  *

# Markovian description of a wide class of feedback-controlled systems:
Application to the feedback flashing ratchet

Natalia Ruiz-Pino and Antonio Prados  Física Teórica, Universidad de Sevilla,
Apartado de Correos 1065, E-41080 Sevilla, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In feedback-controlled systems, an external agent—the feedback
controller—measures the state of the system and modifies its subsequent
dynamics depending on the outcome of the measurement. In this paper, we build
a Markovian description for the joint stochastic process that comprises both
the system and the controller variables. This Markovian description is valid
for a wide class of feedback-controlled systems, allowing for the inclusion of
errors in the measurement. The general framework is motivated and illustrated
with the paradigmatic example of the feedback flashing ratchet.

##  1 Introduction

The dynamics of a physical system typically depends on the system variables
plus some additional quantities, the time dependence of which can often be
externally controlled. For instance, the evolution of the position of an
overdamped Brownian particle in an optical harmonic trap depends both on the
stiffness of the trap and the temperature of the bath. The stiffness of the
trap can be tuned by varying the intensity of the laser beam, and the
temperature of the bath can be effectively increased by using charged
particles and introducing an additional random electric field—both procedures
are now standard in the manipulation of colloidal particles [1, 2].

By engineering the time evolution of these additional quantities—the so-called
control functions, one usually aims at controlling the time evolution of the
system. The simplest possibility is to impose a certain time protocol for
these control functions, independently of the instantaneous state of the
system: this is what is known as open-loop control. For example, this is the
case of the shortcuts to adiabaticity in quantum systems [3] and the analogous
swift state-to-state transformations in classical and stochastic systems [4],
in which open-loop control is employed to accelerate the system dynamics.

A more sophisticated approach is that of feedback control, in which the time
protocol of the control function depends on the state of the system [5, 6, 7,
8]. Ideally, the feedback controller measures without any error the
instantaneous state of the system, without any time delay, and thus acts on
the system dynamics without any uncertainty. Feedback control allows for
improving the performance of the system, e.g. increasing the flux in Brownian
ratchets [9, 10, 11]. Still, this improvement of the performance has a
thermodynamic cost, since the information gathered by controlled when
measuring the system state has to be incorporated to the thermodynamic balance
of the system [12, 13, 14, 6, 7, 15, 16, 17].

The action of the feedback controller makes the system dynamics non-Markovian,
since the instantaneous values of the system variables do not suffice to
univocally determine its subsequent evolution: this evolution also depends on
the values of the variables characterising the state of the controller, which
in turn depend on the value of the system variables at the last time that the
control measured the system state. Note that this is so even for feedback
controllers without any time delay, which are sometimes called Markovian
feedback controllers. The non-Markovianity of the system dynamics complicates
its mathematical description. For example, the full chain of control actions
must be taken into account in the thermodynamic balance [17].

Despite the above, in this paper we build a Markovian description for a wide
class of feedback controlled systems, by considering the joint process of the
system and controller variables. The spirit of our approach is thus similar to
recent works on feedback-controlled quantum systems [8]. For the sake of
simplicity, we present the framework for the one-dimensional case, but the
extension to $d$ dimensions is straightforward. We consider an overdamped
particle with position $x$ moving in a fluctuating potential $V(x,c)$, the
shape of which depends on an additional variable $c$. The feedback controller
measures the instantaneous state of the system just before certain given times
$t_{k}$, $x_{k}\equiv x(t_{k}^{-})$, $k=1,2,\ldots$, and the controller
variable changes to the value $c_{k}$ following a certain probability
distribution $\Theta(c_{k}|x_{k})$—the flashing protocol. Then, the potential
acting on the particle in the time interval between the $k$-th and $(k+1)$-th
measurements, $I_{k}=(t_{k},t_{k+1})$, is given by $V(x,c_{k})$. The above
framework entails that there is no time delay, since the potential
$V(x,c_{k})$ is already felt by the particle for $t=t_{k}^{+}$. Still, our
framework allows for the introduction of error, or uncertainty, in the
measurement—a possibility that has been considered in the literature [10, 18].

Our work is motivated by, and later applied to, a feedback-controlled flashing
ratchet. Brownian ratchets rectify Brownian motion by creating flux in a given
direction, by breaking the left-right symmetry [19, 20, 21, 22]. In addition
to their theoretical importance, they are also relevant in biosystems, as
models for molecular motors [23, 24, 25, 26, 27, 28]. In the open-loop (no
feedback) case, the breaking of spatial symmetry can be brought about by the
consideration of an asymmetric potential, which is periodically switched on
and off [29, 30, 31, 32]. Instead of switching on and off the potential, the
breaking of spatial symmetry can also be brought about by the variation of the
amplitude and period of a force acting on the particle, as in the
deterministic ratchet of Ref. [33]. Several studies have been carried out to
elucidate how the degree of symmetry breaking impacts on the optimisation of
transport in both deterministic [34] and Brownian [35] ratchets.

In general, closed-loop or feedback control makes it possible to increase the
performance of ratchets and other stochastic systems [36, 37, 9, 10, 17, 14,
38]. Therein, spatial symmetry can be broken even for symmetric potentials,
the feedback mechanism breaks it by switching on or off the potential
depending on the sign of the force at the particle’s measured position. In the
specific example of the feedback flashing ratchet that we employ to illustrate
the general framework developed this work, we consider a symmetric V-shaped
potential and a flashing protocol $\Theta(c|x)$ that incorporates error in the
measurement in a realistic way. Yet, analytical predictions can be obtained
for the time evolution of the system, which compare excellently with numerical
simulations of the dynamics.

Our paper is organised as follows. Section 2 motivates our work by considering
the feedback flashing ratchet, neatly showing that $X$ is not Markovian but
the joint process $(X,C)$ is Markovian. The general framework is presented in
Sec. 3, in which a differential Chapman-Kolmogorov equation for the joint
process is derived. In Sec. 4, we prove an $H$-theorem for the joint process
$(X,C)$, which entails the emergence of a long-time regime independent of the
initial conditions. Section 5 considers a coarse-grained timescale, in which
the position $X$ becomes Markovian. Therein, we also consider some relevant
limit cases and discuss the connection between the fine and coarse timescales.
The general framework developed is illustrated in Sec. 6 with the particular
case of the feedback flashing ratchet. Numerical simulations and theoretical
results are compared, finding an excellent agreement between them. The main
results of the paper are discussed in Sec. 7, together with perspectives for
future work. The Appendices deal with some technical details that are omitted
in the main text.

##  2 Motivation: feedback flashing ratchet

Our work is motivated by the so-called feedback flashing ratchet [9, 39, 40,
11, 41, 42, 17]. This system comprises an overdamped Brownian particle
immersed in a fluid at equilibrium with temperature $T$. The friction
coefficient is $\gamma$, which is related to the diffusion coefficient $D$ by
the Einstein fluctuation-dissipation relation $D=k_{B}T/\gamma$. The Brownian
particle moves on a line under the action of a conservative force, deriving
from a periodic potential $V(x)$, and a non-conservative force. The motion of
the particle can be studied either in the whole real line $-\infty<x<+\infty$,
or in a finite interval $-L\leq x\leq L$—in the latter case with periodic
boundary conditions, note that the period of the potential is $2L$, see A.

The potential is switched on or off by an external feedback controller, which
measures the instantaneous position of the particle at regular time intervals
$\Delta t_{m}$ and makes the switching depending thereon. More specifically,
the feedback controller measures the position of the particle $x_{k}^{-}\equiv
x(t_{k}^{-})$ just before $t_{k}=k\Delta t_{m}$, $k=1,2,\ldots$. These
measurements naturally divide the timeline into intervals
$I_{k}=(t_{k},t_{k+1})$ of length $\Delta t_{m}$, so that $t_{k}^{-}\in
I_{k-1}$. After the measurement, the potential is on (off) in the next time
interval $I_{k}$ is the force $-V^{\prime}(x_{k}^{-})>0$
($-V^{\prime}(x_{k}^{-})<0$). This decision on the switching creates the
ratchet effect, i.e. a directed movement to the right[20, 21, 19]. This may be
used to exert work against the external non-conservative force, which
typically opposes the average direction of motion.

The state of the controller can be characterised by a dichotomic variable $C$,
with possible values $c=(0,1)$, such that the potential is on (off) for $c=1$
($c=0$). In other words, the potential felt by the particle is
$V(x,c)=c\,V(x)$. We may define two sets $\mathcal{X}_{1}$ and
$\mathcal{X}_{0}$, such that:

$\displaystyle x\in\mathcal{X}_{1}\text{ if }-V^{\prime}(x)>0,\quad
x\in\mathcal{X}_{0}\text{ if }-V^{\prime}(x)<0.$ (1)

It is clear that $\mathcal{X}_{0}\cup\mathcal{X}_{1}=[x_{L},x_{R}]$, where
$[x_{L},x_{R}]$ is the complete interval in which the movement is studied. The
specific case of a V-shaped potential, which we later use in Sec. 6 for the
numerical check of our theoretical framework, is illustrated on the right
panel of Fig. 1.

Figure 1: Intervals $\mathcal{X}_{0}$ and $\mathcal{X}_{1}$ for the flashing
protocol for different potentials. The interval $\mathcal{X}_{1}$ corresponds
to those positions for which the force is positive, $-\partial_{x}V>0$, and
$\mathcal{X}_{0}$ for those for which the force is negative,
$-\partial_{x}V<0$. The external controller measures the particle position and
the potential is switched on (off) afterwards, i.e. the potential felt by the
particle is $c\,V(x)$, with $c=1$ ($c=0$), if the position of the particle
belongs to $\mathcal{X}_{1}$ ($\mathcal{X}_{0}$).

The above picture entails that, in the time interval $I_{k}$ between control
updates, the Brownian particle moves under the potential
$V(x,c_{k})=c_{k}V(x)$, where $x(t_{k}^{-})\in\mathcal{X}_{c_{k}}$. Therefore,
it obeys the Langevin equation

$\gamma\dot{x}=-c_{k}\,V^{\prime}(x)-F_{\text{ext}}+\xi(t),\quad t\in I_{k},$
(2)

where $\xi(t)$ is the Gaussian white noise:

$\langle\xi(t)\rangle=0,\quad\langle\xi(t)\xi(t^{\prime})\rangle=2k_{B}T\gamma\,\delta(t-t^{\prime}),$
(3)

with $k_{B}$ being the Boltzmann constant. We recall that the non-conservative
force faces the opposite direction of the average motion of the particle to
the right. This is the reason why we write $-F_{\text{ext}}$ in Eq. (2), with
$F_{\text{ext}}>0$.

Despite obeying the Langevin equation (2) between control updates, the
stochastic process $X$ is not Markovian: its time evolution has memory of
earlier times through the value of the control variable $c_{k}$, which depends
on $x(t_{k}^{-})$. The value of $x(t_{k}^{-})$ depends in turn on the previous
value of the control $c_{k-1}$, and so on. Therefore, in order to analyse the
thermodynamic balance of the system, the whole chain of control actions has
been typically considered—which complicates the description of the system
[17].

Here, we argue a fact that seems to have been overlooked in previous analysis
of the feedback flashing ratchet: although the process $X$ alone is not
Markovian, the joint process $(X,C)$ is Markovian. Moreover, this Markovian
description for the joint process is valid for a wide class of feedback
controlled systems, not only for error-free measurements but also for systems
where the action of the controller allows for errors or has some degree of
randomness. This is the reason why, in the following section, we develop a
general theoretical framework, which is afterwards applied to the feedback
flashing ratchet.

##  3 General framework

In this section, we put forward our general formalism for describing a wide
class of feedback-controlled systems. This formalism includes, as a particular
case, the feedback flashing ratchet with error-free measurements described in
the previous section, but also allows for the inclusion of errors in the
measurement [10, 18]. In addition, more complex situations in which the
variable characterising the controller is not dichotomic fit into the general
framework developed below, i.e. we can incorporate feedback to situations in
which the potential fluctuates between more than two states, as in Refs. [43,
44, 45].

Let us consider an overdamped Brownian particle, immersed in a fluid at
equilibrium with temperature $T$. The friction coefficient is $\gamma$, which
is related to the diffusion coefficient $D$ by the Einstein fluctuation-
dissipation relation $D=k_{B}T/\gamma$. The motion of the particle takes place
in a certain interval $[x_{L},x_{R}]$—which may be the whole line,
$x_{L}=-\infty$, $x_{R}=+\infty$, or a finite interval of size $2L$, e.g.
$x_{L}=-L$, $x_{R}=L$.

The particle moves under the action of a conservative force, deriving from a
fluctuating potential, and a non-conservative force. The fluctuation of the
potential is accounted for by a variable $c$, $V(x,c)\neq V(x,c^{\prime})$ if
$c\neq c^{\prime}$. This fluctuation is controlled by an external feedback
mechanism, which uses information about the position of the particle $x$ to
decide the value of $c$—which thus characterises the state of the controller.
When the controller variable equals $c$, the total force acting on the
particle is

$-\partial_{x}\tilde{V}(x,c)=-\partial_{x}V(x,c)-F_{\text{ext}}.$ (4)

Here, we assume that the value of the control variable $c$ is updated at
regular times $t_{k}=k\Delta t_{m}$, $k=0,1,2,\ldots$. This naturally divides
the timeline into intervals $I_{k}=(t_{k},t_{k+1})$, over which the control
variable $c$ takes a constant value $c_{k}=c(t_{k}^{+})$. Over each trajectory
of the stochastic dynamics, the controller measures the position of the
particle just before the update, $x_{k}^{-}\equiv x(t_{k}^{-})$, and the
potential takes the shape $V(x,c_{k})$ with a certain probability
$\Theta(c_{k}|x_{k}^{-})$ throughout the next time interval $I_{k}$.
Normalisation entails that

$\sum_{c}\Theta(c|x)=1,\quad\forall x.$ (5)

In the typical example of the feedback flashing ratchet, described in the
previous section, the potential is switched on or off without any error [9,
12, 17] , depending on the sign of the conservative force—see Eq. (1). In that
case, we have

$\displaystyle\Theta(c|x)=\begin{cases}1,&\text{if }x\in\mathcal{X}_{c},\\\
0,&\text{if }x\not\in\mathcal{X}_{c}.\end{cases}$ (6)

This expression also applies to any error-free feedback mechanism, which
measures the position of the particle $x_{k}^{-}$ and always chooses one of
the potentials $V(x,c)$ during the subsequent time interval $I_{k}$, depending
on the interval $\mathcal{X}_{c}$ to which $x_{k}^{-}$ belongs.

More realistically, the particle position is measured with a certain
uncertainty $\Delta x$, i.e. we know that it lies on a segment $(x-\Delta
x,x+\Delta x)$. With this information, the controller chooses the potential
$V(x,c)$ with a probability proportional to the length of the intersection
between $(x-\Delta x,x+\Delta x)$ and $\mathcal{X}_{c}$:

$\displaystyle\Theta(c|x)=\frac{1}{2\Delta
x}\text{length\;}\\{\mathcal{X}_{c}\cap(x-\Delta x,x+\Delta x)\\}$ (7)

For $\Delta x=0$, Eq. (7) reduces to the error-free case in Eq. (6). On the
other hand, for the periodic case in which the particle moves in a finite
interval $[-L,L]$, the case $\Delta x=L$ corresponds to an open-loop protocol:
the control switches the potential with a probability proportional to the
length of the interval $\mathcal{X}_{c}$, independently of the particle
position. This choice for $\Theta(c|x)$ is illustrated in Fig. 2, for the case
of the simple V-shaped potential depicted in Fig. 1.

Figure 2: Feedback control protocols for different values of the uncertainty
in the measurement. When $\Delta x=0$, the feedback control protocol
corresponds to an error-free one (blue line). As the uncertainty $\Delta x$
increases, the protocol continuously converges to the open-loop protocol,
which is attained for $\Delta x=L$ (green line). At this point, the
probability of switching on the potential becomes independent of the position
and equals the probability of switching it off, i.e. it is equal to $1/2$.

We would like to remark that the general framework developed in the next
section is valid for an arbitrary choice of $\Theta(c|x)$, as long as Eq. (5)
holds. This includes non-dichotomic feedback controllers, whose variable $c$
takes several values corresponding to different states of the potential. It
also allows for open-loop control, which occurs when $\Theta(c|x)$ is
independent of $x$, i.e. $\Theta(c|x)=\Theta(c)$.

###  3.1 Markovian description for the joint position-control stochastic
process

Now we write down an evolution equation for the joint probability distribution
$P(x,c,t)$. In the time intervals $I_{k}$, i.e. between control updates, the
control is “static” and thus $P(x,c,t)$ obeys the Fokker-Planck equation with
potential $V(x,c)$, i.e.

$\partial_{t}P(x,c,t)=-\partial_{x}J(x,c,t),\quad t\in I_{k},$ (8)

where the probability flux is defined from

$\displaystyle-\gamma J(x,c,t)\equiv$
$\displaystyle\partial_{x}\tilde{V}(x,c)\,P(x,c,t)+k_{B}T\,\partial_{x}P(x,c,t).$
(9)

On top of the dynamics over the time interval $I_{k}$, we have the update of
the control values at times $t_{k}$. After measuring the position of the
particle $x_{k}^{-}$ at time $t_{k}^{-}$, we set the value $c$ with
probability $\Theta(c|x_{k}^{-})$. We build $P(x,c,t_{k}^{+})$ in terms of
$P(x,c,t_{k}^{-})$ by thinking of the joint probability $P(x,c,t)$ as the
density of walkers at $(x,t)$ with the control variable equal to $c$. Since
the decision on the value of $c$ at $t_{k}^{+}$ only depends on the value of
the position at $t_{k}^{-}$, we have that

$\displaystyle P(x,c,t_{k}^{+})=P(x,t_{k}^{-})\Theta(c|x),$ (10)

where

$P(x,t)=\sum_{c}P(x,c,t)$ (11)

is the marginal distribution of the particle position. Note that
$P(x,c,t_{k}^{+})$ is normalised as a consequence of $P(x,t_{k}^{-})$ and
$\Theta(c|x)$ being normalised.

The meaning of Eq. (10) is clear: at position $x$ and time $t_{k}^{+}$, the
number of walkers moving under the potential $V(x,c)$ equals the number of
walkers at position $x$ at time $t_{k}^{-}$ multiplied by the probability
$\Theta(c|x)$. Note that the marginal distribution $P(x,t)$ is continuous at
time $t_{k}$,

$P(x,t_{k}^{+})=P(x,t_{k}^{-})\sum_{c}\Theta(c|x)=P(x,t_{k}^{-}),$ (12)

which is logical: the density of walkers at $x$ is unaffected by the update of
the control value. Henceforth, we can thus write $P(x,t_{k})$ without any
ambiguity. The conditional probability of the control values just after the
control update is precisely $\Theta(c|x)$,

$P(c|x,t_{k}^{+})=\frac{P(x,c,t_{k}^{+})}{P(x,t_{k}^{+})}=\Theta(c|x).$ (13)

Thus, for the error-free measurement (6), the value of the control at
$t_{k}^{+}$ is enslaved to the value of the position.

At the control updates, between $t=t_{k}^{-}$ and $t=t_{k}^{+}$, the joint
probability distribution evolves according to

$\displaystyle
P(x,c,t_{k}^{+})-P(x,c,t_{k}^{-})=\Theta(c|x)\sum_{c^{\prime}\neq
c}P(x,c^{\prime},t_{k}^{-})-\left[1-\Theta(c|x)\right]P(x,c,t_{k}^{-}),$ (14)

where we have made use of Eqs. (10) and (11). The first term on the rhs is the
typical gain term, due to transitions from $c^{\prime}$ to $c$, weighted with
probability $\Theta(c|x)$, whereas the second term is the tipical loss term,
weighted with probability $1-\Theta(c|x)$.

Therefore, we can write the following evolution equation for the joint
probability $P(x,c,t)$:

$\displaystyle\partial_{t}P(x,c,t)=$ $\displaystyle-\partial_{x}J(x,c,t)$
$\displaystyle+\sum_{k}\delta(t-t_{k}^{-})\Big{[}\Theta(c|x)\sum_{c^{\prime}\neq
c}P(x,c^{\prime},t^{\prime})-\left[1-\Theta(c|x)\right]P(x,c,t)\Big{]}.$ (15)

This equation has the form of the differential Chapman-Kolmogorov equation
(DCKE)—see section 3.4 of Ref. [46]—and it is thus the mathematical expression
of the fact that $(X,C)$ is a Markov process. On the one hand, the first term
on the rhs corresponds to the Fokker-Planck evolution between control updates.
On the other hand, the second term on the rhs is a master equation term with
time-dependent transition rates, which are time-periodic with period $\Delta
t_{m}$. Therefore, one expects $P(x,c,t)$ to tend to a time-periodic state
with the same period in the long-time limit, when initial conditions are
forgotten. This physical expectation is confirmed in Sec. 4 by our deriving an
$H$-theorem for Eq. (15).

Equation (15) must be complemented with appropriate boundary conditions. If
the movement takes place over the whole real line, we have that
$\lim_{x\to\pm\infty}P(x,c,t)=0$. If the movement takes place over a finite
interval $[-L,L]$ with periodic boundary conditions, we have

$\displaystyle P(-L,c,t)$ $\displaystyle=P(L,c,t),$ $\displaystyle J(-L,c,t)$
$\displaystyle=J(L,c,t).$ (16)

The case with periodic boundary condition is especially relevant for the
feedback flashing ratchet, in which the potential $V(x,c)$ is spatially
periodic. See A for a brief discussion on the relation between the problems
over the whole real line and the problem with periodic boundary conditions.

The evolution equation for $P(x,c,t)$ can be compactly written in operator
form

$\displaystyle\partial_{t}P=$
$\displaystyle\hat{\mathcal{L}}^{(c)}P+\sum_{k}\delta(t-t_{k})\hat{W}P,$ (17)

where $\hat{\mathcal{L}}^{(c)}$ is the Fokker-Planck operator for the
potential $V(x,c)$,

$\displaystyle\hat{\mathcal{L}}^{(c)}P\equiv-\partial_{x}J(x,c,t),$ (18)

and $\hat{W}$ is the master equation operator for the control updates

$\hat{W}P\equiv\sum_{k}\Big{[}\hat{\Theta}_{c}\sum_{c^{\prime}\neq
c}P(x,c^{\prime},t)-(I-\hat{\Theta}_{c})P(x,c,t)\Big{]},$ (19)

where we have defined linear operators $\hat{\Theta}_{c}$,

$\displaystyle\hat{\Theta}_{c}f(x)\equiv\Theta(c|x)\,f(x),\quad\forall f(x),$
(20a) $\displaystyle\sum_{c}\hat{\Theta}_{c}=I.$ (20b)

In terms of them, Eq. (10) can be written as

$P(x,c,t_{k}^{+})=\hat{\Theta}_{c}P(x,t_{k}).$ (21)

For error-free measurements, it is clear that the operators $\hat{\Theta}_{c}$
are projectors, verifying $\hat{\Theta}_{c}^{2}=\hat{\Theta}_{c}$. Their
action on a function $f(x)$ is to restrict it to the interval
$\mathcal{X}_{c}$, cutting its remainder.

###  3.2 Formal solution of the differential Chapman-Kolmogorov equation (15)

Our general framework allows us to write a formal solution of the differential
Chapman-Kolmogorov equation (15), or Eq. (17) in operator form. In the
intervals $I_{k}$, the joint distribution $P(x,c,t)$ evolves following the
Fokker-Planck equation with potential $V(x,c)$: denoting the evolution
operator for such Fokker-Planck equation by
$\hat{U}^{(c)}(t)\equiv\exp(t\hat{\mathcal{L}}^{(c)})$,

$P(x,c,t)=\hat{U}^{(c)}(t-t_{k})P(x,c,t_{k}^{+}),\quad t\in I_{k}.$ (22)

More specifically, in terms of the Green function $K(x,\Delta t|x^{\prime},c)$
of the Fokker-Planck operator $\hat{\mathcal{L}}^{(c)}$:

$\displaystyle P(x,c,t)=\int
dx^{\prime}K(x,t-t_{k}^{+}|x^{\prime},c)\,P(x^{\prime},c,t_{k}^{+}),\quad t\in
I_{k}.$ (23)

In particular, at the time $t_{k}^{-}$ just before the measurement,

$\displaystyle P(x,c,t_{k}^{-})=\int dx^{\prime}K(x,\Delta
t_{m}|x^{\prime},c)\,P(x^{\prime},c,t_{k-1}^{+}).$ (24)

At the measurement times, we resort to Eq. (10),

$\displaystyle P(x,c,t_{k}^{+})=\Theta(c|x)\sum_{c^{\prime}}\int
dx^{\prime}\;K(x,\Delta
t_{m}|x^{\prime},c^{\prime})P(x^{\prime},c^{\prime},t_{k-1}^{+}).$ (25)

The Green function can be written as an expansion in the eigenfunctions of the
Fokker-Planck operator $\hat{\mathcal{L}}^{(c)}$. We define the eigenvalues
and eigenfunctions of $\hat{\mathcal{L}}^{(c)}$:

$\hat{\mathcal{L}}^{(c)}P_{\text{s}}^{(c)}=0,\qquad\hat{\mathcal{L}}^{(c)}\phi_{n}^{(c)}=-\lambda_{n}^{(c)}\phi_{n}^{(c)},\quad\lambda_{n}^{(c)}>0.$
(26)

Therefore, $P_{\text{s}}^{(c)}(x)$ is the stationary distribution of the
particle moving under the action of the potential $V(x,c)$. For the case of
the feedback flashing ratchet, this distribution is computed in B. We recall
that the Fokker-Planck operator $\hat{\mathcal{L}}^{(c)}$ is Hermitian when
the scalar product includes $1/P_{\text{s}}^{(c)}$ as a weight function [46,
47], which makes it possible to write the Green function as

$K(x,t|x^{\prime},c)=P_{\text{s}}^{(c)}(x)+\frac{1}{P_{\text{s}}^{(c)}(x^{\prime})}\sum_{n=1}\phi^{(c)}_{n}(x^{\prime})\phi^{(c)}_{n}(x)e^{-\lambda^{(c)}_{n}t}.$
(27)

This completes the formal solution of the problem: concatenating Eqs. (23) and
(25), with the Green function being given by Eq. (27), $P(x,c,t)$ for any $t$
is written in terms of the initial condition $P(x,c,t=0)$.

##  4 $H$-theorem and long-time regime

In this section, we derive an H-theorem that guarantees the convergence of the
system to a unique long-time state. We start by considering two solutions
$P(x,c,t)$ and $\tilde{P}(x,c,t)$ of the DCKE (15) and defining the functional

$H(t)=\sum_{c}\int_{-\infty}^{+\infty}dxP(x,c,t)\ln\frac{P(x,c,t)}{\tilde{P}(x,c,t)}.$
(28)

Note that this functional is the Kullback-Leibler divergence or the relative
entropy. In the following, we prove that $H$ is a Lyapunov functional for the
DCKE (15), which implies that all its solutions tend to a unique asymptotic
behaviour for long enough times, i.e.

$\lim_{t\to\infty}\frac{P(x,c,t)}{\tilde{P}(x,c,t)}=1.$ (29)

For the sake of concreteness, the proof is presented for periodic boundary
conditions, as given by Eq. (16).

###  4.1 $H$ is a decreasing function of time

Let us prove the $H$-theorem in two steps. First, we explore the behaviour of
$H$ in the time interval $I_{k}$ between control updates. Second, we
investigate how $H$ changes at the measurement times, i.e.
$H(t_{k}^{+})-H(t_{k}^{-})$.

####  4.1.1 During $I_{k}$.-

Taking the time derivative of $H$, we have

$\displaystyle\dot{H}(t)=$ $\displaystyle\sum_{c}\int
dx\,\partial_{t}P(x,c,t)\ln\frac{P(x,c,t)}{\tilde{P}(x,c,t)}-\sum_{c}\int
dx\,\frac{P(x,c,t)}{\tilde{P}(x,c,t)}\partial_{t}\tilde{P}(x,c,t)$
$\displaystyle=$ $\displaystyle-\sum_{c}\int
dx\,\partial_{x}J(x,c,t)\ln\frac{P(x,c,t)}{\tilde{P}(x,c,t)}+\sum_{c}\int
dx\,\frac{P(x,c,t)}{\tilde{P}(x,c,t)}\partial_{x}\tilde{J}(x,c,t)$ (30)

Integrating by parts and bringing to bear the periodic boundary conditions
(16), we get

$\displaystyle\dot{H}(t)=$ $\displaystyle\sum_{c}\int
dx\,P(x,c,t)\left[\frac{J(x,c,t)}{P(x,c,t)}-\frac{\tilde{J}(x,c,t)}{\tilde{P}(x,c,t)}\right]\partial_{x}\ln\frac{P(x,c,t)}{\tilde{P}(x,c,t)}$
(31)

Equation (9) is equivalent to

$\partial_{x}\ln
P(x,c,t)=-\beta\left(\gamma\frac{J(x,c,t)}{P(x,c,t)}+cV^{\prime}(x)+F_{\text{ext}}\right),$
(32)

and an analogous equation for $\partial_{x}\ln\tilde{P}(x,c,t)$ and
$\tilde{J}(x,c,t)$. Therefore, we have

$\frac{J(x,c,t)}{P(x,c,t)}-\frac{\tilde{J}(x,c,t)}{\tilde{P}(x,c,t)}=-(\beta\gamma)^{-1}\partial_{x}\ln\frac{P(x,c,t)}{\tilde{P}(x,c,t)}.$
(33)

Inserting this equation into Eq. (31), we get

$\displaystyle\dot{H}(t)$ $\displaystyle=-(\beta\gamma)^{-1}\sum_{c}\int
dx\,P(x,c,t)\left(\partial_{x}\ln\frac{P(x,c,t)}{\tilde{P}(x,c,t)}\right)^{2}.$
(34)

By defining

$\sigma_{c}(t)\equiv(\beta\gamma)^{-1}\int
dx\,P(x,c,t)\left(\partial_{x}\ln\frac{P(x,c,t)}{\tilde{P}(x,c,t)}\right)^{2}\geq
0,$ (35)

we rewrite Eq. (34) as

$\dot{H}(t)=-\sum_{c}\sigma_{c}(t)\leq 0.$ (36)

Since $\sigma_{c}(t)\geq 0$, $\forall c$, it is clear that $\dot{H}=0$ implies
that $\sigma_{c}(t)$ must vanish, $\forall c$, and this in turn implies that
$P(x,c,t)=k(c,t)\tilde{P}(x,c,t)$. Integrating over $x$,
$P(c,t)=k(c,t)\tilde{P}(c,t)$, which is valid between control updates, i.e.
for $t\in I_{k}$. But neither $P(c,t)$ nor $\tilde{P}(c,t)$ depends on time
between control updates, which entails that $k(c,t)$ only depends on $c$
throughout $I_{k}$. Therefore, we conclude that

$\dot{H}=0\text{ for }t\in I_{k}\iff P(x,c,t)=k(c)\tilde{P}(x,c,t).$ (37)

####  4.1.2 At the control updates.-

Let us compute the change of the $H$ functional at the control updates, i.e.
$H(t_{k}^{+})-H(t_{k}^{-})$. On the one hand,

$\displaystyle H(t_{k}^{+})$ $\displaystyle=\sum_{c}\int
dx\,P(x,c,t_{k}^{+})\ln\frac{P(x,c,t_{k}^{+})}{\tilde{P}(x,c,t_{k}^{+})}=\int
dx\,P(x,t_{k}^{-})\ln\frac{P(x,t_{k}^{-})}{\tilde{P}(x,t_{k}^{-})}$

where we have brought to bear Eq. (10). On the other hand,

$\displaystyle H(t_{k}^{-})$ $\displaystyle=\sum_{c}\int
dxP(x,c,t_{k}^{-})\ln\frac{P(x,c,t_{k}^{-})}{\tilde{P}(x,c,t_{k}^{-})}.$ (38)

Repeated use of Bayes rule leads to

$\displaystyle\delta_{k}\equiv H($ $\displaystyle
t_{k}^{+})-H(t_{k}^{-})=-\int
dx\,P(x,t_{k}^{-})\sum_{c}P(c|x,t_{k}^{-})\ln\frac{P(c|x,t_{k}^{-})}{\tilde{P}(c|x,t_{k}^{-})}\leq
0.$ (39)

Moreover, we have 111Note that $\sum p_{1}\ln\frac{p_{1}}{p_{2}}=\sum
p_{2}\left(\frac{p_{1}}{p_{2}}\ln\frac{p_{1}}{p_{2}}+1-\frac{p_{1}}{p_{2}}\right)$
and $x\ln x+1-x>0$ and it equals $0$ if and only if $x=1$. Therefore, $\sum
p_{1}\ln\frac{p_{1}}{p_{2}}\geq 0$, and $\sum
p_{1}\ln\frac{p_{1}}{p_{2}}=0\iff p_{1}=p_{2}$.

$\delta_{k}=0\iff P(c|x,t_{k}^{-})=\tilde{P}(c|x,t_{k}^{-}),\;\forall(x,c).$
(40)

The above discussion entails that $H$ is discontinuous at the control updates,
for the wide class of feedback controlled systems we are considering. Yet, it
must be noted that the discontinuity of $H$ persists for open-loop
controllers, in which the control does not use information about the system
state, since in general $P(c|x,t_{k}^{-})\neq\tilde{P}(c|x,t_{k}^{-})$.

###  4.2 Long-time limit and tendency towards a universal behaviour

In the previous sections, we have shown that $H$ is non-increasing both during
the time intervals $I_{k}$ and the time instants $t_{k}$ at which the control
is updated. Here we prove that, since $H$ is bounded from below, this means
that (i) $\dot{H}\to 0$ for $t\in I_{k}$, $t_{k}^{+}<t<t_{k+1}^{-}$, and (ii)
$\delta_{k}=H(t_{k}^{+})-H(t_{k}^{-})\to 0$, for large enough $k$.

The total decrease of the $H$-functional would be

$\Delta H=\sum_{k=1}^{\infty}\underbrace{\delta_{k}}_{\leq
0}+\sum_{k=0}^{\infty}\underbrace{\int_{t_{k}^{+}}^{t_{k+1}^{-}}dt\,\dot{H}(t)}_{\equiv\varepsilon_{k}\leq
0}\leq 0.$ (41)

Since $H$ is bounded from below, the two series on the rhs must be convergent,
which implies that

$\lim_{k\to\infty}\delta_{k}=0,\quad\lim_{k\to\infty}\varepsilon_{k}=0.$

The second condition, $\varepsilon_{k}\to 0$ for $k\to\infty$ is equivalent to
stating that $\dot{H}\to 0$ over the time interval $I_{k}$, for large enough
$k$. Let us consider Eq. (37) at $t=t_{k}^{-}$, the condition $\dot{H}=0$
entails that

$\displaystyle P(x,c,t_{k}^{-})=k(c)\tilde{P}(x,c,t_{k}^{-}).$ (42)

Once more, we resort to the Bayes rule to write

$\displaystyle\cancel{P(c|x,t_{k}^{-})}P(x,t_{k}^{-})=k(c)\cancel{\tilde{P}(c|x,t_{k}^{-})}\tilde{P}(x,t_{k}^{-});$
(43)

the cancelled terms on both sides follow from the condition $\delta_{k}=0$, as
expressed by Eq. (40). Integrating over all $x$, we obtain

$k(c)=1\implies P(x,c,t)=\tilde{P}(x,c,t),\;\forall t\in I_{k},$ (44)

bringing to bear again Eq. (37).222The equality of the joint probabilities
$P(x,c,t)$ and $\tilde{P}(x,c,t)$ at $t=t_{k}^{-}$ entails that the
conditional probabilities in Eq. (40) are also equal, consistently with Eq.
(40).

To conclude, we have proven that

$\displaystyle\lim_{t\to\infty}\frac{P(x,c,t)}{\tilde{P}(x,c,t)}=1\quad\text{or}\quad
P(x,c,t)\sim\tilde{P}(x,c,t),\;t\to+\infty.$ (45)

That is, all the solutions of the differential Chapman-Kolmogorov equation
(15) tend towards a universal behaviour, independent of the initial
conditions, for long enough times. It is this universal long-time regime that
we are especially interested in.

##  5 Coarse-graining: Markovian equation for the position

Just after the control updates, the conditional probability of $c$ for given
$x$ is determined by the function $\Theta(c|x)$, which characterises the
action of the control—as expressed by Eq. (13). We can take advantage of this
fact to write down a closed equation for $P(x,t_{k+1})$ in terms of
$P(x,t_{k})$, i.e. to derive a Markov chain in the coarse-grained times
$t_{k}=k\Delta t_{m}$. Interestingly, it is not necessary that the measurement
is error-free: the Markovian description for the position in this coarse-
grained timescale is also valid for measurements involving a given uncertainty
$\Delta x$, as those in Eq. (7).

We introduce the notation

$P^{(k)}(x)\equiv P(x,t_{k})$ (46)

for the position distribution at the measurement times—recall that the
marginal distribution $P(x,t)$ is continuous thereat. Marginalising Eq. (24),
we get

$\displaystyle P^{(k)}(x)$ $\displaystyle=\sum_{c}\int dx^{\prime}\,K(x,\Delta
t_{m}|x^{\prime},c)P(x^{\prime},c,t_{k-1}^{+})$ $\displaystyle=\sum_{c}\int
dx^{\prime}\,K(x,\Delta
t_{m}|x^{\prime},c)\Theta(c|x^{\prime})P^{(k-1)}(x^{\prime}),$ (47)

where we have also made use of (10). Formally, we can write

$P^{(k)}(x)=\Big{[}\sum_{c}\hat{U}^{(c)}(\Delta
t_{m})\hat{\Theta}_{c}\Big{]}P^{(k-1)}(x)$ (48)

or, more specifically,

$\displaystyle P^{(k)}(x)$ $\displaystyle=\int dx^{\prime}\,T(x,\Delta
t_{m}|x^{\prime})P^{(k-1)}(x^{\prime}),$ (49a) $\displaystyle T(x,\Delta
t_{m}|x^{\prime})$ $\displaystyle\equiv\sum_{c}K(x,\Delta
t_{m}|x^{\prime},c)\Theta(c|x^{\prime}).$ (49b)

Of course, Eq. (49) conserves the total probability: since the Green function
conserves the spatial integral of the function to its right, $\int
dx\,P^{(k)}(x)=\int dx\,P^{(k-1)}(x)$.

Equation (49) defines a Markov chain for the distribution of the position at
the discrete times of the control updates. Our coarse-graining of time has
made $X$ Markovian, whereas the joint process $(X,C)$ is Markovian over the
“finer” timescale, which includes both the control updates and the time
evolution between them. Over this fine timescale, $X$ is not Markovian; it is
only over a “rougher” timescale that $X$ alone becomes a Markov process—in
complete analogy with the situation in other physical contexts, such as
Brownian motion [46, 47].

Our (continuous) stochastic matrix $T(x,\Delta t_{m}|x^{\prime})$ has a left
eigenvector with all its components equal to unity: if $\phi(x)\equiv 1$,
$\forall x$,

$\displaystyle\int dx\,\phi(x)T(x,\Delta
t_{m}|x^{\prime})=\sum_{c}\Theta(c|x^{\prime})=1=\phi(x^{\prime}),$ (50)

since $\int dx\,K(x,t|x^{\prime},c)=1$, $\forall(t,x^{\prime},c)$. As a
consequence, there should be a right eigenvector corresponding to the unit
eigenvalue, i.e. a time-independent solution $P_{\text{s}}(x)$,

$\int dx^{\prime}\,T(x,\Delta
t_{m}|x^{\prime})P_{\text{s}}(x^{\prime})=P_{\text{s}}(x).$ (51)

On physical grounds, we expect $P_{\text{s}}(x)$ to be a meaningful
probability distribution, with $P_{\text{s}}(x)\geq 0$ $\forall x$, when
working in the finite interval $x\in[-L,L]$. On the whole real line, there
would not be in general a normalisable stationary distribution.333The non-
negativity of $P_{\text{s}}(x)$ could be also argued on the basis of the
Perron-Frobenius theorem, but rigorously this theorem holds for a finite set
of discrete states, not for an infinite set with a continuous “index” $x$ as
we have here.

The $H$-theorem for the joint process $(X,C)$ over the fine timescale shows
that the system reaches a long-time “normal” state [48], independent of
initial conditions. Physically, one expects this solution to be time-periodic,
with period $\Delta t_{m}$. Over the rougher timescale for which $X$ becomes
Markovian, we have only the instants just after the control is updated, which
are precisely separated by $\Delta t_{m}$. In this description, the long-time
behaviour of $P^{(k)}(x)$ should thus correspond to a stationary, time-
independent, state. This reinforces our expectation of having a meaningful
solution of Eq. (51).

As shown in Sec. 3.2, it is necessary to sequentially concatenate the action
of the Fokker-Planck and measurement operators to obtain $P(x,c,t)$ from a
given initial condition. However, if we are just interested in the long-time
state, and not in the transitory relaxation thereto, the stationary marginal
distribution $P_{\text{s}}(x)$ makes it possible to derive the long-time state
for the joint probability $P(x,c,t)$. By combining Eqs. (23) and (10),
particularised for the steady marginal distribution $P_{\text{s}}(x)$, we get

$P_{\text{per}}(x,c,t)=\int
dx^{\prime}\;K(x,t-t_{k}^{+}|x^{\prime},c)\Theta(c|x^{\prime})P_{\text{s}}(x^{\prime}),\quad
t\in I_{k}.$ (52)

Note that, by construction, $P_{\text{per}}(x,c,t)$ is indeed a periodic
solution, with period $\Delta t_{m}$, of the DCKE (15). Our $H$-theorem
ensures that the joint process $(X,C)$ monotonically approaches this periodic
solution for long enough times. This is an interesting point, since we are
able to compute the long-time regime in the finer time scale, which is non-
stationary, from the steady state that emerges in the coarse-grained scale.

Now we consider the time evolution of the marginal probability for the control
$P(c,t)$ over this coarse-grained timescale, and show that it is not
Markovian. The marginal probability of the control is

$P^{(k)}(c)\equiv P(c,t^{+}_{k}).$ (53)

Over each time interval $t\in I_{k}$, the probability of the control does not
evolve and is given by $P^{(k)}(c)$. Notice that, unlike in the case of the
position, this marginal probability is not continuous at the control updates,
in general $P(c,t_{k}^{-})\neq P(c,t_{k}^{+})$. By recalling Eqs. (10) and
(46), we have

$P^{(k)}(c)=\int dx\,P^{(k)}(x)\Theta(c|x).$ (54)

The evolution of $P^{(k)}(c)$ is obtained from that of $P^{(k)}(x)$. Taking
into account Eq. (49), we get

$\displaystyle P^{(k+1)}(c)=\sum_{c^{\prime}}\int\\!dx\\!\int\\!dx^{\prime}$
$\displaystyle\Theta(c|x)\Theta(c^{\prime}|x^{\prime})K(x,\Delta
t_{m}|x^{\prime},c^{\prime})P^{(k)}(x^{\prime}),$ (55)

which cannot be cast into a closed equation for the evolution of $P^{(k)}(c)$,
due to the dependence on $x^{\prime}$ of $K(x,\Delta
t_{m}|x^{\prime},c^{\prime})$. It is only in the limit as $\Delta
t_{m}\to\infty$ that $K(x,\Delta t_{m}|x^{\prime},c^{\prime})$ becomes
independent of $x^{\prime}$ and $P^{(k)}(c)$ obeys a closed evolution
equation—see Sec 5.1. Yet, the tendency of $P^{(k)}(x)$ to a steady state
implies that also $P^{(k)}(c)$ tends to a steady distribution
$P_{\text{s}}(c)$. Equation (54) tells us that

$P_{\text{s}}(c)=\int dx\,P_{\text{s}}(x)\,\Theta(c|x),$ (56)

the stationary probability of each value of $c$ is given, logically, by the
steady probability of finding the position of the particle in the
corresponding interval.

The physical picture stemming from the above discussion is appealing. Over the
coarse-grained timescale, $X$ becomes a Markov process but $C$ is not: $C$
does not have enough information. If you know the position just after the
control update, you know the conditional probability of the control—but not
the other way round.

In the next sections, we focus on two relevant situations: $\Delta
t_{m}\to\infty$ and $\Delta t_{m}\to 0^{+}$, where we can work out explicit
expressions for $P_{\text{s}}(x)$.

###  5.1 Limit $\Delta t_{m}\to\infty$

Let us analyse the case in which the frequency of control updates is very low,
$\Delta t_{m}\to\infty$. Physically, this limit appears when the time between
control updates is much longer than all the rest of the characteristic times
of the system.

In this situation, the Green function verifies

$K(x,\Delta t_{m}\to\infty|x^{\prime},c)\to P_{\text{s}}^{(c)}(x).$ (57)

Therefore, we have the following equation for $P_{\text{s}}(x)$,

$\displaystyle P_{\text{s}}(x)$ $\displaystyle=\sum_{c}\left(\int
dx^{\prime}\Theta(c|x^{\prime})P_{\text{s}}(x^{\prime})\right)P_{\text{s}}^{(c)}(x)=\sum_{c}P_{\text{s}}(c)P_{\text{s}}^{(c)}(x).$
(58)

where we have made use of Eq. (56). The stationary solution $P_{\text{s}}(x)$
is a weighted average of the steady solutions for the problems with the
different potentials $V(x,c)$, that is, $P_{\text{s}}^{(c)}(x)$.

To solve Eq. (58) for $P_{\text{s}}(x)$, it is, therefore, necessary to derive
an equation for $P_{\text{s}}(c)$. It is handy to introduce a matrix $B$ with
elements

$B_{cc^{\prime}}\\!\equiv\int dx\,\Theta(c|x)P_{\text{s}}^{(c^{\prime})}(x).$
(59)

In the limit $\Delta t_{m}\to\infty$ we are considering, Eq. (55) can be
rewritten as

$P^{(k+1)}(c)=\sum_{c^{\prime}}B_{cc^{\prime}}P^{(k)}(c^{\prime}),$ (60)

which is the evolution equation of a Markov chain. As already commented in the
previous section, it is only in this limit that the marginal process $C$
becomes Markovian. The matrix $B$ with elements $B_{cc^{\prime}}$ is its
corresponding stochastic matrix, such that

$B_{cc^{\prime}}>0,\quad\sum_{c}B_{cc^{\prime}}=1.$ (61)

The matrix $B$ has thus the unit eigenvalue and, by virtue of the Perron-
Frobenius theorem, the corresponding right eigenvector, which verifies

$P_{\text{s}}(c)=\sum_{c^{\prime}}B_{cc^{\prime}}P_{\text{s}}(c^{\prime}),$
(62)

is unique and all its components $P_{\text{s}}(c)$ are positive.444This
equation can also be obtained from (58), multiplying it by $\Theta(c|x)$ and
integrating over $x$. Normalising this eigenvector in the usual sense, we have

$P_{\text{s}}(c)>0,\quad\sum_{c}P_{\text{s}}(c)=1,$ (63)

which ensures that $P_{\text{s}}(x)$ in Eq. (58) is a meaningful probability
distribution, moreover correctly normalised.

###  5.2 Limit $\Delta t_{m}\to 0^{+}$

Another relevant physical situation is that of a high frequency of control
updates, i.e. $\Delta t_{m}\to 0^{+}$. Physically, this limit appears when the
time between control updates is much shorter than all the rest of the
characteristic times of the system.

In this limit, it is useful to introduce the short-time expansion

$\hat{U}^{(c)}(\Delta t_{m})=I+\Delta t_{m}\hat{\mathcal{L}}^{(c)}.$ (64)

Then, Eq. (48) is approximated as

$\displaystyle P(x,t_{k})$ $\displaystyle=\sum_{c}\left[I+\Delta
t_{m}\hat{\mathcal{L}^{(c)}}\right]\hat{\Theta}_{c}P(x,t_{k-1})=P(x,t_{k-1})+\Delta
t_{m}\sum_{c}\hat{\mathcal{L}^{(c)}}\hat{\Theta}_{c}P(x,t_{k-1}),$ (65)

which can be cast as a first-order differential equation in time, recalling
that $t_{k}=k\Delta t_{m}$,

$\partial_{t}P(x,t)=\sum_{c}\hat{\mathcal{L}^{(c)}}\hat{\Theta}_{c}P(x,t),$
(66)

or

$\displaystyle\partial_{t}P(x,t)$
$\displaystyle=\sum_{c}\partial_{x}\Big{[}\partial_{x}\tilde{V}(x,c)\,\Theta(c|x)P(x,t)+k_{B}T\,\partial_{x}\Theta(c|x)P(x,t)\Big{]}$
(67)

By defining the effective potential and force

$\displaystyle V_{\text{eff}}(x)$
$\displaystyle\equiv\sum_{c}\int_{0}^{x}dx^{\prime}\,\partial_{x^{\prime}}V(x^{\prime},c)\Theta(c|x^{\prime}),$
(68a) $\displaystyle-\partial_{x}V_{\text{eff}}(x)$
$\displaystyle=-\sum_{c}\partial_{x}V(x,c)\Theta(c|x),$ (68b)

we have that

$\partial_{t}P(x,t)=\partial_{x}\Big{[}(\partial_{x}V_{\text{eff}}(x)+F_{\text{ext}})P(x,t)+k_{B}T\,\partial_{x}P(x,t)\Big{]}.$
(69)

Equation (69) is the Fokker-Planck equation for an overdamped particle moving
under the action of a conservative force field
$-\partial_{x}V_{\text{eff}}(x)$ plus the non-conservative force
$-F_{\text{ext}}$. In the long-time limit, the system reaches the
corresponding steady solution—see B for details, in particular Eq. (96). For
error-free measurements, the effective potential $V_{\text{eff}}(x)$ can be
seen as the continuous concatenation of the different potentials $V(x,c)$,
each one restricted to the region $\mathcal{X}_{c}$ for which $\Theta(c|x)=1$.
This may be used, as in the feedback flashing ratchet considered below, to
generate a directed motion in a certain direction.

##  6 Feedback flashing ratchet

###  6.1 Description of the system

In order to illustrate the formalism developed in the previous sections, now
we apply it to a specific system: the feedback flashing ratchet sketched as a
motivation for our work in Sec. 2. To be concrete, we consider the particle to
move under the action of a V-shaped potential in the absence of external
force:

$V(x)=F\absolutevalue{x},\quad x\in[-L,L],\qquad F_{\text{ext}}=0.$ (70)

The feedback controller switches on or off the potential $V(x)$ depending on
the particle position $x(t_{k}^{-})$ just before the measurement time, as
described in Eq. (7). Therefore, the fluctuating potential $V(x,c)$ introduced
in Sec. 3 is

$V(x,c)=c\,F\absolutevalue{x},$ (71)

and the corresponding conservative force acting on the particle is

$-\partial_{x}V(x,c)=-c\,F\operatorname{sgn}(x)=c\,F\operatorname{sgn}(-x).$
(72)

Therefore, the sets $\mathcal{X}_{1}$ and $\mathcal{X}_{0}$ defined by
respectively having a positive and negative value of the force are given by
$[-L,0)$ and $(0,L]$—see the right panel of Fig. 1.

The most general formulation of the mathematical problem for the specific
system we are considering can be written as follows. We have to solve the DCKE
(15) for the joint probability $P(x,c,t)$, particularised for the feedback
flashing ratchet with the V-shaped potential:

$\displaystyle\partial_{t}P(x,c,t)=$
$\displaystyle\gamma^{-1}c\,F\,\partial_{x}\left[\,\operatorname{sgn}(x)P(x,c,t)\right]+(\gamma\beta)^{-1}\partial_{x}^{2}P(x,c,t)$
$\displaystyle+\sum_{k}\delta(t-t_{k}^{-})\Big{[}\Theta(c|x)P(x,1-c,t^{\prime})-\Theta(1-c|x)P(x,c,t)\Big{]},$
(73)

with the feedback control protocol that takes into account an uncertainty in
the particle position $\Delta x$:

$\displaystyle\Theta(c|x)=\begin{cases}\text{length\;}\\{[-L,0]\cap(x-\Delta
x,x+\Delta x)\\}/(2\Delta x),&\text{if }c=1,\\\
\text{length\;}\\{[0,L]\cap(x-\Delta x,x+\Delta x)\\}/(2\Delta x),&\text{if
}c=0.\end{cases}$ (74)

The error-free version of this protocol, for $\Delta x=0$, switches on (off)
the potential if the force at the instantaneous position $x$ points to the
right (left). For $\Delta x\neq 0$, we have localised the position of the
particle in the interval $(x-\Delta x,x+\Delta x)$, so the potential is
switched on (off) with probability proportional to the length of this segment
that lies inside the interval $\mathcal{X}_{1}$ ($\mathcal{X}_{0}$) in which
the force points to the right (left). The explicit expressions for
$\Theta(c|x)$ can be found in C.

The boundary conditions for this problem are periodic, as given by Eq. (16).
For the initial conditions, we have chosen

$\displaystyle P(x,1,0)=0,\quad P(x,0,0)=(2L)^{-1},$ (75)

which corresponds to having the potential switched off at $t=0$, with a flat
distribution of the particles in the interval $[-L,L]$. Note that, in the
notation introduced in Sec. 3.1, $P(x,0,0)=P_{\text{s}}^{(0)}(x)$—i.e. it is
the stationary distribution for the potential $V(x,0)=0$.

###  6.2 Analytical results

For this system, we can analytically find the joint probability $P(x,c,t)$,
solution of the problem over the fine timescale that contains both the control
updates and the evolution between them. This is achieved by employing the
formal solution derived in Sec. 3.2, i.e. by concatenating Eqs. (23) and (25),
since we can exactly calculate the Green function $K(x,c|x^{\prime},t)$.555The
expression for the Green function as an infinite series is not especially
illuminating, it is thus relegated to D.

Over the coarse-grained timescale that arises for the times $t_{k}^{+}$ just
after the control updates, the system reaches a steady state and we are able
to derive additional results. For example, in the two limiting cases
introduced in Sec. 5, $\Delta t_{m}\to\infty$ and $\Delta t_{m}\to 0^{+}$, we
can exactly calculate the steady marginal distribution of the position
$P_{\text{s}}(x)$ over this coarse-grained timescale.

For $\Delta t_{m}\to\infty$, we have that Green function $K(x,\Delta
t_{m}\to\infty|x^{\prime},c)\to P_{\text{s}}^{(c)}(x)$, which is independent
of $x^{\prime}$—see Eq. (99) in B. In Sec. 5.1, we have taken advantage of
this property to derive a closed system of equations for the marginal
probability of the control at the steady state $P_{\text{s}}(c)$, given by Eq.
(58). For the case we are considering, we can derive explicit expressions for
the coefficients $B_{cc^{\prime}}$, defined in Eq. (59). By taking into
account that

$\displaystyle P_{\text{s}}^{(0)}(x)=\frac{1}{2L},\quad
P_{\text{s}}^{(1)}(x)=\frac{\beta F}{2(1-e^{-\beta FL})}e^{-\beta
F\absolutevalue{x}},$ (76)

a straightforward calculation leads to

$B_{cc^{\prime}}=\frac{1}{2},\quad\forall c,c^{\prime}=0,1.$ (77)

This implies that

$P_{\text{s}}(c)=\frac{1}{2},\quad\forall c=0,1,$ (78)

and making use of Eq. (58), we finally obtain the steady distribution of the
position over the coarse-grained timescale:

$P_{\text{s}}(x)=\frac{1}{2}\left[P_{\text{s}}^{(0)}(x)+P_{\text{s}}^{(1)}(x)\right],\quad\Delta
t_{m}\to\infty.$ (79)

This result is also independent of the uncertainty in the measurement $\Delta
x$.666In fact, the results in Eqs. (77)–(79) apply to a broad class of
flashing ratchets, see E.

For $\Delta t_{m}\to 0^{+}$, we have shown in Sec. 5.2 that the particle
effectively moves under the action of an effective potential. In the specific
case we are considering, Eq. (68) particularises to

$V_{\text{eff}}(x)=\int_{0}^{x}dx^{\prime}\,\partial_{x^{\prime}}V(x^{\prime},1)\Theta(1|x^{\prime})=F\int_{0}^{x}dx^{\prime}\,\operatorname{sgn}(x^{\prime})\,\Theta(1|x^{\prime}),$
(80)

since the potential is switched off for $c=0$. The shape of this potential
thus depends on uncertainty $\Delta x$ of the measurement, as illustrated in
Fig. 3. The corresponding force is
$-\partial_{x}V_{\text{eff}}(x)=-\partial_{x}V(x)\Theta(1|x)=F\operatorname{sgn}(-x)\Theta(1|x)$.
For the free-error protocol, $-\partial_{x}V_{\text{eff}}=F\eta(-x)$, where
$\eta(x)$ is the Heaviside function. The stationary distribution for the
position is

$P_{\text{s}}(x)=K\left[e^{-\beta V_{\text{eff}}(x)}+\frac{e^{-\beta
FL}-1}{\int_{-L}^{L}dx^{\prime}e^{\beta
V_{\text{eff}}(x^{\prime})}}\int_{-L}^{x}dx^{\prime}e^{-\beta\left[V_{\text{eff}}(x)-V_{\text{eff}}(x^{\prime})\right]}\right],\quad\Delta
t_{m}\to 0^{+},$ (81)

where $K$ is a normalisation constant—see B.

Figure 3: Effective potentials for different values of $\Delta x$. The colour
code is the same as that in Fig. 2. When $\Delta x=0$ (error-free protocol),
the effective potential (blue line) faced by the particle when $\Delta
t_{m}\to 0^{+}$ corresponds to a constant positive force for $x<0$, and no
force for $x>0$. As the uncertainty of the position $\Delta x$ increases, the
protocol converges to the open-loop protocol and thus the effective potential
tends to the V-shaped potential (green line), weighted by the probability
$\Theta(1|\cancel{x})=1/2$ of switching it on.

We recall that Eq. (52) gives the long-time periodic behaviour of the joint
probability distribution $P_{\text{per}}(x,c,t)$ over the fine timescale in
terms of the stationary marginal distribution $P_{\text{s}}(x)$ over the
coarse-grained timescale. Therefore, Eqs. (79) amd (81), inserted into Eq.
(52), also provide analytical predictions for the long-time periodic joint
probability distribution $P_{\text{per}}(x,c,t)$.

###  6.3 Numerical results

Now we compare the analytical results stemming from our Markovian description
of the feedback flashing ratchet with the numerical results obtained from the
simulation of the system dynamics. In particular, we are interested in
elucidating (a) the accuracy of the Markovian description we have developed in
this paper, (b) the emergence of a long-time regime, as ensured by the
$H$-theorem in Sec. 4, in which we expect the system to reach a time-periodic
state, and (c) the specific results obtained in the limiting cases $\Delta
t_{m}\to\infty$ and $\Delta t_{m}\to 0^{+}$, for which we have derived closed
expressions.

Units have been chosen as follows: $L$ as the length unit and $k_{B}T$ as the
energy unit, i.e. we take $L=1$ and $k_{B}T=1$, whereas the time unit has been
determined by taking $\gamma=1$—i.e. the time unit is
$L^{2}/D=L^{2}\gamma/(k_{B}T)$. In general, we have employed a value of the
force $F=5$ and several flashing protocols with different values of the
uncertainty in the measurement of the position, specifically $\Delta x=0$,
$\Delta x=0.5$, $\Delta x=0.75$, and $\Delta x=1$. We recall that $\Delta x=0$
corresponds to an error-free measurement, whereas $\Delta x=1$ corresponds to
the open-loop protocol, since the uncertainty in the measurement extends to
the whole interval due to the periodic boundary conditions.

The theoretical results for the probability distributions have been obtained
with the procedure developed in Sec. 3.2, i.e. by concatenating Eqs. (23),
with the analytical expression for the Green function (27) as an infinite
series, and Eq. (25). The specific expression of the Green function for the
feedback flashing ratchet with the V-shaped potential is derived in D, and we
have truncated it after the $20$-th term for plotting the probability
distributions. The simulation results have been obtained by combining the
numerical integration of the Langevin equation (2) with the Euler method
between the control updates, and the corresponding update of the control
variable at the times $t_{k}=k\Delta t_{m}$, as given by Eq. (74). All the
simulation results have been averaged over $10^{6}$ trajectories of the
dynamics.

We start by checking the accuracy of our Markovian description, along with the
convergence to (i) a time-periodic state over the fine timescale and (ii) the
tendency to a steady state over the coarse-grained timescale.

In Fig. 4, we present the analytical solution for the joint probability
distribution of the position of the control $P(x,c,t)$. This analytical
solution has been built for the initial condition in Eq. (75), in particular
for a system with parameters $\Delta t_{m}=0.1$, $F=5$, and $\Delta x=0.5$,
during a time interval in which there have been $10$ control updates at times
$t_{k}=k\Delta t_{m}$, i.e. from $t=0$ to $t=10\Delta t_{m}$,. Due to the
specific initial conditions chosen, there is no time evolution in the first
time interval $I_{0}=(0,\Delta t_{m})$. Once the control is first updated at
$t_{1}=\Delta t_{m}$, the system quickly evolves towards a time-periodic state
with period $\Delta t_{m}$ inherited from the periodicity of the control
updates. This time-periodic state is the long-time behaviour expected from the
$H$-theorem.

Figure 4: Analytical prediction for the time evolution of the joint
probability distribution $P(x,c,t)$. Specifically, the plotted solution
corresponds to parameters $\Delta t_{m}=0.1$,$F=5$ and $\Delta x=0.5$, in the
time interval $[0,10\Delta t_{m}]$. The tendency of the system to approach a
time-periodic state is clearly observed. Despite the even parity of the
V-shaped potential, $P(x,c,t)$ is not left-right symmetric due to the action
of the feedback controller.

In Fig. 5, we compare the analytical solution for the joint probability
distribution $P(x,c,t)$ with the numerical integration of the dynamics.
Specifically, we consider three different points in the spatial interval
$[1-,1]$, $x=0$ and $x=\pm 0.7$, during the same time interval as in Fig. 4.
The times $t_{k}$ at which the control is updated are marked with the vertical
lines. The agreement between the numerical results (symbols) and the
theoretical prediction (lines) is excellent. Again, the time-periodic
behaviour reached by the system after a large enough number of control updates
$k$ is neatly observed.

Figure 5: Time evolution of the joint probability $P(x,c,t)$. Four flashing
protocols with different uncertainties have been considered in the different
panels: (a) $\Delta x=0$, (b) $0.5$, (c) $0.75$, and (d) $1$. In each panel,
we plot $P(x,c,t)$, with $c=0,1$, for three different values of $x$, $x=-0.7$
, $0$, and $0.7$. Numerical results (symbols) are compared with theoretical
predictions (lines) for both $P(x,0,t)$ (filled circles, solid line)
$P(x,1,t)$ (empty squares, dashed line). It is clearly observed the tendency
of the system towards a time-periodic state with periodicity $\Delta t_{m}$
after a large enough number of control updates. Vertical lines represent the
measurement times. The parameters used have been $\Delta t_{m}=0.1$ and $F=5$.

Fig. 6 illustrates the compatibility of the descriptions over the fine and
coarse-grained timescales. We employ the same parameters as in Fig. 5. The top
panel shows the evolution of $P(c=1,t)$, which only changes when the value of
the control is updated at $t_{k}=k\Delta t_{m}$ after the measurement of the
position at $t_{k}^{-}$. After a few measurements, $P(c=1,t)$ reaches a steady
state $P_{\text{s}}(c=1)$. In the bottom panels, we focus on the time
evolution of $P(x,c,t)$ and $P(x,t)$ in one time interval
$I_{k}=(t_{k},t_{k+1})$ between control updates, in particular the last one,
$I_{9}=(t_{9},t_{10})$. At the end of the previous interval
$I_{8}=(t_{8},t_{9})$, $P^{(k)}(x)\equiv P(x,t_{k}^{-})$ has already reached
the steady state $P_{\text{s}}(x)$ over the coarse-grained timescale for large
enough $k$. However, over the finer timescale, both $P(x,c,t)$ and $P(x,t)$
evolve. When a measurement of the particle position takes place at the time
$t_{k}^{-}$, the joint distribution changes abruptly from $P(x,c,t_{k}^{-})$
to $P(x,c,t_{k}^{+})$, following Eq. (21). This can also be seen in the bottom
panels, where both $P(x,0,t)$ (green dashed line) and $P(x,1,t)$ (red dashed
line) are shown. However, $P(x,t_{k}^{-})$ (blue solid line), is continuous at
the control updates, i.e. $P(x,t_{k}^{-})=P(x,t_{k}^{+})$, as predicted by Eq.
(12)—see panels (a) and (b). After the measurement, the joint probability
$P(x,c,t)$ evolves from its initial state following its corresponding Fokker-
Planck equation, i.e. under the action of the potential $V(x,c)=c\,V(x)$, and
the marginal distribution $P(x,t)$ evolves too—see panel (c). At the final
instant of the interval $I_{9}$, i.e. at $t_{10}^{-}$, $P(x,t)$ recovers the
steady state shape $P_{\text{s}}(x)$ over the coarse-grained timescale—panel
(d).

Figure 6: Compatibility of the evolution over the fine and coarse-grained
timescales. System parameters are the same as in Fig. 5. The upper panel shows
the evolution of $P^{(k)}(c)$ from $k=0$ to $k=10$, i.e. from $t=0$ to
$t=10\Delta t_{m}$. The bottom panels focus on the evolution of $P(x,t)$ (blue
solid line), $P(x,0,t)$ (green dashed line) and $P(x,1,t)$ (red dotted line)
in the last time interval $I_{9}=(t_{9},t_{10})$ (shaded in the upper panel),
i.e. between $t=9\Delta t_{m}$ to $t=10\Delta t_{m}$. Each panel collects the
probabilities for different times, from (a) to (d): $t_{9}^{-}$, $t_{9}^{+}$,
$1.1t_{9}$, and $t_{10}^{-}$. We appreciate how, if we just pay attention to
the time instants $t_{9}^{-}$ and $t_{9}^{+}$ of panels (a) and (b), just
before and after the measurement time, the marginal distribution $P(x,t)$ is
continuous thereat, while the joint distribution $P(x,c,t)$ is not. Panel (c)
shows a snapshot of the time evolution of the probabilities in the time
interval $I_{9}$, specifically at $t=1.1t_{9}$. The steady state reached over
the coarse-grained timescale is illustrated in panel (d), compare the profiles
in panels (a) and (d).

Now we proceed to inspect the specific results we have obtained in the limits
$\Delta t_{m}\to\infty$ and $\Delta t_{m}\to 0^{+}$. Once more, we have
employed the feedback flashing protocol (74), for which we have analytical
predictions for the steady distribution of the position over the coarse-
grained timescale, as given by Eqs. (79) ($\Delta t_{m}\to\infty$) and (81)
($\Delta t_{m}\to 0^{+}$). In Fig. 7, we have plotted the marginal probability
$P^{(k)}(x)\equiv P(x,t_{k})$, as defined in Eq. (46), for large enough $k$,
$k>k_{r}$. Note that “enough” here depends on the time between control
updates: $k_{r}$ decreases as $\Delta t_{m}$ increases, and $k_{r}\to 1$ for
$\Delta t_{m}\to\infty$ because the steady state is reached after only one
control update. In both cases, the numerical results for large $k$ (symbols)
are very close to the analytical expressions for the steady distributions
(lines).

Figure 7: Stationary marginal probability of the position over the coarge-
grained timescale. In particular, we show the simulation result for
$P(x,t_{k}^{-})$ (symbols) for large enough $k$, and compare them with our
analytical prediction for the stationary distribution $P_{\text{s}}(x)$
(lines). As in Fig. 5, panels (a)–(d) correspond to different flashing
protocols, with increasing uncertainty from $\Delta x=0$ to $\Delta x=1$, for
a force $F=5$. Here, two intervals between control updates are used: a short
one $\Delta t_{m}=10^{-3}$ (filled circles, solid line), and a long one
$\Delta t_{m}=1$ (open squares, solid line), to check the limits $\Delta
t_{m}\to 0^{+}$ and $\Delta t_{m}\to\infty$, respectively.

Finally, we present results for the average velocity of the particle. With
periodic boundary conditions, the average velocity of the particle is
calculated by considering the average of the velocity field $v(x,c,t)\equiv
J(x,c,t)/P(x,c,t)$—see A. Specifically, one has

$\expectationvalue{v}\equiv\sum_{c}\int_{-L}^{L}dx\,v(x,c,t)P(x,c,t)=\sum_{c}\int_{-L}^{L}dx\,J(x,c,t).$
(82)

Our family of flashing protocols depends on the uncertainty $\Delta x$ in the
position measurement, varying from error-free feedback when $\Delta x=0$ to
open-loop control when $\Delta x=1$. Fig. 8 presents $\expectationvalue{v}$ as
a function of time, for different values of $\Delta x$; $\expectationvalue{v}$
decreases with $\Delta x$, vanishing for $\Delta x=1$. This trend can be
understood as follows: the physical mechanism behind the creation of a
directed flux is the breaking of the spatial symmetry. Since the potential in
left-right symmetric in our system, the symmetry breaking must come from the
feedback control, which uses information from the position to update the
potential state. For $\Delta x=1$, we have an open-loop protocol that no
longer gathers spatial information from the system and, therefore, no flux is
created.

Figure 8: Average velocity $\expectationvalue{v}$ of the particle as a
function of time. System parameters are the same as in Figs. 5 and 7.
Simulation (symbols) and theoretical (lines) results have been plotted for
different values of the uncertainty, as usual $\Delta x=0$ (diamonds), $0.5$
(squares), $0.75$ (circles), and $1$ (stars). As $\Delta x$ increases,
$\expectationvalue{v}$ decreases from its maximum value for $\Delta x=0$
(error-free feedback) to zero for $\Delta x=1$ (open-loop).

##  7 Summary and conclusions

We have considered a wide class of feedback controlled systems, in which an
overdamped particle moves in a fluctuating potential that is changed by an
external controller depending on the instantaneous position of the particle.
The controller acts regularly at times $t_{k}=k\Delta t_{m}$, $k=1,2,\ldots$,
which naturally divides the time evolution into intervals
$I_{k}=(t_{k},t_{k+1})$. In each trajectory of the dynamics, the controller
measures the position of the particle $x_{k}^{-}\equiv x(t_{k}^{-})$ just
before $t_{k}$, and the potential felt by the particle at time $t_{k}^{+}$
(and during the whole interval $I_{k}$) is $V(x,c_{k})$ with probability given
by a generic function $\Theta(c_{k}|x_{k}^{-})$. This allows for the inclusion
of imperfect measurements with a certain uncertainty $\Delta x$. As a
consequence, our framework includes as limiting cases error-free feedback
control, where $\Theta(c|x)$ depends on $x$ but equals either $0$ or $1$ for
any $x$, and open-loop, i.e. no actual feedback, control, where $\Theta(c|x)$
is independent of $x$.

In general, neither the position $X$ nor the controller variable $C$ is a
Markov process. However, the joint stochastic process $(X,C)$ is Markovian,
even for imperfect control. The joint probability distribution $P(x,c,t)$
evolves following a differential Chapman-Kolmogorov equation, which has two
contributions: (i) a Fokker-Planck contribution, which accounts for the time
evolution under the potential $V(x,c)$ between control updates, i.e. in the
time intervals $I_{k}$, and (ii) a master equation contribution, which
accounts for the change of the potential at the times of the control updates
$t_{k}$. This general Markovian framework allows us to give a formal
analytical solution for the time evolution of the joint probability
distribution, in terms of an expansion in the eigenmodes and eigenfunctions of
the potentials $V(x,c)$, for an arbitrary frequency of the control updates,
i.e. for arbitrary $\Delta t_{m}$.

We have proved an $H$-theorem for the joint stochastic process $(X,C)$, by
constructing a Lyapunov functional $H(t)$ that is the Kullback-Leibler
divergence between two solutions of the differential Chapman-Kolmogorov
equation. We have shown that $H(t)$ is monotonically decreasing, both during
the Fokker-Planck evolution in the time intervals $I_{k}$ and at the control
updates at the times $t_{k}$ delimiting them. This $H$-theorem ensures that
the joint probability distribution $P(x,c,t)$ reaches a long-time regime,
independent of the initial conditions. On a physical basis, the long-time
behaviour of the joint distribution, $P_{\text{per}}(x,c,t)$, is expected to
be time-periodic with period $\Delta t_{m}$, inherited from the control.

The above Markovian description holds over the (fine) timescale that includes
both the times $t_{k}$ at which the control is updated and the time intervals
$I_{k}$ between control updates. One may go to a coarse-grained timescale that
only considers the times $t_{k}^{+}$ just after the control updates. Even for
imperfect measurements, the control is enslaved to the position at these times
and thus the position $X$ alone becomes Markovian. Over this coarse-grained
timescale, the probability distribution of the position approaches a
stationary distribution $P_{\text{s}}(x)$. Interestingly, finding this
stationary distribution over the coarse-grained timescale makes it possible to
derive the time-periodic distribution $P_{\text{per}}(x,c,t)$ over the fine
timescale, which the system approaches for long times as a consequence of the
$H$-theorem. In the limits of very frequent ($\Delta t_{m}\to 0^{+}$) or very
infrequent ($\Delta t_{m}\to\infty$) control updates, explicit expressions for
the stationary distribution $P_{\text{s}}(x)$—and thus for
$P_{\text{per}}(x,c,t)$—have been derived.

The just described theoretical framework has been applied to a specific model:
the feedback flashing ratchet. Therein, we have compared the theoretical
expressions obtained with numerical simulations of the system dynamics—the
latter consisting in the concatenation of the numerical integration of the
Langevin equation in the time intervals $I_{k}$ with the control updates at
the times $t_{k}$. The agreement between the numerics and the theoretical
results has been excellent in all cases.

The impact of the information gathered by a feedback controller in the
thermodynamic balance of the system, since it reduces the entropy of the
particle, is an active line of research [12, 13, 6, 14, 7, 15, 16, 17]. For
the particular situation of the feedback flashing ratchet, this has recently
been done [17] by considering the full history of control actions
$(c_{1},c_{2},\ldots)$—due to the non-Markovianity of the position. The
Markovian description for the joint process $(X,C)$ opens new perspectives in
this direction, since one may consider the thermodynamic balance for the
Shannon entropy of the joint process,

$S=-\sum_{c}\int dx\,P(x,c,t)\ln P(x,c,t).$ (83)

We would like to stress that the connection between this Shannon entropy and
the entropy of the full history of control actions is not trivial, and
deserves to be further investigated.

The use of the Shannon entropy (83) for the Markovian joint process $(X,C)$
opens other interesting perspectives for future research. For example, it
seems relevant to seek possible refinements of the second principle, for
instance by splitting the entropy production into “excess” and “housekeeping”
contributions [49, 50, 51, 52]. In this context, it is also worth looking into
the possible relevance of information geometry ideas, which may lead to better
refinements of the second principle—by considering alternative splittings of
the entropy production [53, 54, 55, 56, 57, 58, 59].

## Acknowledgments

We acknowledge financial support from Grant PID2021-122588NB-I00 funded by
MCIN/AEI/10.13039/501100011033/ and by “ERDF A way of making Europe”, and from
Grant ProyExcel_00796 funded by Junta de Andalucía’s PAIDI 2020 programme. NRP
acknowledges support from the FPU programme through Grant FPU2021/01764. We
also thank C. A. Plata and F. J. Cao for really useful discussions.

## Data availability

The codes employed for generating the data that support the findings of this
study, together with the Mathematica notebooks employed for producing the
figures presented in the paper, are openly available in the GitHub page of
University of Sevilla’s FINE research group.

##  Appendix A Periodic boundary conditions

In our system, the particle is moving under a periodic force with spatial
period $2L$. We can thus study the system on the whole real line,
$-\infty<x<+\infty$, or only in one period of the potential, e.g. $-L\leq
x\leq L$. In this section, we would like to clarify the relation between the
two descriptions.

Our starting point is the Fokker-Planck equation (8) in the time intervals
$I_{k}$, i.e.

$\displaystyle\partial_{t}P(x,c,t)$ $\displaystyle=-\partial_{x}J(x,c,t),$
$\displaystyle
J(x,c,t)=-\gamma^{-1}\left[\tilde{V}^{\prime}(x,c)P(x,c,t)+k_{B}T\,\partial_{x}P(x,c,t)\right],$
(84)

The force is periodic, so that
$\tilde{V}^{\prime}(x)=\tilde{V}^{\prime}(x+2L)$, and therefore
$P(x+2nL,c,t)$, with $n\in\mathbb{Z}$, is also a solution of the Fokker-Planck
equation. Then, we can construct the probability of $x$ modulo $L$,

$P_{\text{mod}}(x,c,t)\equiv\sum_{n}P(x+2nL,c,t),\quad\absolutevalue{x}\leq
L.$ (85)

i.e. we treat all the points that differ by $2nL$ as equivalent and thus sum
all their probabilities.

When constructing $P_{\text{mod}}(x,t)$, we restrict the problem to one period
$[-L,L]$ and then expect that this is equivalent to considering periodic
boundary conditions. And, in fact, this is basically what is being done, but
let us understand it on a solid basis. Equation (85) entails that
$P_{\text{mod}}(-L,c,t)=P_{\text{mod}}(L,c,t)$. Moreover, we have

$0=\frac{d}{dt}\int_{-\infty}^{+\infty}dxP(x,c,t)=\frac{d}{dt}\int_{-L}^{L}dxP_{\text{mod}}(x,c,t)=J_{\text{mod}}(-L,c,t)-J_{\text{mod}}(L,c,t),$
(86)

where

$J_{\text{mod}}(x,c,t)\equiv\sum_{n}J(x+2nL,c,t)=-\gamma^{-1}\left[\tilde{V}^{\prime}(x,c)P_{\text{mod}}(x,c,t)+k_{B}T\,\partial_{x}P_{\text{mod}}(x,c,t)\right],$
(87)

for $\absolutevalue{x}\leq L$. Therefore, the boundary conditions for the
problem in $[-L,L]$ are

$P_{\text{mod}}(-L,c,t)=P_{\text{mod}}(L,c,t),\quad
J_{\text{mod}}(-L,c,t)=J_{\text{mod}}(L,c,t).$ (88)

###  A.1 Particle velocity in the periodic picture

It is also interesting to link the average velocity of the particle on the
whole real line with the average flux in the periodic picture. The evolution
equation for the average position on the whole real line is readily derived
from the Fokker-Planck equation:

$\left\langle
v\right\rangle\equiv\frac{d}{dt}\sum_{c}\int_{-\infty}^{+\infty}dx\,xP(x,c,t)=-\frac{1}{\gamma}\sum_{c}\int_{-\infty}^{+\infty}dx\,\tilde{V}^{\prime}(x,c)P(x,c,t)=-\frac{1}{\gamma}\left\langle\tilde{V}^{\prime}(x,c)\right\rangle,$
(89)

where we have integrated by parts. When one goes to the interval $[-L,L]$ with
periodic boundary conditions, $\left\langle
v\right\rangle\neq\sum_{c}\int_{-L}^{L}dx\,xP_{\text{mod}}(x,c,t)$. Where is
the information of the average velocity stored? The answer is that it is
stored in the spatial average of the current:

$\displaystyle\sum_{c}\int_{-L}^{L}dxJ_{\text{mod}}(x,c,t)=$
$\displaystyle-\frac{1}{\gamma}\sum_{c}\int_{-L}^{L}dx\,\tilde{V}^{\prime}(x,c)P_{\text{mod}}(x,c,t)$
$\displaystyle-\frac{k_{B}T}{\gamma}\sum_{c}\cancelto{0}{\left[P_{\text{mod}}(L,c,t)-P_{\text{mod}}(-L,c,t)\right]}.$
(90)

Taking into account that $\tilde{V}^{\prime}(x,c)$ is spatially periodic,

$\displaystyle\sum_{c}\int_{-L}^{L}dxJ_{\text{mod}}(x,c,t)$
$\displaystyle=-\frac{1}{\gamma}\sum_{c}\int_{-\infty}^{+\infty}dx\,\tilde{V}^{\prime}(x,c)P(x,c,t)=-\frac{1}{\gamma}\left\langle\tilde{V}^{\prime}(x,c)\right\rangle=\left\langle
v\right\rangle,$ (91)

which completes the proof.

##  Appendix B Stationary distributions

Let us consider the stationary state for the Fokker-Planck equation with force
$-\tilde{V}^{\prime}(x)$ and periodic boundary conditions—consistently with
our notation in the main text, we omit the subindex “mod” from now on. At the
steady state, the current is uniform in the whole system:

$J_{\text{s}}=-\gamma^{-1}\left[\tilde{V}^{\prime}(x)P_{\text{s}}(x)+k_{B}T\,\partial_{x}P_{\text{s}}(x)\right].$
(92)

Integrating this equation, we get

$P_{\text{s}}(x)=Ke^{-\beta\tilde{V}(x)}-\beta\gamma
J_{\text{s}}\int_{0}^{x}dx^{\prime}e^{-\beta\left[\tilde{V}(x)-\tilde{V}(x^{\prime})\right]},$
(93)

where $K$ is a normalisation constant. By imposing the periodicity of $P$,
$P_{\text{s}}(-L)=P_{\text{s}}(L)$, we get

$Ke^{-\beta\tilde{V}(-L)}=Ke^{-\beta\tilde{V}(L)}-\beta\gamma
J_{\text{s}}\int_{-L}^{L}dxe^{-\beta\left[\tilde{V}(L)-\tilde{V}(x)\right]}.$
(94)

This equation tells us that $J_{\text{s}}\neq 0$ if and only if
$\tilde{V}(-L)\neq\tilde{V}(L)$. Solving for $J_{\text{s}}$,

$\beta\gamma
J_{\text{s}}=K\frac{e^{-\beta\tilde{V}(L)}-e^{-\beta\tilde{V}(-L)}}{\int_{-L}^{L}dx\,e^{-\beta\left[\tilde{V}(L)-\tilde{V}(x)\right]}},$
(95)

and one has finally that

$P_{\text{s}}(x)=K\left[e^{-\beta\tilde{V}(x)}+\frac{e^{-\beta\tilde{V}(-L)}-e^{-\beta\tilde{V}(L)}}{\int_{-L}^{L}dx^{\prime}e^{-\beta\left[\tilde{V}(L)-\tilde{V}(x^{\prime})\right]}}\int_{-L}^{x}dx^{\prime}e^{-\beta\left[\tilde{V}(x)-\tilde{V}(x^{\prime})\right]}\right].$
(96)

Note that $K>0$ in order to have a meaningful stationary distribution, because
$P_{\text{s}}(-L)=Ke^{-\beta\tilde{V}(-L)}$. This implies that
$J_{\text{s}}>0$ if $\tilde{V}(L)<\tilde{V}(-L)$, in agreement with the
physical intuition. The sign of $J_{\text{s}}$ is reversed if the potential is
reflected through the midpoint of the interval, i.e. if we have a potential
$\tilde{V}_{2}(x)=\tilde{V}(-x)$. In that case, it is easy to see that
$P_{2}(x,t)=P(-x,t)$ is a solution of the Fokker-Planck equation with force
$-{\tilde{V}_{2}}^{\prime}(x)$. In the steady state,
$P_{2}^{\text{s}}(x)=P_{\text{s}}(-x)$ and thus
$J_{2,\text{s}}=-J_{\text{s}}$.

###  B.1 Conditional steady distributions

Let us consider the time evolution of the marginal distribution for the
control $P(c,t)$, which we obtain by integrating (15) over $x$:

$\displaystyle\partial_{t}P(c,t)=\\!\\!\sum_{k}\delta(t-t_{k}^{-})\\!\\!\int\\!\\!dx\Big{[}\Theta(c|x)\sum_{c^{\prime}\neq
c}P(x,c^{\prime},t)-\left[1-\Theta(c|x)\right]P(x,c,t)\Big{]}.$ (97)

This is not a closed equation, $C$ alone is not enough to characterise the
stochastic process. The delta functions express the fact that the marginal
distribution only changes, logically, at the control updates.

In the main text, we have introduced the notation $P^{(k)}(c)\equiv
P(c,t_{k}^{+})$, which gives the probability of the control values throughout
the whole time interval $I_{k}$. The constancy of $P^{(k)}(c)$ over the time
interval $I_{k}$ entails that the conditional probability $P(x,t|c)$ obeys the
same Fokker-Planck equation as $P(x,c,t)$ between control updates. Therefore,
$P(x,t|c)$ irreversible evolves towards the stationary solution
$P_{\text{s}}^{(c)}(x)$ of the Fokker-Planck equation corresponding to the
potential $V(x,c)$, which verifies

$\tilde{V}^{\prime}(x,c)P_{\text{s}}^{(c)}(x)+k_{B}T\,\partial_{x}P_{\text{s}}^{(c)}(x)=-\gamma
J_{\text{s}}^{(c)},$ (98)

with $J_{\text{s}}^{(c)}$ being the uniform steady current with the control
variable equal to $c$.

For later reference, we write here the expression for $P_{\text{s}}^{(c)}(x)$
and $J_{\text{s}}^{(c)}$ for the periodic boundary conditions in Eq. (16),
which are basically the transposition to the case considered here of the
formulas just derived above:

$P_{\text{s}}^{(c)}(x)=K(c)\left[e^{-\beta\tilde{V}(x,c)}+\frac{e^{-\beta\tilde{V}(\textasciiacute
L,c)}-e^{-\beta\tilde{V}(-L,c)}}{\int_{-L}^{L}dx^{\prime}e^{-\beta\left[\tilde{V}(L,c)-\tilde{V}(x^{\prime},c)\right]}}\int_{-L}^{x}dx^{\prime}e^{-\beta\left[\tilde{V}(x,c)-\tilde{V}(x^{\prime},c)\right]}\right],$
(99)

where $K(c)>0$ is a normalisation constant, such that
$\int_{-L}^{L}dx\,P_{\text{s}}^{(c)}(x)=1$. The corresponding steady current
is

$\beta\gamma
J_{\text{s}}^{(c)}=K(c)\frac{e^{-\beta\tilde{V}(L,c)}-e^{-\beta\tilde{V}(-L,c)}}{\int_{-L}^{L}dx\,e^{-\beta\left[\tilde{V}(L,c)-\tilde{V}(x,c)\right]}}.$
(100)

The average velocity of the particle in the steady state corresponding to
$\tilde{V}(x,c)$ is $\left\langle
v\right\rangle_{\text{s}}(c)=LJ_{\text{s}}^{(c)}$. In agreement with physical
intuition, the particle moves to the right in the steady state for
$\tilde{V}(x,c)$, i.e. $\left\langle v\right\rangle_{\text{s}}(c)>0$ (or
$J_{\text{s}}^{(c)}>0$), when $\tilde{V}(L,c)<\tilde{V}(-L,c)$.777In the
absence of external force, $F_{\text{ext}}=0$, the periodicity of the
potential $V(x,c)$ ensures that $J_{\text{s}}^{(c)}=0$ and the stationary
distributions have the canonical form,
$P_{\text{s}}^{(c)}(x)=P^{\text{eq}}(x|c)\propto\exp[-\beta V(x,c)]$.

In general, $P(x,t|c)$ does not reach $P_{\text{s}}^{(c)}(x)$ at the end of
the interval $I_{k}$. At that instant, the Fokker-Planck evolution is
interrupted by the action of the control, which changes discontinuously the
value of $P(x,c,t)$. It is only in the limit as $\Delta t_{m}\to\infty$ that
$P(x,t|c)$ reaches $P_{\text{s}}^{(c)}(x)$—see Sec. 5.1.

##  Appendix C Feedback flashing protocol

Here, we explicitly write the expression for the flashing protocol presented
in Sec. 6, with uncertainty $\Delta x$ in the measurement of the position. In
order to obtain an easier expression in terms of the modulo function
$\text{mod}(x,L)$, we shift the spatial interval by $+L$, i.e. we consider
periodic boundary conditions in $[0,2L]$. With this choice, for a generic saw-
tooth potential $V(x)$ with minimum at $x=a\in(0,2L)$, we have

$\Theta(1|x)=\begin{cases}1,&\text{if }\text{mod}(x-\Delta
x,L)<\text{mod}(x+\Delta x,L)\leq a,\\\ \frac{a-\text{mod}(x-\Delta
x,L)}{2\Delta x},&\text{if }\text{mod}(x-\Delta x,L)\leq a<\text{mod}(x+\Delta
x,L),\\\ 0,&\text{if }a<\text{mod}(x-\Delta x,L)\leq\text{mod}(x+\Delta
x,L),\\\ \frac{a+\text{mod}(x+\Delta x,L)-\text{mod}(x-\Delta x,L)}{2\Delta
x},&\text{if }\text{mod}(x+\Delta x,L)\leq a<\text{mod}(x-\Delta x,L),\\\
\frac{\text{mod}(x+\Delta x,L)}{2\Delta x},&\text{if }\text{mod}(x+\Delta
x,L)\leq a<\text{mod}(x-\Delta x,L),\\\ \frac{a}{2\Delta x},&\text{if
}a<\text{mod}(x+\Delta x,L)\leq\text{mod}(x-\Delta x,L).\end{cases}$ (101)

For the symmetric V-shaped potential considered in our numerical simulations,
$a=L$. Once we have specified $\Theta(1|x)$, it is straightforward to define
$\Theta(0|x)=1-\Theta(1|x)$.

##  Appendix D Eigenvalues and eigenfunctions of the Fokker-Planck equation
with the V-shaped potential

Let us consider the Green function $K(x,t|x^{\prime},c)$ for the Fokker-Planck
equation particularised for the V-shaped potential defined in Eq. (70). Then,
we have to solve

$\displaystyle\gamma\,\partial_{t}K(x,t|x^{\prime},c)=$
$\displaystyle\partial_{x}\left[c\,F\operatorname{sgn}(x)\,K(x,t|x^{\prime},c)\right]+\beta^{-1}\partial^{2}_{x}K(x,t|x^{\prime},c),$
(102)

with initial condition

$K(x,t=0|x^{\prime},c)=\delta(x-x^{\prime}),$ (103)

and periodic boundary conditions.

The Green function is given in general by Eq. (27) of the main text in terms
of the the eigenvalues and eigenfunctions of the Fokker-Planck equation with
operator $\mathcal{L}^{(c)}$, i.e. for the potential $V(x,c)$. The stationary
distributions $P_{\text{s}}^{(c)}(x)$, corresponding to the non-degenerate
eigenvalue $\lambda_{0}^{(c)}=0$, have been derived in B.1 for a general
potential $V(x,c)$. Particularising these expressions for our V-shaped
potential $V(x,c)=c\,F\absolutevalue{x}$, Eq. (76) of the main text is
obtained. The remainder of eigenvalues $\lambda_{n}^{(c)}>0$ are doubly
degenerate and, therefore, the Green function can be written, more
specifically, as

$\displaystyle
K(x,c|x^{\prime},t)=P_{\text{s}}^{(c)}(x)+\frac{1}{P_{\text{s}}^{(c)}(x^{\prime})}\sum_{n=1}^{\infty}e^{-\lambda^{(c)}_{n}t}\Big{[}\phi^{(c)}_{n,1}(x^{\prime})\phi^{(c)}_{n,1}(x)+\phi^{(c)}_{n,2}(x^{\prime})\phi^{(c)}_{n,2}(x)\Big{]}$
(104)

with eigenvalues

$\displaystyle\lambda_{0}^{(1)}=0,\quad\lambda_{n}^{(1)}=\frac{\beta
F^{2}}{4}+\frac{n^{2}\pi^{2}}{\beta L^{2}},$ $\displaystyle n=1,\dots,$ (105)
$\displaystyle\lambda_{n}^{(0)}=\frac{n^{2}\pi^{2}}{\beta L^{2}},$
$\displaystyle n=0,1,\dots,$ (106)

and eigenfunctions

$\displaystyle\phi^{(1)}_{0}(x)$ $\displaystyle=\frac{F\beta}{2(1-e^{-F\beta
L})}e^{-F\beta\absolutevalue{x}},$
$\displaystyle\phi^{(0)}_{0}(x)=\frac{1}{2L},$ (107a)
$\displaystyle\phi^{(1)}_{n,1}(x)$
$\displaystyle=\frac{\sqrt{\phi^{(1)}_{0}(x)}}{\sqrt{L}}\sin\left(\frac{n\pi
x}{L}\right),$
$\displaystyle\phi^{(0)}_{n,1}(x)=\frac{1}{\sqrt{2}L}\cos\left(\frac{n\pi
x}{L}\right),$ (107b) $\displaystyle\phi^{(1)}_{n,2}(x)$
$\displaystyle=\frac{2\sqrt{\phi^{(1)}_{0}(x)}}{\sqrt{L\beta\lambda^{(1)}_{n}}}\left[\frac{2n\pi}{L}\cos\left(\frac{n\pi
x}{L}\right)-F\beta\sin\left(\frac{n\pi\absolutevalue{x}}{L}\right)\right],$
$\displaystyle\phi^{(0)}_{n,2}(x)=\frac{1}{\sqrt{2}L}\sin\left(\frac{n\pi
x}{L}\right).$ (107c)

##  Appendix E Calculation of the coefficients $B_{cc^{\prime}}$

In Sec. 6, we have computed the coefficients $B_{cc^{\prime}}$ defined in Eq.
(59), for the particular protocol we have considered, with error $\Delta x$.
All the coefficients $B_{cc^{\prime}}$ are equal to $1/2$ in that case.
Interestingly, this result is more general and applies to symmetric potentials
and flashing protocols with the following structure:

$\displaystyle\Theta(1|x)=f_{1}(x)\,\eta(x)+(1-f_{1}(-x))\,\eta(-x),$ (108)

where $\eta(x)$ represents the Heaviside function, and $0\leq f_{1}(x)\leq
1$.888The latter condition ensures that $0\leq\Theta(1|x)\leq 1$. Of course,
$\Theta(0|x)=1-\Theta(1|x)=(1-f_{1}(x))\,\eta(x)+f_{1}(-x)\,\eta(-x)$. Let us
calculate $B_{1c}$, assuming that $\Theta(1|x)$ has the structure in Eq.
(108):

$\displaystyle B_{1c}$ $\displaystyle=\int
dx\,\Theta(1|x)P_{\text{s}}^{(c)}(x)=\int_{0}^{L}dx\,f_{1}(x)P_{\text{s}}^{(c)}(x)+\int_{-L}^{0}dx\,(1-f_{1}(-x))P_{\text{s}}^{(c)}(x)$
$\displaystyle=\int_{0}^{L}dx\,f_{1}(x)P_{\text{s}}^{(c)}(x)+\int_{0}^{L}dx\,(1-f_{1}(x))P_{\text{s}}^{(c)}(x)=\int_{0}^{L}dx\,P_{\text{s}}^{(c)}(x)=1/2,$

where we have employed that $P_{\text{s}}^{(c)}(x)=P_{\text{s}}^{(c)}(-x)$,
due to the symmetry of the potential. An analogous calculation shows that
$B_{0c}=1/2$, since $\Theta(0|x)=1-\Theta(1|x)$.

The class of flashing protocols in Eq. (108) has a neat physical
interpretation. For the V-shaped potential, we recall that the error-free case
corresponds to activating the potential for $x<0$. Therefore, Eq. (108) tells
us that the probability of wrongly switching on the potential is
$\Theta(1|x>0)=f_{1}(x)$ for $x>0$. Also, it tells us that the probability of
correctly switching on the potential is
$\Theta(1|x<0)=1-f_{1}(\absolutevalue{x})$. Therefore, the probability of
success—switching on the potential correctly—and the probability of
error—switching on the potential wrongly—are balanced, since both of them add
to one. Note that this condition is not necessary for $\Theta(c|x)$ to be a
well-defined probability, but it seems to be a reasonable requirement at the
physical level that is fulfilled by the protocol that we have analysed in this
paper. In Fig. 2, this condition is clearly illustrated, since the protocol is
odd with respect to the $y=1/2$ axis.

Finally, we would like to remark that the structure (108) is also present in
other “noisy” flashing protocols considered in the literature. For example,
the flashing protocol of Ref. [10] is written in our notation as

$\displaystyle\Theta(1|x)=p\,\eta(x)+(1-p)\,\eta(-x),$ (109)

i.e. $f_{1}(x)=p$, independent of $x$. Therefore, the error in this flashing
protocol is simpler than the one considered in this paper: the probability of
wrongly switching on the potential is equal to $p$, regardless of the position
of the particle. Note that in our flashing protocol with error in the
measurement $\Delta x$, the probability of wrongly switching on the potential
increases as the particle approaches the point $x=0$ at which the force
changes sign.

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