[i]}P_{R}^{[i]}(x,t),$ (6b)

where each $V^{\prime[i]}$ is spatially independent, and
$J_{R}^{[2]}(L^{-},t)-J_{R}^{[1]}(0^{+},t)$ is the net flux directed into the
origin $x=0$ in the resetting phase, which quantifies the rate at which the
particle arrives at the resetting point. We always have
$J_{R}^{[2]}(L^{-},t)\geq 0$ and $J_{R}^{[1]}(0^{+},t)\leq 0$, regardless of
the system parameters, since the individual terms of Eq. (6b) are both
positive for $J_{R}^{[2]}(L^{-},t)$, and both negative for
$J_{R}^{[1]}(0^{+},t)$, as will become clear once $P_{R}(x,t)$ is derived
later. Hence, the term $(J_{R}^{[2]}(L^{-},t)-J_{R}^{[1]}(0^{+},t))\delta(x)$
in Eq. (5) always represents a source term for the diffusion phase, while the
terms $-J_{R}^{[2]}(L^{-},t)\delta(x)$ and $J_{R}^{[1]}(0^{+},t)\delta(x)$
always represent sink terms for each region in the resetting phase.

Upon summing the probabilities in Eq. (5), we recover the continuity equation

$\frac{\partial P(x,t)}{\partial t}=-\frac{\partial J(x,t)}{\partial x},$ (7)

where $P(x,t)=P_{D}(x,t)+P_{R}(x,t)$ is the overall probability density,
$J(x,t)=J_{D}(x,t)+J_{R}(x,t)$ is the net current, and recall that
$P_{R}(x,t)$ is composed of components $P_{R}^{[i]}(x,t)$ that are valid only
for specific regions of space as per Eq. (4). Hence, the sink term
$-rP_{D}(x,t)$ from Eq. (5a) is cancelled in Eq. (7) by the source terms
$rP_{D}(x,t)$ from Eqs. (5b) and (5c) in a piecewise manner, first by that of
Eq. (5b) in the region $x\in[0,a)$, and then by that of Eq. (5c) in the region
$x\in[a,L)$.

Our setup generalises the mod-linear resetting model of Ref. [30] by allowing
the resetting potential to be asymmetric and periodic in order to generate a
non-zero net current $J(x,t)$. Thus, our model reduces to that of Ref. [30] by
setting $a=L/2$ while taking $L\to\infty$ and $h\to\infty$ in such a way that
the gradients $V^{\prime[i]}$ remain constant. The periodicity of the
potential imposes the following periodic boundary conditions,

$\displaystyle P_{D}(0,t)$ $\displaystyle=P_{D}(L,t),$ (8a) $\displaystyle
P_{R}^{[1]}(0,t)$ $\displaystyle=P_{R}^{[2]}(L,t)=0,$ (8b)

where Eq. (8b) also imposes that a particle in the resetting phase can never
be found at the origin because it would instantly transmute to the diffusion
phase.

We are interested in the steady-state behaviour of the system, for which the
left-hand sides of Eq. (5) vanish, i.e.
$\lim_{t\to\infty}\partial_{t}P_{D,R}(x,t)=0$. Henceforth, we will represent
all steady-state quantities by dropping the ‘$t$’ from their arguments, e.g.
$P_{D}(x)\equiv\lim_{t\to\infty}P_{D}(x,t)$. The steady-state net current $J$
is independent of position as a result of the steady-state condition
$\partial_{t}P(x)=-\partial_{x}J=0$, where $P(x)=P_{D}(x)+P_{R}(x)$ is the
overall probability density at stationarity.

We now recast the dynamics in dimensionless form. This will allow us to
explore the full parameter space more efficiently by reducing the description
of the system from five dimensionful parameters $h,a,L,r,D_{0}$, to three
dimensionless parameters, to be specified below. Defining the typical
diffusion length scale between resets $\sigma\coloneq\sqrt{D_{0}/r}$ and the
average time for the potential to switch on $\tau\coloneq 1/r$ as our units of
distance and time, respectively, the dimensionless Fokker-Planck equation at
stationarity is

$\displaystyle 0$
$\displaystyle=\frac{\partial^{2}\tilde{P}_{D}(\tilde{x})}{\partial\tilde{x}^{2}}-\tilde{P}_{D}(\tilde{x})+\left(\tilde{J}_{R}^{[2]}(\ell^{-})-\tilde{J}_{R}^{[1]}(0^{+})\right)\delta(\tilde{x}),$
(9a) $\displaystyle 0$
$\displaystyle=\frac{\partial^{2}\tilde{P}_{R}^{[1]}(\tilde{x})}{\partial\tilde{x}^{2}}+\tilde{V}^{\prime[1]}\frac{\partial\tilde{P}_{R}^{[1]}(\tilde{x})}{\partial\tilde{x}}+\tilde{P}_{D}(\tilde{x})+\tilde{J}_{R}^{[1]}(0^{+})\delta(\tilde{x}),$
(9b) $\displaystyle 0$
$\displaystyle=\frac{\partial^{2}\tilde{P}_{R}^{[2]}(\tilde{x})}{\partial\tilde{x}^{2}}+\tilde{V}^{\prime[2]}\frac{\partial\tilde{P}_{R}^{[2]}(\tilde{x})}{\partial\tilde{x}}+\tilde{P}_{D}(\tilde{x})-\tilde{J}_{R}^{[2]}(\ell^{-})\delta(\tilde{x}),$
(9c)

where we have represented non-dimensionalised quantities with a tilde, e.g.
$\tilde{x}=x/\sigma$, $\tilde{P}_{D,R}=\sigma P_{D,R}$, and
$\tilde{J}_{R}=\tau J_{R}$. The gradients of the potential in dimensionless
form are given by

$\tilde{V}^{\prime}(\tilde{x})=\begin{cases}\tilde{V}^{\prime[1]}=\frac{\kappa}{\delta\ell},&0\leq\tilde{x}<\delta\ell\\\
\tilde{V}^{\prime[2]}=-\frac{\kappa}{(1-\delta)\ell},&\delta\ell\leq\tilde{x}<\ell\end{cases},$
(10)

where $\ell\coloneq L/\sigma$ is the rescaled system length, and we have
introduced relevant dimensionless parameters, namely the coupling strength
$\kappa\coloneq h/D_{0}$ of the particle to the potential, and the potential
asymmetry parameter $\delta\coloneq a/L\in[0,1]$, i.e. the position of the
ratchet apex for an interval normalised to unit length. In dimensionless form,
the periodic boundary conditions (8) become

$\displaystyle\tilde{P}_{D}(0,t)$ $\displaystyle=\tilde{P}_{D}(\ell,t),$ (11a)
$\displaystyle\tilde{P}_{R}^{[1]}(0,t)$
$\displaystyle=\tilde{P}_{R}^{[2]}(\ell,t)=0.$ (11b)

Figure 2: (a)-(b) Time evolution of the probability densities for each phase
$\tilde{P}_{D,R}(\tilde{x},\tilde{t})$, as determined from simulations.
Simulation parameters are $\ell=4$, $\kappa=1$, $\delta=0.2$ for a particle
initialised at the origin $\tilde{x}=0$ at time $\tilde{t}=0$, i.e.
$\tilde{P}_{D}(\tilde{x},0)=\delta(\tilde{x})$. Full simulation details are
given in Appendix A. The time-dependent densities converge to the theoretical
steady-state densities, $\tilde{P}_{D}(\tilde{x})$ and
$\tilde{P}_{R}(\tilde{x})$, given in Eqs. (14) and (19), respectively. (c)
Probability of being in the diffusion phase $p_{D}(\tilde{t})$ as a function
of time for various values of $\ell$, as obtained from simulations for a
single particle initialised at the origin $\tilde{x}=0$ in the diffusion phase
at time $\tilde{t}=0$, with $\kappa=1$ and $\delta=0.2$.

To obtain the steady-state probability density for the diffusion phase
$\tilde{P}_{D}(\tilde{x})$, we first solve Eq. (9a) away from the resetting
point, i.e. in the region $\tilde{x}\in(0,\ell)$, where the term proportional
to the Dirac delta function vanishes. This results in

$\tilde{P}_{D}(\tilde{x})=A_{D}e^{\tilde{x}}+B_{D}e^{-\tilde{x}}.$ (12)

By integrating Eq. (9a) over the region $\tilde{x}\in[-\epsilon,\epsilon]$,
recalling that $-\epsilon$ is identified with $\ell-\epsilon$, and then taking
$\epsilon\to 0^{+}$, we obtain the following jump condition at the origin

$\left.\frac{\partial\tilde{P}_{D}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to
0^{+}}-\left.\frac{\partial\tilde{P}_{D}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to\ell^{-}}=-\left(\tilde{J}_{R}^{[2]}(\ell^{-})-\tilde{J}_{R}^{[1]}(0^{+})\right).$
(13)

Using this jump condition (13) in combination with the periodic boundary
condition (11a), we can fix the constants $A_{D}$ and $B_{D}$ in Eq. (12),
eventually obtaining

$\tilde{P}_{D}(\tilde{x})=\frac{p_{D}}{2}\frac{\cosh(\tilde{x}-\frac{\ell}{2})}{\sinh(\frac{\ell}{2})},$
(14)

where we have integrated Eq. (14) over the entire interval
$\tilde{x}\in[0,\ell]$ to yield
$\tilde{J}_{R}^{[2]}(\ell^{-})-\tilde{J}_{R}^{[1]}(0^{+})=p_{D}$, where

$p_{D}=\int_{0}^{\ell}d\tilde{x}~{}\tilde{P}_{D}(\tilde{x})$ (15)

is the probability to be in the diffusion phase at stationarity. We similarly
define $p_{R}=p_{R}^{[1]}+p_{R}^{[2]}$ as the probability to be in the
resetting phase at stationarity, where

$\displaystyle p_{R}^{[1]}$
$\displaystyle=\int_{0}^{\delta\ell}d\tilde{x}~{}\tilde{P}_{R}^{[1]}(\tilde{x}),$
(16a) $\displaystyle p_{R}^{[2]}$
$\displaystyle=\int_{\delta\ell}^{\ell}d\tilde{x}~{}\tilde{P}_{R}^{[2]}(\tilde{x}),$
(16b)

are the probabilities for the particle to be found in the different regions of
the ratchet and in the resetting phase.

Figure 3: Steady-state probability densities $\tilde{P}_{D}(\tilde{x})$, Eq.
(14), $\tilde{P}_{R}(\tilde{x})$, Eq. (19), and
$\tilde{P}(\tilde{x})=\tilde{P}_{D}(\tilde{x})+\tilde{P}_{R}(\tilde{x})$, as a
function of particle position $\tilde{x}$ for parameters (a) $\delta=0$,
$\ell=4$, $\kappa=1$, (b) $\delta=0.2$, $\ell=4$, $\kappa=1$, (c)
$\delta=0.5$, $\ell=4$, $\kappa=1$, and (d) $\delta=0$, $\ell=1$, $\kappa=4$.
Vertical dotted lines indicate the position of the ratchet apex
$\tilde{x}=\delta\ell$ (equivalent to $x=a$) for the cases where $\delta>0$.
Simulation data are plotted as symbols and are in excellent agreement with the
theoretical results. Full simulation details are given in Appendix A.

We now proceed to obtain the steady-state probability density for the
resetting phase $\tilde{P}_{R}(\tilde{x})$. Due to the discontinuity in the
potential at the apex $\tilde{x}=\delta\ell$, we also have continuity of the
probability density and current for the resetting phase at this position, i.e.

$\displaystyle\tilde{P}_{R}^{[1]}(\delta\ell)$
$\displaystyle=\tilde{P}_{R}^{[2]}(\delta\ell),$ (17a)
$\displaystyle\tilde{J}_{R}^{[1]}(\delta\ell)$
$\displaystyle=\tilde{J}_{R}^{[2]}(\delta\ell),$ (17b)

where the current for the resetting phase is defined in Eq. (6b). We also have
the jump condition at the origin,

$\left.\frac{\partial\tilde{P}_{R}^{[1]}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to
0^{+}}-\left.\frac{\partial\tilde{P}_{R}^{[2]}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to\ell^{-}}=\tilde{J}_{R}^{[2]}(\ell^{-})-\tilde{J}_{R}^{[1]}(0^{+})=p_{D}.$
(18)

As above, we first solve Eqs. (9b)-(9c) in the region $\tilde{x}\in(0,\ell)$,
where the terms proportional to the Dirac delta function vanish. After
inserting the solution for $\tilde{P}_{D}(\tilde{x})$, Eq. (14), into Eqs.
(9b)-(9c) and then solving, we eventually obtain

$\tilde{P}_{R}(\tilde{x})=\begin{cases}\frac{p_{D}}{2}\frac{\delta\ell}{e^{\ell}-1}\left(\frac{e^{\ell-\tilde{x}}}{\lambda^{[1]}_{-}}-\frac{e^{\tilde{x}}}{\lambda^{[1]}_{+}}\right)+A_{R}^{[1]}e^{-\frac{\kappa\tilde{x}}{\delta\ell}}+B_{R}^{[1]},&0\leq\tilde{x}<\delta\ell\\\
\frac{p_{D}}{2}\frac{(1-\delta)\ell}{e^{\ell}-1}\left(\frac{e^{\tilde{x}}}{\lambda^{[2]}_{-}}-\frac{e^{\ell-\tilde{x}}}{\lambda^{[2]}_{+}}\right)+A_{R}^{[2]}e^{\frac{\kappa\tilde{x}}{(1-\delta)\ell}}+B_{R}^{[2]},&\delta\ell\leq\tilde{x}<\ell\end{cases},$
(19)

where we have written the solution in terms of the compact notation
$\lambda^{[1]}_{\pm}=\kappa\pm\delta\ell$,
$\lambda^{[2]}_{\pm}=\kappa\pm(1-\delta)\ell$. Equation (19) is valid only
when both $\lambda_{-}^{[1]}\neq 0$ and $\lambda_{-}^{[2]}\neq 0$. We provide
the solutions for the edge cases (i)
$\lambda_{-}^{[1]}=0\neq\lambda_{-}^{[2]}$, (ii)
$\lambda_{-}^{[2]}=0\neq\lambda_{-}^{[1]}$, and (iii)
$\lambda_{-}^{[1]}=\lambda_{-}^{[2]}=0$ at the end of Appendix B. The
constants $A_{R}^{[1]},A_{R}^{[2]},B_{R}^{[1]},B_{R}^{[2]}$ in Eq. (19) are
fixed by the periodic boundary condition (11b), continuity conditions (17),
and the jump condition (18). The diffusion-phase probability $p_{D}$ can be
fixed in terms of the system parameters $\ell,\kappa,\delta$ through the
normalisation condition,

$p_{D}+p_{R}^{[1]}+p_{R}^{[2]}=1,$ (20)

where the probabilities for the particle to be found in each phase $p_{D,R}$
are defined in Eqs. (15)-(16). In principle, applying all of the
aforementioned conditions yields closed-form solutions for the densities
$\tilde{P}_{D}(\tilde{x})$, Eq. (14), and $\tilde{P}_{R}(\tilde{x})$, Eq.
(19). However, we find the full expressions are too cumbersome to write down
here, in particular that of the overall normalisation $p_{D}$. Nevertheless,
it turns out that in certain (relevant) limits, other observables such as the
net current $\tilde{J}$ are compact enough to display explicitly in terms of
the system parameters. We demonstrate this in Sec. IV.

As noted in Refs. [29, 30], an alternative way to fix $p_{D}$ is to relate the
probability of being in either phase to the mean first-passage time
$T_{R}(x_{0})$ for the particle to reach the origin $x=0$ (or $x=L$) from a
position $x_{0}$ in the resetting phase. Specifically,

$p_{D}=\frac{\tau}{\tau+\int_{0}^{L}dx_{0}~{}T_{R}(x_{0})P(x_{0}|D)},$ (21)

where $\tau=1/r$ is the average time spent in the diffusion phase in a single
cycle, and $P(x_{0}|D)=P_{D}(x_{0})/p_{D}$ (the probability to be at position
$x_{0}$ given the particle is in the diffusion phase) provides the weighting
for initially being found at position $x_{0}$ in the resetting phase, such
that $\int_{0}^{L}dx_{0}~{}T_{R}(x_{0})P(x_{0}|D)$ is the average time spent
in the resetting phase each cycle. However, Eqs. (20) and (21) are not
independent conditions, since we show in Appendix D that Eq. (21) can be
derived from Eq. (20) using only the definition of the mean first-passage
time. For completeness, we derive the mean first-passage time $T_{R}(x_{0})$
in Appendix E. The time-dependent behaviour of $p_{D}(t)$ is plotted in figure
2c, where it can be seen that $p_{D}(t)$ saturates to a steady-state value
$p_{D}$ determined by Eq. (20) or, equivalently, Eq. (21).

In addition to the steady-state solutions derived in this section, we
demonstrate how one can derive the fully time-dependent Laplace-space
solutions in Appendix B. The time-dependent densities,
$\tilde{P}_{D}(\tilde{x},\tilde{t})$ and $\tilde{P}_{R}(\tilde{x},\tilde{t})$,
are plotted in figures 2a and 2b, respectively. It can be seen that the time-
dependent densities converge to the expressions for the theoretical steady-
state densities, $\tilde{P}_{D}(\tilde{x})$ and $\tilde{P}_{R}(\tilde{x})$,
given in Eqs. (14) and (19), respectively.

The solutions for the steady-state densities $\tilde{P}_{D}(\tilde{x})$, Eq.
(14), and $\tilde{P}_{R}(\tilde{x})$, Eq. (19), are plotted in figure 3
alongside simulation results for a few different sets of parameters. Data from
the simulations are in excellent agreement with the theoretical results. We
observe that the probability density in the diffusion phase
$\tilde{P}_{D}(\tilde{x})$ has a symmetric exponential form about the origin
$x=0$, as is typical for a resetting model. The diffusion-phase density is
also periodic, as required by the boundary conditions, with a minimum at the
centre of the ratchet potential $\tilde{x}=\ell/2$, such that it is also
symmetric about $\tilde{x}=\ell/2$. The shape of the probability density in
the resetting phase $\tilde{P}_{R}(\tilde{x})$ heavily depends on the position
of the ratchet potential’s apex $\delta\ell$. In particular, the resetting
density is symmetric if the potential is symmetric, i.e. $\delta=1/2$, see
figure 3c, but is otherwise asymmetric and generally has a prominent corner at
the apex as a result of the different potential gradients either side of
$\tilde{x}=\delta\ell$.

Figure 4: Steady-state probability to be in the diffusion phase $p_{D}$, Eq.
(15), as a function of the dimensionless parameters $\ell$, $\kappa$, and
$\delta$. The constant parameter in each subfigure is (a) $\ell=8$, (b)
$\kappa=4$, and (c) $\delta=0.2$.

In figure 4, we plot the probability of the particle being in the diffusion
phase at stationarity, Eq. (15). This captures the competition between the
relevant timescales of the system, namely the average times spent in the
diffusion and resetting phases, as illustrated by Eq. (21). The particle has a
greater chance of being in the diffusion phase for larger $\kappa$ and smaller
$\ell$. Larger $\kappa$ indicates a stronger coupling to the potential, and
therefore the potential resets the particle to the diffusion phase quicker,
thus spending less time in the resetting phase and more time in the diffusion
phase. In fact, $p_{D}\to 1$ as $\kappa\to\infty$. Similarly, smaller $\ell$
indicates the particle has been able to explore more of the interval between
resetting phases, i.e. the density is more equilibrated between resets, and so
the particle spends a greater proportion of its time in the diffusion phase.
As above, $p_{D}\to 1$ as $\ell\to 0$.

In figure 4, we observe that the particle spends more time in the resetting
phase for a fully asymmetric ratchet, i.e. $\delta=0$ or $\delta=1$. As this
is not immediately clear from intuition alone, we explain this behaviour with
the following heuristic argument. We will consider the case where the particle
has had sufficient time between resetting phases to spread out uniformly
across the interval as this allows us to make several simplifying assumptions
about the dynamics. In this case, the probability of being found in either
region upon entering the resetting phase is given by the fraction of the
interval the region covers, i.e. $\delta$ for $i=1$ and $1-\delta$ for $i=2$.
For a sufficiently strong potential, we can further approximate the particle
as being returned to the origin deterministically by the force from the
ratchet. In other words, the average time spent in region $i=1$ or region
$i=2$ during the resetting phase is simply the average distance the particle
travels to the origin (half the length of the region) divided by the force in
that region, resulting in $\delta^{2}\ell^{2}/2\kappa$ for $i=1$ and
$(1-\delta)^{2}\ell^{2}/2\kappa$ for $i=2$. Weighting these times by the
probability to be found in either region, we find the average time spent in
the resetting phase each cycle under these approximations is
$(3\delta^{2}-3\delta+1)\ell^{2}/2\kappa$, which is minimised for a symmetric
ratchet, $\delta=1/2$, and maximised for a fully asymmetric ratchet,
$\delta=0$ or $\delta=1$.

##  IV Current

The currents for the diffusion $\tilde{J}_{D}(\tilde{x})$ and resetting
$\tilde{J}_{R}(\tilde{x})$ phases are related to the probability densities
calculated in Sec. III through Eqs. (6a) and (6b), respectively. The net
current is then given by $\tilde{J}=\tilde{J}_{D}(x)+\tilde{J}_{R}(x)$,
yielding

$\begin{split}\tilde{J}&=\frac{p_{D}}{2}\frac{\kappa^{2}}{(e^{\ell}-1)(1-e^{\kappa})\lambda_{+}^{[1]}\lambda_{-}^{[1]}\lambda_{+}^{[2]}\lambda_{-}^{[2]}}\left(\lambda_{-}^{[1]}\lambda_{+}^{[2]}e^{\lambda_{+}^{[1]}}+(2\delta-1)\left(\lambda_{+}^{[1]}\lambda_{+}^{[2]}e^{\ell}-\lambda_{-}^{[1]}\lambda_{-}^{[2]}\right)-\lambda_{+}^{[1]}\lambda_{-}^{[2]}e^{\lambda_{+}^{[2]}}\right),\\\
\end{split}$ (22)

where $p_{D}$ is fixed from either Eq. (20) or Eq. (21), and we have reused
the same compact notation $\lambda^{[1]}_{\pm}=\kappa\pm\delta\ell$,
$\lambda^{[2]}_{\pm}=\kappa\pm(1-\delta)\ell$ introduced in Eq. (19). In the
present section, we explore the parameter space of the net current
$\tilde{J}$. Given that Eq. (22) is rather opaque, we focus on exploring its
more relevant limiting behaviour and general properties, as well as that of
the currents associated with each phase, $\tilde{J}_{D}(x)$ and
$\tilde{J}_{R}(x)$.

Figure 5: Steady-state currents $\tilde{J}_{D}(x)$, Eq. (6a),
$\tilde{J}_{R}(x)$, Eq. (6b), and
$\tilde{J}=\tilde{J}_{D}(x)+\tilde{J}_{R}(x)$, Eq. (22), plotted for the same
sets of parameters used in figure 3, i.e. (a) $\delta=0$, $\ell=4$,
$\kappa=1$, (b) $\delta=0.2$, $\ell=4$, $\kappa=1$, (c) $\delta=0.5$,
$\ell=4$, $\kappa=1$, and (d) $\delta=0$, $\ell=1$, $\kappa=4$. Dashed grey
lines are shown for $\tilde{J}=0$ and $\tilde{x}=\ell/2$, and act as a guide
for the eye.

The net current $\tilde{J}$, Eq. (22), and currents for each phase
$\tilde{J}_{D,R}(x)$, are plotted in figure 5 for the same sets of parameters
used to plot the corresponding probability densities in figure 3. We observe
that the diffusion-phase current is always zero at the halfway point of the
interval, i.e. $\tilde{J}_{D}(\ell/2)=0$, and that the spatially averaged
diffusion-phase current vanishes, i.e.
$\int_{0}^{\ell}d\tilde{x}~{}\tilde{J}_{D}(\tilde{x})/\ell=0$. More generally,
the diffusion-phase current is antisymmetric about the origin $\tilde{x}=0$,
i.e. $\tilde{J}_{D}(\tilde{x})=-\tilde{J}_{D}(\ell-\tilde{x})$, where we have
identified $-\tilde{x}\sim\ell-\tilde{x}$. This immediately implies the
diffusion-phase current is also antisymmetric about the halfway point
$\tilde{x}=\ell/2$, which can be easily seen by redefining
$\tilde{x}=\tilde{x}^{\prime}-\ell/2$ in the above relation. These symmetry
properties arise because the diffusion phase is not biased by any potential,
and so every time the particle is reset to the origin $\tilde{x}=0$, its
density proceeds to diffuse away symmetrically, see figure 2a. Unlike the
diffusion phase, the resetting-phase current $\tilde{J}_{R}(\tilde{x})$ is
generally antisymmetric about the halfway point $\tilde{x}=\ell/2$ and the
origin $\tilde{x}=0$ only after adjusting by a constant shift, i.e.
$\tilde{J}_{R}(\tilde{x})-\tilde{J}=-(\tilde{J}_{R}(\ell-\tilde{x})-\tilde{J})$.
This is expected from the antisymmetry properties of
$\tilde{J}_{D}(\tilde{x})$, since
$\tilde{J}_{D}(\tilde{x})=\tilde{J}-\tilde{J}_{R}(\tilde{x})$ by definition.
As a result of this antisymmetry, the spatially averaged resetting-phase
current is simply the net current, i.e.
$\int_{0}^{\ell}d\tilde{x}~{}\tilde{J}_{R}(\tilde{x})/\ell=\tilde{J}$,
indicating that the resetting-phase current is solely responsible for the non-
zero net current that is generated by an asymmetric ratchet, $\delta\neq 1/2$,
see figures 5a, 5b, and 5d. On the other hand, for a symmetric ratchet,
$\delta=1/2$, the resetting-phase current has the same antisymmetry properties
as the diffusion-phase current and so the net current $\tilde{J}$ vanishes,
see figure 5c.

The role of the ratchet’s symmetry is more clearly elucidated from inserting
the explicit expression for the resetting-phase current
$\tilde{J}_{R}^{[i]}(\tilde{x})=-\partial_{\tilde{x}}\tilde{P}_{R}^{[i]}(\tilde{x})-\tilde{V}^{{}^{\prime}[i]}\tilde{P}_{R}^{[i]}(\tilde{x})$
into the relation
$\tilde{J}=\int_{0}^{\ell}d\tilde{x}~{}\tilde{J}_{R}(\tilde{x})/\ell$ found
above, whence we obtain

$\tilde{J}=-\frac{1}{\ell}\int_{0}^{\ell}d\tilde{x}~{}\tilde{V}^{\prime}(\tilde{x})\tilde{P}_{R}(\tilde{x})=\frac{\kappa}{\ell^{2}}\left(\frac{p_{R}^{[2]}}{1-\delta}-\frac{p_{R}^{[1]}}{\delta}\right),$
(23)

where we have used $\tilde{P}_{R}(0)=\tilde{P}_{R}(\ell)=0$, Eq. (11b), the
definitions for each $p_{R}^{[i]}$, Eq. (16), and have inserted the explicit
expressions for each $\tilde{V}^{\prime[i]}$, Eq. (10). Equation (23) relates
the current $\tilde{J}$ to the probabilities of being found in either region
of the ratchet and shows, in particular, that the current vanishes when there
is an equal chance of being found in either region, $p_{R}^{[1]}=p_{R}^{[2]}$,
i.e. when the ratchet is symmetric $\delta=1/2$.

Figure 6: Steady-state current $\tilde{J}$, Eq. (22), as a function of the
dimensionless parameters $\ell$, $\kappa$, and $\delta$. The constant
parameter in each subfigure is (a) $\ell=4$, (b) $\kappa=4$, and (c)
$\delta=0.2$.

The net current $\tilde{J}$, Eq. (22), is plotted in figure 6 as a function of
the three dimensionless parameters $\ell,\kappa,\delta$. We observe that
maximum positive current is obtained for a fully asymmetric ratchet
$\delta=0$, since this removes the region of the ratchet that imparts an
unfavourable negative drift. Figure 6 also shows that a stronger coupling to
the potential $\kappa$ leads to a larger current. This is because larger
$\kappa$ reduces the time it takes to reset the particle and reduces the
chance of unfavourable trajectories where the particle diffuses up the
gradient of the potential. Larger $\ell$, indicating a relatively small
diffusion distance $\sigma=\sqrt{D_{0}/r}$ between resetting phases, is seen
to decrease the current. If $\ell$ satisfies either $1/\ell<\delta<1/2$ or
$1/\ell<1-\delta<1/2$, equivalent to $\sigma<a<L/2$ and $\sigma<L-a<L/2$,
respectively, then the majority of resetting cycles will result in the
particle being reset without “completing a lap” of the system (on an extended
periodic landscape, this would be equivalent to the particle being reset to
the same resetting point each time). This results in an effective confinement
of the particle close to the origin $\tilde{x}=0$, reducing any current
generated to rare noise-induced escapes from this effective confinement.

We now consider several relevant limiting cases of the net current
$\tilde{J}$, Eq. (22). First, we consider instantaneous stochastic resetting
$\kappa\to\infty$. In this limit, the force from the potential is infinitely
strong and so the particle is instantaneously reset to the origin
$\tilde{x}=0$ every time the potential is switched on. We find the current in
the instantaneous resetting limit $\kappa\to\infty$ is given by

$\tilde{J}\xrightarrow{\kappa\to\infty}\frac{\sinh\left(\frac{(1-2\delta)\ell}{2}\right)}{2\sinh\left(\frac{\ell}{2}\right)},$
(24)

which is bounded by $1/2-\delta$ from above or below depending on whether
$\delta<1/2$ or $\delta>1/2$, respectively, and decays monotonically to zero
as $\ell\to\infty$. Equation (24) could have equally been derived from the
diffusion-phase density, Eq. (14), alone using the following argument. For
instantaneous resetting $\kappa\to\infty$, the net current depends solely on
where the particle enters the resetting phase, as this determines the distance
the particle travels upon being reset. Hence, we can deduce that the current
will be given by

$\tilde{J}\xrightarrow{\kappa\to\infty}\frac{1}{\ell}\int_{\delta\ell}^{\ell}dy~{}(\ell-y)\tilde{P}_{D}(y|D)-\frac{1}{\ell}\int_{0}^{\delta\ell}dy~{}y\tilde{P}_{D}(y|D),$
(25)

where the first term arises from trajectories where the particle ends its
diffusion phase in the $i=2$ region of the ratchet at position $y$, resulting
in a jump forward by a distance $\ell-y$ upon being reset. The second term
comes from the particle ending its diffusion phase in the $i=1$ region, in
which case it jumps backwards a distance $y$ upon being reset. By using that
$\tilde{P}_{D}(y|D)=\tilde{P}_{D}(y)$ for $\kappa\to\infty$, since $p_{D}=1$
in this case, then inserting the explicit expression for $\tilde{P}_{D}(y)$
from Eq. (14) into Eq. (25), and finally evaluating the integrals, we recover
the instantaneous resetting limit, Eq. (24). In the limit $\kappa\to 0$, we
find the current vanishes because there is no potential to rectify the
particle’s motion, and so its dynamics reduce to purely diffusive motion on an
unbiased interval.

As illustrated in figure 6, the net current $\tilde{J}$, Eq. (22), is larger
for a fully asymmetric ratchet, $\delta\to 0$, since the drift imparted by the
potential is always positive in this case. In this “ideal ratchet” limit
$\delta\to 0$, we obtain the more compact expression

$\begin{split}\tilde{J}&\xrightarrow{\delta\to
0}\frac{p_{D}}{2}\frac{\kappa^{2}(e^{\ell}-1)(e^{\kappa}+1)-\kappa\ell(e^{\ell}+1)(e^{\kappa}-1)}{(\kappa^{2}-\ell^{2})(e^{\kappa}-1)(e^{\ell}-1)}\\\
&=\frac{\kappa^{2}(e^{\ell}-1)(e^{\kappa}+1)-\kappa\ell(e^{\ell}+1)(e^{\kappa}-1)}{2\kappa^{2}(e^{\kappa}-1)(e^{\ell}-1)+\kappa\ell^{2}(e^{\kappa}+1)(e^{\ell}-1)-\ell^{2}(e^{\kappa}-1)(\ell-2+e^{\ell}(\ell+2))},\end{split}$
(26)

while the opposite limit $\delta\to 1$ simply returns the above expression
with an overall minus sign.

Figure 6a suggests the maximum possible current is obtained for a ratchet that
is both fully asymmetric $\delta\to 0$, and infinitely strong
$\kappa\to\infty$. This corresponds to an instantaneous resetting model in
which a particle jumps forward every resetting event. Taking both
$\kappa\to\infty$ and $\delta\to 0$ in Eq. (22), we find $\tilde{J}\to 1/2$,
which also establishes an upper bound on the current that can be generated.
This could have equally been derived through the following physical argument.
Suppose the diffusion phase ends with the particle located a distance $y$ from
the origin $\tilde{x}=0$. Then, upon being reset, the particle will jump
forward by either $y$ or $\ell-y$, i.e. an average distance $\ell/2$. Given
the diffusion-phase dynamics are symmetric and hence do not contribute to the
net current, the average number of times the particle passes a point in the
interval of length $\ell$ is therefore $\tilde{J}=1/2$. In dimensionful units,
this equates to $J=r/2$, and so the physical current can be made arbitrarily
large by increasing the resetting rate, since this corresponds to an
infinitely frequent series of instantaneous jumps forward.

As seen in figure 6, the current’s dependence on the system parameters is
monotonic. However, the average drift speed of the particle’s rectified
motion, given by

$\langle\tilde{v}\rangle=\tilde{J}\ell$ (27)

in dimensionless units, has non-monotonic dependence on the system length
$\ell$ for finite $\kappa$ or $\delta$, as shown in figure 7. For
$\kappa\to\infty$ and $\delta\to 0$, however, we find
$\langle\tilde{v}\rangle\to\ell/2$, suggesting the particle’s drift speed
diverges as $\ell\to\infty$. In this case, it is advantageous to have a larger
system length $\ell$ for obtaining greater drift speed, as the particle always
jumps forward upon being reset and the distance it jumps is greater for larger
$\ell$.

Figure 7: Steady-state average drift speed $\langle\tilde{v}\rangle$, Eq.
(27), as a function of the dimensionless parameters $\ell$, $\kappa$, and
$\delta$. The constant parameter in each subfigure is (a) $\kappa=4$ and (b)
$\delta=0.2$.

##  V Energetic cost and efficiency of resetting

In the previous section, we found the dimensionless current $\tilde{J}$ is
trivially maximised by taking $\delta\to 0$ and $\kappa\to\infty$, the latter
being equivalent to an infinitely strong potential. The physical current
$J=r\tilde{J}$ is then maximised by an infinite resetting rate $r\to\infty$.
While maximising the current is a physically relevant objective, it does not
take into account the cost associated with using the potential to reset the
particle. In particular, we expect a potential of greater amplitude to incur a
greater energetic cost. A larger resetting rate $r$ would similarly represent
a larger rate with which energy needs to be supplied to the system.

To quantify the energetic cost associated with producing a current in this
system, we are motivated to study a suitable efficiency parameter. For non-
equilibrium systems, the thermodynamic efficiency
$\eta_{\mathrm{th}}=\dot{W}/(\dot{W}+D_{0}\dot{S})$ is typically studied [49,
50], where $\dot{W}$ is the useful power that can be extracted from the system
(such as by doing work against an external force [49, 50, 59, 60, 61]) and
$\dot{S}$ is the entropy production rate [53], which quantifies the
irreversibility of the dynamics. As for many resetting models, the entropy
production of the present system diverges due to the resetting process being
unidirectional [62]. Hence, the thermodynamic efficiency vanishes and
therefore cannot be used to distinguish between different parameters. This is
avoided in approximate resetting schemes, such as motion in a fluctuating
confining potential [18, 63], or resetting to a distribution [bressloff2024,
65], for which the corresponding entropy production rates are finite. However,
for single-point resetting, an alternative way of quantifying the energetic
cost must be sought.

Figure 8: Energetic cost of resetting compared to simulations. (a) Cumulative
energy input for a single simulated particle against time. The analytical
prediction for the cumulative energy input is obtained by multiplying the
exact analytic expression for energy input rate, Eq. (29), by the elapsed
time, and is plotted as a dashed line. In the simulation, the cumulative
energy input experiences discontinuous jumps of $\tilde{V}(\tilde{x})$ each
time the potential is switched on, i.e. when the particle goes from the
diffusion phase to the resetting phase. The simulation parameters are
$\ell=4$, $\kappa=1$, $\delta=0.5$. (b)-(e) Rate of energy input
$\dot{E}_{i}$, Eq. (28), compared to simulations. The theoretical results from
Eq. (29) are shown as solid lines. The simulation data are plotted as symbols
for $\kappa=1,2,3,4$, against various values of $\delta\in(0,1)$, with
$\ell=4$. The simulations show very good agreement with the analytical
results. The simulated energy input rates were calculated by averaging
$10^{4}$ particle realisations up to a total time $\tilde{t}=500$. The
simulations of the energetic cost were performed with a smaller timestep
$\delta\tilde{t}=10^{-5}$ than the other simulations in this work due to the
small variation in energetic cost (see, in particular, subfigure (c))
necessitating a higher degree of accuracy.

One recent proposal for an appropriate cost function associated with resetting
is based on the distance the particle travels upon being reset [66]. A
drawback of this is that it does not take into account the nature of the
mechanism by which the particle is reset. A different proposal that is more
grounded in thermodynamics is based on an experimental implementation of
stochastic resetting, where the cost of a reset is defined as the total energy
supplied to an experimental trapping device, such as an optical trap, over the
duration of the resetting phase [14]. In the same vein, we define an
experimentally motivated, yet analytically tractable, cost function given by
the average rate of work done by the potential in order to repeatedly return
the particle to the origin. The work done over the course of a single
resetting phase is equal to the difference in potential energy of the particle
at the beginning and end of the phase [18, 63, 35]. Since every stochastic
trajectory in the resetting phase ends at $x=0$, where $V(0)=0$, then the net
work done in a resetting phase starting at position $x=x_{s}$ is simply
$V(x_{s})$. Said another way, the average rate of work done is equal to the
power supplied to the system to sustain the repeated jumps in the particle’s
potential energy from switching on the resetting potential, see figure 8a.
Therefore, the average work done over a complete cycle of diffusion and
resetting phases is $\int_{0}^{L}dx~{}V(x)P(x|D)$, where
$P(x|D)=P_{D}(x)/p_{D}$ is the conditional probability density for the
particle to be found at position $x$ given it is in the diffusion phase.
Meanwhile, the average time for a resetting cycle is
$\tau+\int_{0}^{L}dx~{}T_{R}(x)P(x|D)=1/(rp_{D})$, see Eq. (21). Hence, by
dividing the average work done over a cycle by the average time for a cycle,
we obtain for the steady-state rate of energy input/work done,

$\dot{E}_{i}=r\int_{0}^{L}dx~{}V(x)P_{D}(x).$ (28)

Equation (28) vanishes for either $h\to 0$ or $r\to 0$, and diverges in the
instantaneous resetting limit $h\to\infty$, as we would expect for a
physically meaningful measure of the energetic cost associated with resetting.

By evaluating the integral in Eq. (28), we obtain the following expression for
the energy input rate in terms of the dimensionless parameters and $p_{D}$,

$\dot{\tilde{E}}_{i}=p_{D}\frac{\kappa\sinh\left(\frac{\delta\ell}{2}\right)\sinh\left(\frac{(1-\delta)\ell}{2}\right)}{\delta\ell(1-\delta)\sinh\left(\frac{\ell}{2}\right)}.$
(29)

In figure 8, we show that Eq. (29) is in good agreement with the simulation
results. In figure 9, we observe that the energy input rate generally
increases with $\kappa$ and decreases with $\ell$. The energy input rate
decreases with $\ell$ because larger $\ell$ represents a relatively smaller
diffusion distance $\sigma=\sqrt{D_{0}/r}$ between resets, meaning the
particle is effectively confined closer to the origin $\tilde{x}=0$ between
resets, and so the potential does relatively little work to reset the particle
each cycle. Varying the asymmetry parameter $\delta$ has comparatively little
effect on the energy input rate, but there is an interesting non-monotonic
dependence on $\delta$ nonetheless, as can be seen in figures 8b-e.

Figure 9: Steady-state energy input rate $\dot{\tilde{E}}_{i}$, Eq. (29), as a
function of the dimensionless parameters $\ell$, $\kappa$, and $\delta$. The
constant parameter in each subfigure is (a) $\ell=2$, (b) $\kappa=10$, and (c)
$\delta=0.2$.

In the fully asymmetric limit $\delta\to 0$, the energy input rate can be
written out exactly in terms of the dimensionless parameters, i.e.

$\dot{\tilde{E}}_{i}\xrightarrow{\delta\to
0}p_{D}\frac{\kappa}{2}=\frac{\kappa(\kappa^{2}-\ell^{2})(e^{\kappa}-1)(e^{\ell}-1)}{2\kappa^{2}(e^{\kappa}-1)(e^{\ell}-1)+\kappa\ell^{2}(e^{\kappa}+1)(e^{\ell}-1)-\ell^{2}(e^{\kappa}-1)(\ell-2+e^{\ell}(\ell+2))},$
(30)

and so, in this case, the rate that energy is supplied to the system is simply
equal to the average height of the potential, $\kappa/2$, divided by the total
time for a complete cycle, $1/p_{D}$, in dimensionless units.

We now wish to compare the energetic cost of resetting against the return we
get in current, similarly to the cost-weighted efficacy [67] and payoff [68]
functions recently defined for diffusive search processes with stochastic
resetting. We define our efficiency parameter as

$\eta=\frac{\langle\tilde{v}\rangle^{2}}{\dot{\tilde{E}}_{i}},$ (31)

where, in our units in which the friction coefficient has been set to unity,
$\langle\tilde{v}\rangle^{2}$ is the energy per unit time associated with the
current flow. Thus, $\eta$, which is bounded by $0$ and $1$, can be thought of
as the ratio of useful power output $\langle v\rangle^{2}$ to power input
$\dot{E}_{i}$. The efficiency can be expressed compactly in terms of the
probabilities to be in either phase and the dimensionless parameters,

$\eta=\frac{1}{p_{D}}\left(\frac{p_{R}^{[2]}}{1-\delta}-\frac{p_{R}^{[1]}}{\delta}\right)^{2}\frac{\kappa\delta(1-\delta)\sinh\left(\frac{\ell}{2}\right)}{\ell\sinh\left(\frac{\delta\ell}{2}\right)\sinh\left(\frac{(1-\delta)\ell}{2}\right)},$
(32)

where we stress again that $p_{D}$ and $p_{R}$ have closed-form expressions,
determined from Eq. (20), but they are too complicated to write out explicitly
here. Equation (32) is plotted in figure 10. We observe that the efficiency
has a non-monotonic dependence on $\kappa$ and $\ell$ due to a compromise
between larger $\kappa$ and smaller $\ell$ resulting in greater current, but
also incurring a greater energetic cost. The efficiency is generally greater
for a fully asymmetric ratchet, $\delta\to 0$ or $\delta\to 1$, due to the
magnitude of the particle’s average drift speed $\langle\tilde{v}\rangle$
increasing faster than the energetic cost $\dot{\tilde{E}}_{i}$ as $\delta\to
0$ or $\delta\to 1$. In the limit $\delta\to 0$, the efficiency reduces to the
exact expression,

$\eta\xrightarrow{\delta\to
0}\frac{\kappa\ell^{2}(\ell\coth(\ell/2)-\kappa\coth(\kappa/2))^{2}}{(\ell^{2}-\kappa^{2})(2(\ell^{2}-\kappa^{2})+\ell^{3}\coth(\ell/2)-\ell^{2}\kappa\coth(\kappa/2))},$
(33)

which is plotted in figure 11, and could have equally been obtained by taking
$\delta\to 1$ in Eq. (32). One can now find values for the remaining
parameters to optimise the efficiency. For instance, the optimal
$\kappa=\kappa^{*}$ for a given $\ell$ that maximises the efficiency is found
by solving $\left.\partial_{\kappa}\eta\right|_{\kappa=\kappa*}=0$.

Figure 10: Efficiency $\eta$, Eq. (32), as a function of the dimensionless
parameters $\ell$, $\kappa$, and $\delta$. The constant parameter in each
subfigure is (a) $\ell=4$, (b) $\kappa=4$, and (c) $\delta=0.2$. The maximum
efficiency in each subfigure is achieved at (a) $\delta=0$, $\kappa=5.9$; (b)
$\delta=0$, $\ell=5.7$; and (c) $\kappa=6.7$, $\ell=5.1$.

Figure 11 demonstrates that arbitrarily perfect efficiency can be achieved for
a region of parameter space where both $\kappa\to\infty$ and $\ell\to\infty$.
However, naïvely taking $\kappa\to\infty$ or $\ell\to\infty$ in Eq. (33)
returns zero efficiency. Extra care needs to be taken when taking such limits.
Physically, we expect $D_{0}\rightarrow 0$ to be optimal as it removes the
possibility of unfavourable trajectories where the particle diffuses up the
potential gradient. To take $D_{0}\to 0$, we can take $\kappa\to\infty$ and
$\ell\to\infty$ while keeping $\kappa/\ell^{2}=c=h/(L^{2}r)$ a finite
constant. This yields a finite efficiency $\eta=1/(1+2c)$, implying that
decreasing $c$ will result in a better efficiency. One may then be tempted to
set $c=0$ to yield perfect efficiency. However, this breaks our previous
requirement of $c$ being a finite constant. Hence, extra care needs to be
taken when taking these multiple limits. In particular, one needs to ensure
$D_{0}\to 0$ much faster than $c\to 0$. These limits are, of course, purely
mathematical, but give useful insight into how one could optimise the
efficiency of potential-mediated resetting nonetheless.

Figure 11: Efficiency $\eta$ in the limit $\delta\to 0$, Eq. (33), as a
function of $\kappa$ and $\ell$ on a log-log scale.

##  VI Conclusion

In this paper, we considered an experimentally motivated resetting model,
whereby the resetting of a particle to its initial position is mediated by a
periodic ratchet potential, Eq. (3), allowing a steady-state net current to be
generated. In particular, the ratchet potential switches on with a constant
rate, but switches off only upon the particle reaching the minimum of the
potential. Not only does this constitute an optimised variation of the
fluctuating potential ratchet model [40], it also generalises other well-known
resetting models [30, 1], which can be recovered by taking certain limits.

We derived the exact solutions for the steady-state phase-dependent
probability densities, Eqs. (12) and (19), which were analysed in detail in
Sec. III, and corroborated these with particle-based Monte Carlo simulations,
see figures 2 and 3. This was supplemented by a derivation of the time-
dependent densities in Appendix B, as well as a derivation of the steady-state
densities for a similar “stop-and-start” resetting model in Appendix C. From
the densities we derived an exact expression for the steady-state net current,
Eq. (22), in Sec. IV. We showed the net current is tightly bound from above by
$r/2$, where $r$ is the rate the potential is switched on, by considering the
optimal limits of an infinitely strong, Eq. (24), and fully asymmetric, Eq.
(26), potential. As far as we are aware, this is the first analytic study of a
steady-state net current generated through finite-time resetting. We made
connection with experiments and work extraction in Sec. V, where we calculated
the power input that would be required to sustain this system, Eq. (28), and
compared it with simulations, see figure 8. We then went further by defining
an efficiency parameter that quantifies the useful power output in the form of
current for a given power input, Eq. (31). We showed that this efficiency
parameter is bounded between $0$ and $1$, before finding the parameters that
yield arbitrarily perfect efficiency.

We believe this work can be extended in several ways. Firstly, we formulated
the model for overdamped dynamics, and so it could be generalised to
underdamped dynamics by incorporating inertia. In particular, it would be
interesting to see in what parameter regimes inertia improves the current. One
may expect inertia to enhance the current for a fully asymmetric ratchet where
the force from the ratchet is always in the desired direction, but hindered
for finite apex positions as then the ratchet has a region of much larger
force in the reverse direction. Another possible avenue for further research
is to allow the resetting phase to end at a position determined by a
distribution, such as a Gaussian of finite width centred at the minimum of the
potential. Allowing the resetting phase to potentially end at any position
renders the entropy production finite, which would then pave the way for a
study of the thermodynamic efficiency. An alternative approach could be to
associate some inherent precision error with determining when the particle has
reached the potential minimum in the resetting phase. As the error is made
smaller, the associated cost of measurement goes up [60], and so this becomes
an additional consideration for finite-time resetting schemes where the
resetting potential is modulated by a hypothetical external controller. If we
consider the potential to be periodic on an extended domain, then the present
model becomes exactly a finite-time resetting variation of the local resetting
model studied in Ref. [8]. One could then investigate how the addition of a
finite resetting time affects the boost in late-time diffusion seen in this
local resetting model. Finally, an interesting analogue to the stop-and-start
model we considered in Appendix C could be an active particle whose self-
propulsion stochastically switches off with a constant rate, and switches back
on again only upon the particle reaching the potential minimum. Active
particles in static ratchets generally prefer to move up the shallower slope
of the potential [50, 47], whereas the stop-and-start resetting scheme tends
to rectify particles down the shallower slope. An active stop-and-start
resetting model may then have non-monotonic dependence on the system
parameters to reflect this competition, as well as the possibility of
displaying current reversals.

###### Acknowledgements.

The authors thank Henry Alston, Thibault Bertrand, Gunnar Pruessner, and
Letian Chen for useful discussions. C.R. acknowledges support from the
Engineering and Physical Sciences Research Council (Grant No. 2478322). E.L.
was funded by the President’s PhD Scholarship at Imperial College London. E.S.
was funded by the Roth Scholarship at Imperial College London.

##  Appendix A Simulation details

By rescaling space and time by $\sigma=\sqrt{D_{0}/r}$ and $\tau=1/r$,
respectively, we can rewrite the Langevin equations (1)-(2) in terms of the
nondimensional parameters $\ell,\kappa,\delta$ defined in the main text, i.e.

$\displaystyle\dot{\tilde{x}}_{D}(\tilde{t})$
$\displaystyle=\sqrt{2}\xi(\tilde{t}),$ (34a)
$\displaystyle\dot{\tilde{x}}_{R}(\tilde{t})$
$\displaystyle=-\tilde{V}^{\prime}\left(\tilde{x}_{R}(\tilde{t})\right)+\sqrt{2}\xi(\tilde{t}),$
(34b)

where the gradient of the ratchet potential $\tilde{V}^{\prime}(\tilde{x})$ is
given by Eq. (10). The probability at each timestep to go from the diffusion
phase (34a) to the resetting phase (34b) is $\Delta\tilde{t}$, i.e. the
timestep duration, and the particle returns to the diffusion phase from the
resetting phase with probability 1 upon crossing either $\tilde{x}=0$ or
$\tilde{x}=\ell$. We solve Eq. (34) numerically using the Euler-Maruyama
method with timestep $\Delta\tilde{t}=10^{-4}$. Simulations were run up till a
final time $\tilde{t}=20$ for the numerical estimates of the steady-state
probability densities in figure 3, and were averaged over $10^{6}$
realisations.

##  Appendix B Time-dependent solutions for probability densities

In this appendix, we demonstrate how to derive the time-dependent probability
densities $P_{D,R}(x,t)$. For clarity, we will perform the derivation in terms
of the dimensionful variables $h$, $a$, $L$, $D_{0}$, and $r$, whose
definitions can be found in the main text. An analogous derivation for a
different model appears in Ref. [30]. The time-dependent probability densities
obey the coupled Fokker-Planck equations (5) which, for convenience, we repeat
here:

$\displaystyle\frac{\partial P_{D}(x,t)}{\partial t}$
$\displaystyle=D_{0}\frac{\partial^{2}P_{D}(x,t)}{\partial
x^{2}}-rP_{D}(x,t)+\left(J_{R}^{[2]}(L^{-},t)-J_{R}^{[1]}(0^{+},t)\right)\delta(x),$
(35a) $\displaystyle\frac{\partial P_{R}^{[1]}(x,t)}{\partial t}$
$\displaystyle=D_{0}\frac{\partial^{2}P_{R}^{[1]}(x,t)}{\partial
x^{2}}+V^{\prime[1]}\frac{\partial P_{R}^{[1]}(x,t)}{\partial
x}+rP_{D}(x)+J_{R}^{[1]}(0^{+},t)\delta(x),$ (35b)
$\displaystyle\frac{\partial P_{R}^{[2]}(x,t)}{\partial t}$
$\displaystyle=D_{0}\frac{\partial^{2}P_{R}^{[2]}(x,t)}{\partial
x^{2}}+V^{\prime[2]}\frac{\partial P_{R}^{[2]}(x,t)}{\partial
x}+rP_{D}(x)-J_{R}^{[2]}(L^{-},t)\delta(x).$ (35c)

As in the main text, we separate the solution for the resetting-phase density
into two separate components

$P_{R}(x,t)=\begin{cases}P_{R}^{[1]}(x,t),&0\leq x<a,\\\
P_{R}^{[2]}(x,t),&a\leq x<L,\end{cases}$ (36)

to align with the different regions of the potential, which has gradients
given by

$V^{\prime}(x)=\begin{cases}V^{\prime[1]}=\frac{h}{a},&0\leq x<a,\\\
V^{\prime[2]}=-\frac{h}{L-a},&a\leq x<L.\end{cases}$ (37)

We begin with the time-dependent solution for the diffusion-phase probability
density $P_{D}(x,t)$, described by Eq. (35a). In addition to the periodic
boundary conditions,

$\displaystyle P_{D}(0,t)$ $\displaystyle=P_{D}(L,t),$ (38a) $\displaystyle
P_{R}(0,t)$ $\displaystyle=P_{R}(L,t)=0,$ (38b)

we define the initial conditions as

$\displaystyle P_{D}(x,0)$ $\displaystyle=\delta(x),$ (39a) $\displaystyle
P_{R}(x,0)$ $\displaystyle=0,$ (39b)

i.e. we assume the particle is initialised at the origin in the diffusion
phase. From the definition of the Laplace transform,

$\hat{P}_{D,R}(x,s)=\int_{0}^{\infty}dt~{}e^{-st}P_{D,R}(x,t),$ (40)

we obtain for the Laplace-transformed diffusion-phase density,

$D_{0}\frac{\partial^{2}\hat{P}_{D}(x,s)}{\partial
x^{2}}-(s+r)\hat{P}_{D}(x,s)+\left(1+\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s)\right)\delta(x)=0,$
(41)

where we have used that

$\int_{0}^{\infty}dt~{}e^{-st}\frac{\partial P_{D}(x,t)}{\partial
t}=s\hat{P}_{D}(x,s)-\delta(x)$ (42)

after integrating by parts, recalling the initial condition
$P_{D}(x,0)=\delta(x)$, Eq. (39a).

Integrating Eq. (41) over the region $[-\epsilon,\epsilon]$, recalling
$-\epsilon\sim L-\epsilon$, and taking $\epsilon\to 0^{+}$, we obtain the
following jump condition at the origin,

$D_{0}\left(\left.\frac{\partial\hat{P}_{D}(x,s)}{\partial x}\right|_{x\to
0^{+}}-\left.\frac{\partial\hat{P}_{D}(x,s)}{\partial x}\right|_{x\to
L^{-}}\right)=-\left(1+\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s)\right),$
(43)

where $\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s)$ will be fixed in
terms of the parameters later by Eq. (57).

We now consider Eq. (41) in the region $x\in(0,L)$, where the term
proportional to the Dirac delta function vanishes. Solving in this region, we
obtain

$\hat{P}_{D}(x,s)=\hat{B}_{D}(s)e^{\sqrt{\frac{s+r}{D_{0}}}x}+\hat{C}_{D}(s)e^{-\sqrt{\frac{s+r}{D_{0}}}x},$
(44)

where $\hat{B}_{D}(s)$ and $\hat{C}_{D}(s)$ are constants to be determined.
The periodic boundary condition (38a) implies

$\hat{C}_{D}(s)=\frac{\exp\left(\sqrt{\frac{s+r}{D_{0}}}L\right)-1}{1-\exp\left(-\sqrt{\frac{s+r}{D_{0}}}L\right)}\hat{B}_{D}(s).$
(45)

Then, inserting the expression for $\hat{C}_{D}(s)$, Eq. (45), into Eq. (44)
and applying the jump condition (43), we obtain

$\displaystyle\hat{B}_{D}(s)$
$\displaystyle=\frac{1+\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s)}{2\sqrt{D_{0}(r+s)}\left(\exp\left(\sqrt{\frac{s+r}{D_{0}}}L\right)-1\right)},$
(46a) $\displaystyle\hat{C}_{D}(s)$
$\displaystyle=e^{-\sqrt{\frac{s+r}{D_{0}}}L}\frac{1+\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s)}{2\sqrt{D_{0}(r+s)}\left(\exp\left(\sqrt{\frac{s+r}{D_{0}}}L\right)-1\right)}.$
(46b)

Putting these together and defining

$\hat{A}_{D}(s)=\frac{1+\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s)}{2\sqrt{D_{0}(r+s)}},$
(47)

we finally obtain for the Laplace-transformed diffusion-phase density,

$\hat{P}_{D}(x,s)=\hat{A}_{D}(s)\frac{\cosh\left(\sqrt{\frac{s+r}{D_{0}}}\left(x-\frac{L}{2}\right)\right)}{\sinh\left(\sqrt{\frac{s+r}{D_{0}}}\frac{L}{2}\right)}.$
(48)

We now proceed to obtain the time-dependent solution for the resetting-phase
probability density $P_{R}(x,t)$, described by Eqs. (35b)-(35c). Taking the
Laplace transform of Eqs. (35b)-(35c), we obtain

$\displaystyle D_{0}\frac{\partial^{2}\hat{P}_{R}^{[1]}(x,s)}{\partial
x^{2}}+V^{\prime[1]}\frac{\partial\hat{P}_{R}^{[1]}(x,s)}{\partial
x}-s\hat{P}_{R}^{[1]}(x,s)+r\hat{P}_{D}(x,s)+\hat{J}_{R}^{[1]}(0^{+},s)\delta(x)$
$\displaystyle=0,$ (49a) $\displaystyle
D_{0}\frac{\partial^{2}\hat{P}_{R}^{[2]}(x,s)}{\partial
x^{2}}+V^{\prime[2]}\frac{\partial\hat{P}_{R}^{[2]}(x,s)}{\partial
x}-s\hat{P}_{R}^{[2]}(x,s)+r\hat{P}_{D}(x,s)-\hat{J}_{R}^{[2]}(L^{-},s)\delta(x)$
$\displaystyle=0.$ (49b)

As for the diffusion-phase density, we consider Eq. (49) in the region
$x\in(0,L)$ such that, after inserting the solution for $\hat{P}_{D}(x,s)$,
Eq. (48), we obtain the following partial differential equations for each
component of the Laplace-transformed resetting-phase density,

$\displaystyle D_{0}\frac{\partial^{2}\hat{P}_{R}^{[1]}(x,s)}{\partial
x^{2}}+\frac{h}{a}\frac{\partial\hat{P}_{R}^{[1]}(x,s)}{\partial
x}-s\hat{P}_{R}^{[1]}(x,s)+r\hat{A}_{D}(s)\frac{\cosh\left(\sqrt{\frac{s+r}{D_{0}}}\left(x-\frac{L}{2}\right)\right)}{\sinh\left(\sqrt{\frac{s+r}{D_{0}}}\frac{L}{2}\right)}$
$\displaystyle=0,$ (50a) $\displaystyle
D_{0}\frac{\partial^{2}\hat{P}_{R}^{[2]}(x,s)}{\partial
x^{2}}-\frac{h}{L-a}\frac{\partial\hat{P}_{R}^{[2]}(x,s)}{\partial
x}-s\hat{P}_{R}^{[2]}(x,s)+r\hat{A}_{D}(s)\frac{\cosh\left(\sqrt{\frac{s+r}{D_{0}}}\left(x-\frac{L}{2}\right)\right)}{\sinh\left(\sqrt{\frac{s+r}{D_{0}}}\frac{L}{2}\right)}$
$\displaystyle=0.$ (50b)

The solutions for each component of the resetting-phase density are then given
by

$\displaystyle\hat{P}_{R}^{[1]}(x,s)$
$\displaystyle=\hat{g}^{[1]}(x,s)+\hat{A}_{R}^{[1]}(s)\hat{h}_{a_{1}}(x,s)+\hat{B}_{R}^{[1]}(s)\hat{h}_{b_{1}}(x,s),$
(51a) $\displaystyle\hat{P}_{R}^{[2]}(x,s)$
$\displaystyle=\hat{g}^{[2]}(x,s)+\hat{A}_{R}^{[2]}(s)\hat{h}_{a_{2}}(x,s)+\hat{B}_{R}^{[2]}(s)\hat{h}_{b_{2}}(x,s).$
(51b)

where the $\hat{g}^{[i]}(x,s)$ are the particular solutions of Eq. (50),
$(\hat{h}_{a_{i}},\hat{h}_{b_{i}})$ are the homogeneous solutions, and
$(\hat{A}^{[i]}_{R},\hat{B}^{[i]}_{R})$ are constants to be fixed. The
homogeneous solutions are given by

$\displaystyle\hat{h}_{a_{1}}(x,s)$
$\displaystyle=\exp\left(-\frac{x}{2D_{0}a}\left(h+\sqrt{h^{2}+4D_{0}a^{2}s}\right)\right),$
(52a) $\displaystyle\hat{h}_{b_{1}}(x,s)$
$\displaystyle=\exp\left(-\frac{x}{2D_{0}a}\left(h-\sqrt{h^{2}+4D_{0}a^{2}s}\right)\right),$
(52b) $\displaystyle\hat{h}_{a_{2}}(x,s)$
$\displaystyle=\exp\left(\frac{x}{2D_{0}(L-a)}\left(h+\sqrt{h^{2}+4D_{0}a^{2}s}\right)\right),$
(52c) $\displaystyle\hat{h}_{b_{2}}(x,s)$
$\displaystyle=\exp\left(\frac{x}{2D_{0}(L-a)}\left(h-\sqrt{h^{2}+4D_{0}a^{2}s}\right)\right),$
(52d)

while the particular solutions (for the case when both $h/a\neq\sqrt{Dr}$ and
$h/(L-a)\neq\sqrt{Dr}$) are given by

$\displaystyle\hat{g}^{[1]}(x,s)$
$\displaystyle=\hat{A}_{D}(s)\frac{D_{0}ra^{2}}{(r+s)h^{2}-D_{0}ra^{2}}\frac{r\cosh\left(\sqrt{\frac{r+s}{D_{0}}}\left(x-\frac{L}{2}\right)\right)-\sqrt{\frac{r+s}{D_{0}}}\frac{h}{a}\sinh\left(\sqrt{\frac{r+s}{D_{0}}}\left(x-\frac{L}{2}\right)\right)}{\sinh\left(\sqrt{\frac{r+s}{D_{0}}}\frac{L}{2}\right)},$
(53a) $\displaystyle\hat{g}^{[2]}(x,s)$
$\displaystyle=\hat{A}_{D}(s)\frac{D_{0}r(L-a)^{2}}{(r+s)h^{2}-D_{0}r(L-a)^{2}}\frac{r\cosh\left(\sqrt{\frac{r+s}{D_{0}}}\left(x-\frac{L}{2}\right)\right)+\sqrt{\frac{r+s}{D_{0}}}\frac{h}{L-a}\sinh\left(\sqrt{\frac{r+s}{D_{0}}}\left(x-\frac{L}{2}\right)\right)}{\sinh\left(\sqrt{\frac{r+s}{D_{0}}}\frac{L}{2}\right)}.$
(53b)

Next, we fix the constants
$\hat{A}^{[1]}_{R}(s),\hat{A}^{[2]}_{R}(s),\hat{B}^{[1]}_{R}(s),\hat{B}^{[2]}_{R}(s)$
in Eq. (51). In addition to the periodic boundary condition, Eq. (38b), we
also have the following continuity conditions at the apex $x=a$,

$\displaystyle P_{R}^{[1]}(a,t)$ $\displaystyle=P_{R}^{[2]}(a,t),$ (54a)
$\displaystyle J_{R}^{[1]}(a,t)$ $\displaystyle=J_{R}^{[2]}(a,t),$ (54b)

where the latter condition (54b) can be expressed explicitly as the jump
condition

$D_{0}\left(\left.\frac{\partial P_{R}^{[2]}(x,t)}{\partial x}\right|_{x\to
a^{+}}-\left.\frac{\partial P_{R}^{[1]}(x,t)}{\partial x}\right|_{x\to
a^{-}}\right)=\left(V^{\prime[1]}(x\to a^{-})-V^{\prime[2]}(x\to
a^{+})\right)P_{R}^{[1]}(a,t).$ (55)

The boundary conditions for the resetting-phase density
$\hat{P}_{R}^{[i]}(x,s)$ can then be summarised by the matrix equation
$\mathsf{M}(s)\cdot\left(A_{R}^{[1]}(s),B_{R}^{[1]}(s),A_{R}^{[2]}(s),B_{R}^{[2]}(s)\right)=\textbf{u}(s)$,
explicitly

$\displaystyle\begin{bmatrix}\hat{h}_{a_{1}}(0,s)&\hat{h}_{b_{1}}(0,s)&0&0\\\
0&0&\hat{h}_{a_{2}}(L,s)&\hat{h}_{b_{2}}(L,s)\\\
\hat{h}_{a_{1}}(a,s)&\hat{h}_{b_{1}}(a,s)&\hat{h}_{a_{2}}(a,s)&\hat{h}_{b_{2}}(a,s)\\\
\frac{h/a+h/(L-a)}{D_{0}}\hat{h}_{a_{1}}(a,s)+\partial_{x}\hat{h}_{a_{1}}(a,s)&\frac{h/a+h/(L-a)}{D_{0}}\hat{h}_{b_{1}}(a,s)+\partial_{x}\hat{h}_{b_{1}}(a,s)&-\partial_{x}\hat{h}_{a_{2}}(a,s)&-\partial_{x}\hat{h}_{b_{2}}(a,s)\\\
\end{bmatrix}\cdot\begin{bmatrix}\hat{A}_{R}^{[1]}(s)\\\
\hat{B}_{R}^{[1]}(s)\\\ \hat{A}_{R}^{[2]}(s)\\\
\hat{B}_{R}^{[2]}(s)\end{bmatrix}=$ (56)
$\displaystyle\begin{bmatrix}-\hat{g}^{[1]}(0,s)\\\ -\hat{g}^{[2]}(L,s)\\\
-\left(\hat{g}^{[1]}(a,s)-\hat{g}^{[2]}(a,s)\right)\\\
-\left(\partial_{x}\hat{g}^{[1]}(a,s)-\partial_{x}\hat{g}^{[2]}(a,s)+\frac{h/a+h/(L-a)}{D_{0}}\hat{g}^{[1]}(a,s)\right)\end{bmatrix},$

where the expressions arising from the first two rows of Eq. (56) enforce the
periodic boundary condition (38b), the third row enforces Eq. (54a), and the
final row enforces Eq. (54b). We can obtain explicit expressions for the
constants
$\hat{A}^{[1]}_{R}(s),\hat{A}^{[2]}_{R}(s),\hat{B}^{[1]}_{R}(s),\hat{B}^{[2]}_{R}(s)$
in terms of the system parameters by multiplying the matrix inverse with the
vector on the right-hand side of Eq. (56), i.e.
$\left(\hat{A}_{R}^{[1]}(s),\hat{B}_{R}^{[1]}(s),\hat{A}_{R}^{[2]}(s),\hat{B}_{R}^{[2]}(s)\right)=\mathsf{M}^{-1}(s)\cdot\textbf{u}(s)$.
It then remains to find an explicit expression for
$\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s)$ in terms of the system
parameters. This can be achieved using the jump condition of the resetting-
phase density at the origin, obtained by integrating Eq. (49) over the region
$[-\epsilon,\epsilon]$, recalling $-\epsilon\sim L-\epsilon$, and then taking
$\epsilon\to 0^{+}$, i.e.

$D_{0}\left(\left.\frac{\partial\hat{P}_{R}^{[1]}(x,s)}{\partial
x}\right|_{x\to 0^{+}}-\left.\frac{\partial\hat{P}_{R}^{[2]}(x,s)}{\partial
x}\right|_{x\to
L^{-}}\right)=\hat{J}_{R}^{[2]}(L^{-},s)-\hat{J}_{R}^{[1]}(0^{+},s),$ (57)

which concludes the derivation of the solutions for the densities in Laplace
space.

In principle, exact expressions for the time-dependent probability densities
in real space, i.e. $P_{D,R}(x,t)$, can be obtained by taking the inverse
Laplace transform of Eqs. (48) and (51), though the resulting expressions are
too complicated to write down here explicitly. However, we can recover the
steady-state limits of the probability densities $P_{D,R}(x)$ derived in the
main text, Eqs. (14) and (19), through the final value theorem, i.e.

$\lim_{t\to\infty}P_{D,R}(x,t)=\lim_{s\to 0}sP_{D,R}(x,s).$ (58)

It turns out that $\lim_{s\to 0}s\hat{A}_{D}(s)=p_{D}/2$, where
$\hat{A}_{D}(s)$ is given by Eq. (47) and $p_{D}$ is the normalisation of the
diffusion-phase probability density, defined by Eq. (15) in the main text. For
the case when both $h/a\neq\sqrt{Dr}$ and $h/(L-a)\neq\sqrt{Dr}$, we recover
the steady-state solutions of the main text, Eqs. (14) and (19), namely (in
dimensionless form),

$\displaystyle\tilde{P}_{D}(\tilde{x})$
$\displaystyle=\frac{p_{D}}{2}\frac{\cosh(\tilde{x}-\frac{\ell}{2})}{\sinh(\frac{\ell}{2})},$
(59a) $\displaystyle\tilde{P}_{R}(\tilde{x})$
$\displaystyle=\begin{cases}\frac{p_{D}}{2}\frac{\delta\ell}{e^{\ell}-1}\left(\frac{e^{\ell-\tilde{x}}}{\lambda^{[1]}_{-}}-\frac{e^{\tilde{x}}}{\lambda^{[1]}_{+}}\right)+A_{R}^{[1]}e^{-\frac{\kappa\tilde{x}}{\delta\ell}}+B_{R}^{[1]},&0\leq\tilde{x}<\delta\ell,\\\
\frac{p_{D}}{2}\frac{(1-\delta)\ell}{e^{\ell}-1}\left(\frac{e^{\tilde{x}}}{\lambda^{[2]}_{-}}-\frac{e^{\ell-\tilde{x}}}{\lambda^{[2]}_{+}}\right)+A_{R}^{[2]}e^{\frac{\kappa\tilde{x}}{(1-\delta)\ell}}+B_{R}^{[2]},&\delta\ell\leq\tilde{x}<\ell,\end{cases}$
(59b)

For completeness, the edge-case solutions for the resetting-phase probability
density are given by

  * •

Case (i): $h/a=\sqrt{Dr}\neq h/(L-a)$,
$\left(\lambda_{-}^{[1]}=0\neq\lambda_{-}^{[2]}\right)$

$\displaystyle\tilde{P}_{R}(\tilde{x})$
$\displaystyle=\begin{cases*}\frac{p_{D}}{2}\frac{2e^{\ell-\tilde{x}}(1+\tilde{x})-e^{\tilde{x}}}{2(e^{\ell}-1)}+A_{R}^{[1]}e^{-\tilde{x}}+B_{R}^{[1]},&$0\leq\tilde{x}<\delta\ell$,\\\
\frac{p_{D}}{2}\frac{\ell(1-\delta)}{e^{\ell}-1}\left(\frac{e^{\tilde{x}}}{\lambda^{[2]}_{-}}-\frac{e^{\ell-\tilde{x}}}{\lambda^{[2]}_{+}}\right)+A_{R}^{[2]}e^{\frac{\tilde{x}\kappa}{\ell(1-\delta)}}+B_{R}^{[2]},&$\delta\ell\leq\tilde{x}<\ell$,\end{cases*}$
(61a)

  * •

Case (ii): $h/(L-a)=\sqrt{Dr}\neq h/a$,
$\left(\lambda_{-}^{[2]}=0\neq\lambda_{-}^{[1]}\right)$

$\displaystyle\tilde{P}_{R}(\tilde{x})$
$\displaystyle=\begin{cases*}\frac{p_{D}}{2}\frac{\ell\delta}{e^{\ell}-1}\left(\frac{e^{\ell-\tilde{x}}}{\lambda^{[1]}_{-}}-\frac{e^{\tilde{x}}}{\lambda^{[1]}_{+}}\right)+A_{R}^{[1]}e^{-\frac{\tilde{x}\kappa}{\ell\delta}}+B_{R}^{[1]},&$0\leq\tilde{x}<\delta\ell$,\\\
\frac{p_{D}}{2}\frac{-2e^{\tilde{x}}(1-\tilde{x})+e^{\ell-\tilde{x}}}{2(e^{\ell}-1)}+A_{R}^{[2]}e^{\tilde{x}}+B_{R}^{[2]},&$\delta\ell\leq\tilde{x}<\ell$,\end{cases*}$
(61b)

  * •

Case (iii): $h/a=h/(L-a)=\sqrt{Dr}$,
$\left(\lambda_{-}^{[1]}=\lambda_{-}^{[2]}=0\right)$

$\displaystyle\tilde{P}_{R}(\tilde{x})$
$\displaystyle=\begin{cases*}\frac{p_{D}}{2}\frac{2e^{\ell-\tilde{x}}(1+\tilde{x})-e^{\tilde{x}}}{2(e^{\ell}-1)}+A_{R}^{[1]}e^{-\tilde{x}}+B_{R}^{[1]},&$0\leq\tilde{x}<\delta\ell$,\\\
\frac{p_{D}}{2}\frac{-2e^{\tilde{x}}(1-\tilde{x})+e^{\ell-\tilde{x}}}{2(e^{\ell}-1)}+A_{R}^{[2]}e^{\tilde{x}}+B_{R}^{[2]},&$\delta\ell\leq\tilde{x}<\ell$,\end{cases*}$
(61c)

where the constants $A^{[1]}_{R},A^{[2]}_{R},B^{[1]}_{R},B^{[2]}_{R}$ differ
for each case, and can be calculated using the final value theorem, e.g.
$A^{[1]}_{R}=\lim_{s\to 0}s\hat{A}^{[1]}_{R}(s)$.

##  Appendix C Solution for a “stop-and-start” model with deterministic
returns in a static ratchet

In this appendix, we discuss how to obtain the steady-state solution for a
“stop-and-start” resetting model. In this model, while diffusing in a ratchet
potential, a particle stochastically switches off its diffusion. This
commences the “resetting phase”, which ends only upon the particle reaching
the origin, where it then switches its diffusion back on again, repeating the
cycle. Since the particle does not diffuse during the resetting phase, the
returns to the origin are deterministic. The dynamics are captured by the
following Langevin equations,

$\displaystyle\dot{x}_{D}(t)$
$\displaystyle=-V^{\prime}(x_{D}(t))+\sqrt{2D_{0}}\xi(t),$ (62a)
$\displaystyle\dot{x}_{R}(t)$ $\displaystyle=-V^{\prime}(x_{R}(t)),$ (62b)

where the ratchet potential $V(x)$ is the same as the one defined in Eq. (3).
This model is similar to that considered in Ref. [22], for which the authors
obtain numerical results. However, in that work, the ratchet potential is
continuously differentiable, whereas we will continue to consider a piecewise-
linear ratchet potential, Eq. (3).

Figure 12: Steady-state probability densities $\tilde{P}_{D,R}(\tilde{x})$ for
the “stop-and-start” model as a function of particle position $\tilde{x}$ for
parameters $\ell=4$, $\kappa=1$, and $\delta=0.2$. The vertical dotted line
indicates the position of the ratchet apex $\tilde{x}=\delta\ell$. Simulation
data are plotted as symbols and are in excellent agreement with the
theoretical results obtained by the procedure described in Appendix C.

One way to solve this model is to first consider that the diffusion and
resetting phases each have an associated finite diffusion constant, say
$D_{D}$ and $D_{R}$, respectively. After solving this more general model, one
can take the limit $D_{R}\to 0$ to recover the stop-and-start model. It turns
out that solving the model in this way avoids an issue with solving the
$D_{R}=0$ case directly, namely that the usual resetting-phase boundary
conditions $P_{R}(0)=P_{R}(L)=0$ no longer hold for $D_{R}=0$ as the
resetting-phase density has a finite value at the origin in this case [19,
23]. Solving the more general model for finite $D_{R}$ allows us to keep the
usual resetting-phase boundary conditions. However, one needs to take some
care as the limits $x\to 0$ and $D_{R}\to 0$ do not commute [30]. Hence,
taking the limit $D_{R}\to 0$ in the more general model recovers the $D_{R}=0$
result everywhere except at $x=0$, where the solution abruptly drops to $0$
due to the boundary conditions.

The steady-state Fokker-Planck equations for the model with finite $D_{R}$ can
be cast in the following dimensionless form,

$\displaystyle 0$
$\displaystyle=\frac{\partial^{2}\tilde{P}^{[1]}_{D}(\tilde{x})}{\partial\tilde{x}^{2}}+\tilde{V}^{\prime[1]}\frac{\partial\tilde{P}^{[1]}_{D}(\tilde{x})}{\partial\tilde{x}}-\tilde{P}^{[1]}_{D}(\tilde{x})-\tilde{J}_{R}^{[1]}(0^{+})\delta(\tilde{x}),$
(63a) $\displaystyle 0$
$\displaystyle=\frac{\partial^{2}\tilde{P}^{[2]}_{D}(\tilde{x})}{\partial\tilde{x}^{2}}+\tilde{V}^{\prime[2]}\frac{\partial\tilde{P}^{[2]}_{D}(\tilde{x})}{\partial\tilde{x}}-\tilde{P}^{[2]}_{D}(\tilde{x})+\tilde{J}_{R}^{[2]}(\ell^{-})\delta(\tilde{x}),$
(63b) $\displaystyle 0$
$\displaystyle=\alpha\frac{\partial^{2}\tilde{P}^{[1]}_{R}(\tilde{x})}{\partial\tilde{x}^{2}}+\tilde{V}^{\prime[1]}\frac{\partial\tilde{P}^{[1]}_{R}(\tilde{x})}{\partial\tilde{x}}+\tilde{P}_{D}^{[1]}(\tilde{x})+\tilde{J}_{R}^{[1]}(0^{+})\delta(\tilde{x}),$
(63c) $\displaystyle 0$
$\displaystyle=\alpha\frac{\partial^{2}\tilde{P}^{[2]}_{R}(\tilde{x})}{\partial\tilde{x}^{2}}+\tilde{V}^{\prime[2]}\frac{\partial\tilde{P}^{[2]}_{R}(\tilde{x})}{\partial\tilde{x}}+\tilde{P}_{D}^{[2]}(\tilde{x})-\tilde{J}_{R}^{[2]}(\ell^{-})\delta(\tilde{x}),$
(63d)

where we have introduced $\alpha=D_{R}/D_{0}$ as a new dimensionless parameter
and all other symbols retain the same definitions from the main text. One can
verify that the Boltzmann solution, i.e.
$\tilde{P}(\tilde{x})=\tilde{P}_{R}(\tilde{x})+\tilde{P}_{D}(\tilde{x})\propto\exp(-\tilde{V}(\tilde{x}))$
is recovered for $\alpha=1$. The stop-and-start model is recovered in the
limit $\alpha\to 0$.

We now describe how to obtain the solution to Eq. (63). Since the procedure is
mostly analogous to Appendix B, we will omit most of the details. As in
Appendix B, we first solve for the diffusion-phase density, and then fix the
constants of integration using the periodic boundary conditions
$\tilde{P}_{D}^{[1]}(0)=\tilde{P}_{D}^{[2]}(\ell)$, density continuity
condition $\tilde{P}_{D}^{[1]}(\delta\ell)=\tilde{P}_{D}^{[2]}(\delta\ell)$,
current continuity condition
$\tilde{J}_{D}^{[1]}(\delta\ell)=\tilde{J}_{D}^{[2]}(\delta\ell)$, and jump
condition at $\tilde{x}=0$, which can be found by integrating Eqs. (63a)-(63b)
over the region $\tilde{x}\in[-\epsilon,\epsilon]$, recalling
$-\epsilon\sim\ell-\epsilon$, and then taking $\epsilon\to 0^{+}$, resulting
in

$\left.\frac{\partial\tilde{P}_{D}^{[1]}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to
0^{+}}-\left.\frac{\partial\tilde{P}_{D}^{[2]}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to\ell^{-}}+\frac{\kappa}{\ell\delta(1-\delta)}\tilde{P}^{[1]}_{D}(0)=-\left(\tilde{J}_{R}^{[2]}(\ell^{-})-\tilde{J}_{R}^{[1]}(0^{+})\right),$
(64)

where $\tilde{J}_{R}^{[2]}(\ell^{-})-\tilde{J}_{R}^{[1]}(0^{+})$ can be fixed
using the normalisation condition $p_{D}+p_{R}=1$. The solution for
$\tilde{P}_{D}(\tilde{x})$ can then be inserted into Eqs. (63c)-(63d) to solve
for $\tilde{P}_{R}(\tilde{x})$, the procedure of which is analogous to the
above. The resulting expressions for the densities of both phases are too
complicated to write down here. However, we have plotted the densities in
figure 12, where the analytic expressions obtained by the procedure above are
shown to be in good agreement with simulations.

##  Appendix D Relation between normalisation conditions, Eqs. (20) and (21)

In this appendix, we show that Eq. (21) can be derived from Eq. (20) without
the need for an ad-hoc justification based solely on physical arguments.

We begin by multiplying both sides of Eq. (20), $p_{D}+p_{R}=1$, by $\tau$ and
then rearranging to obtain

$p_{D}=\frac{\tau}{\tau+\frac{\tau}{p_{D}}p_{R}}.$ (65)

Next, we derive an expression for $p_{R}$ in terms of the mean first-passage
time $T_{R}(x)$. First, we multiply the dimensionless Fokker-Planck equation
(9b)-(9c) by the dimensionless mean first-passage time $\tilde{T}_{R}(x)$, and
then integrate over the entire region. This results in

$\displaystyle 0$
$\displaystyle=\int_{0}^{\ell}d\tilde{x}~{}\tilde{T}_{R}(x)\left[\frac{\partial^{2}\tilde{P}_{R}(\tilde{x})}{\partial\tilde{x}^{2}}+\frac{\partial}{\partial\tilde{x}}\left(\tilde{V}^{\prime}(\tilde{x})\tilde{P}_{R}(\tilde{x})\right)+\tilde{P}_{D}(\tilde{x})-\left(\tilde{J}_{R}(\ell^{-})-\tilde{J}_{R}(0^{+})\right)\delta(\tilde{x})\right]$
(66a)
$\displaystyle=\int_{0}^{\ell}d\tilde{x}~{}\left[\frac{\partial^{2}\tilde{T}_{R}(\tilde{x})}{\partial\tilde{x}^{2}}-\tilde{V}^{\prime}(\tilde{x})\frac{\partial\tilde{T}_{R}(\tilde{x})}{\partial\tilde{x}}\right]\tilde{P}_{R}(\tilde{x})+\int_{0}^{\ell}d\tilde{x}~{}\tilde{P}_{D}(\tilde{x})\tilde{T}_{R}(\tilde{x})$
(66b)

where, in the second equality, we integrated by parts twice with the resulting
boundary terms vanishing due to the absorbing boundary conditions satisfied by
$\tilde{P}_{R}(\tilde{x})$ and $\tilde{T}_{R}(\tilde{x})$. The term
proportional to $\delta(\tilde{x})$ also vanishes because
$\tilde{T}_{R}(0)=0$. The first integral on the right-hand side of Eq. (66b)
can be simplified using the definition of the mean first-passage time given by
Eq. (69), i.e.

$\frac{\partial^{2}\tilde{T}_{R}(\tilde{x})}{\partial\tilde{x}^{2}}-\tilde{V}^{\prime}(\tilde{x})\frac{\partial\tilde{T}_{R}(\tilde{x})}{\partial\tilde{x}}=-1.$
(67)

Hence, the first integral on the right-hand side of Eq. (66b) simplifies
drastically to $-\int_{0}^{\ell}d\tilde{x}~{}\tilde{P}_{R}(\tilde{x})=-p_{R}$,
whence we obtain

$p_{R}=\int_{0}^{\ell}d\tilde{x}~{}\tilde{P}_{D}(\tilde{x})\tilde{T}_{R}(\tilde{x}).$
(68)

Substituting this expression for $p_{R}$, Eq. (68), into Eq. (65) (which is
just a rearrangement of Eq. (20)) recovers Eq. (21) in the main text, thus
concluding the derivation.

##  Appendix E Mean first-passage time in the resetting phase

In this appendix, we derive the (unconditional) mean first-passage time (MFPT)
$T_{R}(x_{0})$ of the particle in the resetting phase, i.e. the average time
taken for a particle initialised at position $x_{0}\in[0,L]$ in the resetting
phase to exit the resetting phase by reaching either $x=0$ or $x=L$. To be
consistent with the main text, we define our units of length and time as
$\sigma=\sqrt{D_{0}/r}$ and $\tau=1/r$, respectively, despite the MFPT being
independent of $r$. While the solution we derive below depends on the
dimensionless parameters $\kappa$, $\delta$ and $\ell$, the latter of which
depends on $r$, the dependence on $r$ vanishes after converting back to
dimensionful parameters, as should be expected.

Figure 13: Mean first-passage time $\tilde{T}_{R}(\tilde{x})$ to exit the
resetting phase, Eq. (72), as a function of $\tilde{x}$ for $\ell=4$,
$\kappa=4$, and $\delta=0.2$ (blue) or $\delta=0.5$ (orange). Simulation
results are given by markers and are obtained by averaging over $10^{6}$
realizations for each initial position $\tilde{x}/\ell=0,0.05,\dots,1$.

In dimensionless form, the MFPT obeys the following partial differential
equation [69],

$-1=\frac{\partial^{2}\tilde{T}_{R}(\tilde{x})}{\partial\tilde{x}^{2}}-\tilde{V}^{\prime}(\tilde{x})\frac{\partial\tilde{T}_{R}(\tilde{x})}{\partial\tilde{x}},$
(69)

where the advective term has the opposite sign compared to that of the
probability densities (see, for example, Eqs. (9b)-(9c)), due to the MFPT
obeying the backwards Kolmogorov equation. By definition, we have the
following absorbing boundary conditions

$\tilde{T}_{R}^{[1]}(0)=\tilde{T}_{R}^{[2]}(\ell)=0,$ (70)

which impose that a particle initialised at either $\tilde{x}=0$ or
$\tilde{x}=\ell$ is absorbed instantly and thus takes zero time to exit the
resetting phase. The MFPT is also subject to the following continuity
conditions at the apex $\tilde{x}=\delta\ell$ [50],

$\displaystyle\tilde{T}_{R}^{[1]}(\delta\ell)$
$\displaystyle=\tilde{T}_{R}^{[2]}(\delta\ell),$ (71a)
$\displaystyle\left.\frac{\partial\tilde{T}_{R}^{[1]}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to\delta\ell^{-}}$
$\displaystyle=\left.\frac{\partial\tilde{T}_{R}^{[2]}(\tilde{x})}{\partial\tilde{x}}\right|_{\tilde{x}\to\delta\ell^{+}}.$
(71b)

In a similar manner to that of the probability densities in Appendix B, we can
solve Eq. (69), whence we obtain

$\tilde{T}_{R}(\tilde{x})=\begin{cases*}\frac{\ell\delta}{\kappa^{2}(e^{\kappa}-1)}\left(\ell(1-\delta)e^{-\kappa}(e^{\frac{\kappa}{\ell\delta}\tilde{x}}-1)+\kappa\tilde{x}(e^{\kappa}-1)-\ell(e^{\frac{\kappa}{\ell\delta}\tilde{x}}-1)(1-\kappa+\delta(2\kappa-1))\right),&$0\leq\tilde{x}<\delta\ell$,\\\
\frac{\ell(1-\delta)}{\kappa^{2}(e^{\kappa}-1)}\left(\ell\delta\left[e^{-\kappa\frac{\tilde{x}/\ell-\delta}{1-\delta}}-e^{-\kappa}\right]+e^{\kappa}\kappa(\ell-\tilde{x})-\ell
e^{\kappa-\frac{\kappa}{1-\delta}(\tilde{x}/\ell-\delta)}(\delta+\kappa-2\delta\kappa)+(\tilde{x}\kappa+\ell\delta(1-2\kappa))\right),&$\delta\ell\leq
x<\ell$.\end{cases*}$ (72)

The analytical expression for the MFPT, Eq. (72), is plotted in figure 13 and
shows good agreement with simulation results.

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