# On the efficiency of an autonomous dynamic Szilard engine
operating on a single active particle

Luca Cocconi<EMAIL_ADDRESS>Max Planck Institute for Dynamics and
Self-Organization, Göttingen 37073, Germany  Letian Chen  Department of
Mathematics and Centre of Complexity Science, Imperial College London, South
Kensington, London SW7 2BZ, United Kingdom

(August 31, 2024)

###### Abstract

The Szilard engine stands as a compelling illustration of the intricate
interplay between information and thermodynamics. While at thermodynamic
equilibrium, the apparent breach of the second law of thermodynamics was
reconciled by Landauer and Bennett’s insights into memory writing and erasure,
recent extensions of these concepts into regimes featuring active fluctuations
have unveiled the prospect of exceeding Landauer’s bound, capitalizing on
information to divert free energy from dissipation towards useful work. To
explore this question further, we investigate an autonomous dynamic Szilard
engine, addressing the thermodynamic consistency of work extraction and
measurement costs by extending the phase space to incorporate an auxiliary
system, which plays the role of an explicit measurement device. The
nonreciprocal coupling between active particle and measurement device
introduces a feedback control loop, and the cost of measurement is quantified
through excess entropy production. The study considers different measurement
scenarios, highlighting the role of measurement precision in determining
engine efficiency.

##  I Introduction

Starting with Maxwell’s seminal _Gedankenexperiment_ , in which the existence
of an entity capable of sorting individual molecules according to their
velocity would seemingly lead to a violation of the second law of
thermodynamics [1], the physical nature of information and its interplay with
thermodynamics has undergone much scrutiny. This interplay finds its most
vivid illustration in another thought experiment, this time concocted by
Szilard [2]. In Szilard’s _engine_ , a partition is inserted at the mid-plane
of a box containing a single gas particle upon a binary measurement of the
position of the particle relative to the midplane. The volume of the empty
half of the box is then reduced at no energetic cost, resulting, once the
partition is removed, in an increase in free energy of the one-particle gas
which is subsequently converted into up to $k_{B}T\ln 2$ Joules of useful work
via isothermal expansion. Later work by Landauer and Bennet [3, 4] offered a
resolution to the seeming paradox by noting that, particularly when operated
cyclically, all such information engines rely on the writing onto, storing in
and eventual erasure from physical memory, thus demanding irreversible
manipulations which are bound to generate entropy as a by-product. In short,
thermodynamic consistency is recovered upon expanding the phase space to
include the _daemon_ itself [5, 6, 7, 8, 9, 10]. Theoretical efforts to
clarify where precisely dissipation occurs continue to this day [11, 12, 13,
14], in part stimulated by experimental work [15, 16, 17, 18].

The traditional Szilard engine, in its various implementations, is assumed to
operate on systems coupled to an equilibrium thermal reservoir and it is
indeed the measurement-mediated rectification of thermal fluctuations induced
by this coupling that renders work extraction possible. It is thus natural to
wonder how such an engine would perform when allowed to operate on out-of-
equilibrium processes, e.g. active particles, which are subject to non-
negligible active fluctuations with macroscopic persistence times. Remarkably,
as demonstrated in recent theoretical [19] and experimental [16] works, such
activity allows for the violation of Landauer’s bound, potentially granting
access to efficiencies far exceeding the equilibrium limit. Qualitatively,
this can be understood as a consequence of information being used in this case
to redirect part of the free energy that would otherwise be dissipated as heat
(i.e. entropy production) into useful work, rather than to extract the latter
directly from the heat bath.

In particular, the authors of Ref. [19] introduced a hypothetical _dynamic_
Szilard engine which, rather than operating in a quasi-static regime, exploits
the finite correlation time of the velocity of an active Brownian particle
(ABP). Their protocol consists of repeatedly measuring the particle’s position
and direction of motion, subsequently placing a piston that will exert a force
opposite to the active self-propulsion, eventually resulting in positive work
against the former. The cost of measurement is estimated as
$\mathcal{M}\simeq-k_{B}T\ln(\delta/2)$ by analogy with the thermal case, with
$\delta$ the precision error of the position measurement normalized by the
system size. A unified stochastic thermodynamic treatment encompassing the
quantification of both work extraction and measurement costs in such active
information engines in the spirit of Refs. [5, 6, 7, 8, 9, 10], however, is
yet to be established.

Here, we develop such a treatment by studying a minimal Szilard engine
operating autonomously on a single run-and-tumble (RnT) particle in one
dimension, as schematised in Fig. 1. We ensure thermodynamic consistency by
expanding the phase space to explicitly include an auxiliary system playing
the role of the measurement device. The auxiliary system’s dynamics are
designed to induce a correlation between the state of the former and the
particle’s self-propulsion direction. Positive average work is then readily
extracted by applying a time-dependent force smaller than and opposite to the
particle’s estimated self-propulsion [20, 21, 22, 23]. The non-reciprocal
coupling between active particle and measurement device introduces the
feedback loop which underlies this information engine and the cost of
measurement can be computed analytically by evaluating the excess entropy
production of the joint dynamics [24, 25]. We consider in particular two
scenarios: in the first, discussed in Sec. II, the measurement device is a
binary process that couples directly to the RnT motility mode; in the second,
discussed in Sec. III, the motility mode is not accessible to direct
observation and the measurement device is a continuous state process that
couples to the particle position, from whose history the motility mode is
inferred. In both cases, the accuracy of the measurement plays a key role in
determining the engine’s efficiency.

Figure 1: Schematic representation of the dynamic Szilard engine considered in
this work. A single active particle undergoing run-and-tumble motion is
subject to a time-dependent force $F_{\rm ext}$ applied by an external
controller. The force is modulated based on the current state of auxiliary
processes $s$ or $Q$, which are coupled to the internal self-propulsion state
$w$ and particle position $x$, respectively. The nonequilibrium driving of the
auxiliary system results in an increase in the total mean entropy production
rate by an amount $\dot{S}_{\rm exc}$, which captures the thermodynamic cost
of operating this information engine.

##  II Direct measurement

Consider the one-dimensional run-and-tumble (RnT) process governed by the
Langevin equation for the positional coordinate

$\dot{x}(t)=\nu w(t)+\gamma^{-1}F_{\rm ext}(t)+\xi_{x}(t)$ (1)

with $\nu>0$ a self-propulsion speed, $w\in\\{-1,1\\}$ a symmetric dichotomous
noise characterised by a Poisson switching rate $\alpha$, and $\xi_{x}$ a
Gaussian white noise with Dirac-delta covariance,
$\langle\xi_{x}(t)\xi_{x}(t^{\prime})\rangle=2D_{x}\delta(t-t^{\prime})$.
Here, $F_{\rm ext}$ denotes an external force and we work in units such that
the viscosity $\gamma=1$. In practice, $F_{\rm ext}$ could be implemented
using an optical trap [16] or, for a charged active colloid [26], through an
external electric field of time-varying magnitude and direction. As a
preliminary step for the definition of a dynamic Szilard engine operating on a
single RnT particle, we expand the phase space to incorporate an auxiliary
system, playing the role of a measurement device, which is coupled
unidirectionally to the internal state $w$. We thus introduce a binary Markov
process $s\in\\{-1,+1\\}$ with $w$-dependent transition rate matrix
$\Xi_{s}(w(t))$,

$\Xi_{s}(t)=\begin{cases}m\begin{pmatrix}-\epsilon&1-\epsilon\\\
\epsilon&-(1-\epsilon)\end{pmatrix}&\text{if $w(t)=+1$}\\\\[15.0pt]
m\begin{pmatrix}1-\epsilon&\epsilon\\\
-(1-\epsilon)&-\epsilon\end{pmatrix}&\text{if $w(t)=-1$}\end{cases}$ (2)

acting on the vector of probabilities
$\mathbf{P}_{s}(t)=(P_{s=+1}(t),P_{s=-1}(t))$. The rate $m>0$ in Eq. (2)
defines the characteristic timescale of the auxiliary dynamics, while the
dimensionless parameter $\epsilon\in[0,1/2]$ can be understood as a
measurement error probability. This is a thermodynamically consistent
description, in the sense that
$\Delta\mu\equiv\beta^{-1}\ln[(1-\epsilon)/\epsilon]$, with
$\beta^{-1}=k_{B}T$ a thermal energy scale, can be interpreted as the free
energy drawn from some reservoir to bias the coupling, such that effective
decoupling is recovered in the limit $\epsilon\to 1/2$. Indeed, for
$\epsilon=1/2$, the auxiliary process $s(t)$ reduces to a symmetric
dichotomous noise.

The steady-state joint probability of $w$ and $s$, here denoted $\pi_{w,s}$,
can be computed straightforwardly. In particular, we have by symmetry that
$\pi_{1,1}=\pi_{-1,-1}$ and $\pi_{1,-1}=\pi_{-1,1}$ with
$\pi_{1,1}+\pi_{1,-1}=1/2$, whence

$\displaystyle\pi_{1,1}=\pi_{-1,-1}$
$\displaystyle=\frac{\alpha+m(1-\epsilon)}{2(2\alpha+m)}$
$\displaystyle\pi_{1,-1}=\pi_{-1,1}$
$\displaystyle=\frac{\alpha+m\epsilon}{2(2\alpha+m)}$ (3)

which reduces to $\pi_{\alpha,\beta}=1/4$ for all $\alpha,\beta\in\\{1,-1\\}$
in the decoupled limit, $\epsilon=1/2$, as expected.

We will assume the existence of an external controller tasked with the
application of the time-dependent external force $F_{\rm ext}(s(t))$,
henceforth referred to as the _protocol_ , which can be a function of the
current state of the measurement device $s(t)$, but not of the hidden self-
propulsion state $w(t)$.

###  II.1 Excess entropy production

In order to quantify the efficiency of this dynamic Szilard engine, we define
the operational cost of measurement as the excess entropy production rate [27,
28] induced by coupling of the auxiliary degree of freedom $s$ to the RnT
dynamics. In isolation, free RnT motion in one dimension is characterised by
an entropy production rate $\dot{\mathcal{S}}_{\rm RnT}=\nu^{2}/D_{x}$ [28].
Upon coupling of the auxiliary system, we write the entropy production rate as
the Kullback-Leibler divergence per unit time of the ensemble of forward
$(x,w,s)$ trajectories of duration $T$ and their time-reversed counterparts to
obtain

$\displaystyle\dot{\mathcal{S}}$
$\displaystyle\equiv\lim_{T\to\infty}\frac{1}{T}\left\langle\ln\frac{\mathbb{P}_{F}[x,w,s]}{\mathbb{P}_{R}[x,w,s]}\right\rangle$
$\displaystyle=\lim_{T\to\infty}\frac{1}{T}\left[\left\langle\ln\frac{\mathbb{P}_{F}[w,s]}{\mathbb{P}_{R}[w,s]}\right\rangle+\left\langle\ln\frac{\mathbb{P}_{F}[x|w,s]}{\mathbb{P}_{R}[x|w,s]}\right\rangle\right]\leavevmode\nobreak\
,$ (4)

with $\mathbb{P}_{F}$ and $\mathbb{P}_{R}$ denoting the corresponding path
probability densities. The first term in Eq. (4) corresponds to the average
entropy production rate of the four-state Markov process for the joint
dynamics of $w$ and $s$. Using Eqs. (2) and (3), this is given by

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\left\langle\ln\frac{\mathbb{P}_{F}[w,s]}{\mathbb{P}_{R}[w,s]}\right\rangle$
$\displaystyle=2[m\epsilon\pi_{1,1}-m(1-\epsilon)\pi_{1,-1}]\ln\frac{m\epsilon\pi_{1,1}}{m(1-\epsilon)\pi_{1,-1}}$
$\displaystyle\quad+2\alpha(\pi_{1,1}-\pi_{-1,1})\ln\frac{\pi_{1,1}}{\pi_{-1,1}}$
$\displaystyle=\frac{\alpha
m(1-2\epsilon)}{2\alpha+m}\ln\left(\frac{1-\epsilon}{\epsilon}\right)\leavevmode\nobreak\
.$ (5)

It vanishes at $\epsilon=1/2$, where $w$ and $s$ are independent equilibrium
processes, and diverges logarithmically as $\epsilon\to 0$, where state
transitions of $s$ become irreversible. The second term in Eq. (4) can be
computed within the Onsager-Machlup path integral formalism [28, 29]

$\displaystyle\mathbb{P}_{F}[x|w,s]$ $\displaystyle\propto{\rm
exp}\left[-\frac{1}{4D_{x}}\int_{0}^{T}dt\ \Big{(}\dot{x}-\nu w(t)-F_{\rm
ext}(s(t))\Big{)}^{2}\right]$ $\displaystyle\mathbb{P}_{R}[x|w,s]$
$\displaystyle\propto{\rm exp}\left[-\frac{1}{4D_{x}}\int_{0}^{T}dt\
\left(\dot{x}+\nu w(t)+F^{\dagger}_{\rm ext}(s(t))\right)^{2}\right]$ (6)

where $F_{\rm ext}^{\dagger}$ denotes the reversed protocol. Assuming we are
working on a ring, such that the marginal probability density $P(x)$ is
uniform at steady state, and using the dual-reversed convention [30, 27] for
the time-reversed protocol, $F_{\rm ext}^{\dagger}(s)=-F_{\rm ext}(s)$, this
term reduces to the entropy production $\dot{\mathcal{S}}_{\rm RnT}$ of a free
RnT particle. In this instance, the excess entropy $\dot{\mathcal{S}}_{\rm
exc}\equiv\dot{\mathcal{S}}-\dot{\mathcal{S}}_{\rm RnT}$ is thus simply given
by Eq. (5). The same result can be obtained by computing the entropy
production associated with $x$ in the reference frame co-transported by the
protocol, $\dot{x}_{c}\equiv\dot{x}-F_{\rm ext}$.

###  II.2 Naïve protocol

First, let us consider a naïve form of the protocol, where the controller
applies an external force _as if_ the value of the auxiliary system $s$ were
an exact copy of the hidden state $w$. It is straightforward to check that,
when $w$ is directly observable, the average power output
$\langle\dot{W}[F_{\rm ext}]\rangle\equiv-\langle F_{\rm
ext}(t)\dot{x}(t)\rangle$ is maximal for $F^{*}_{\rm ext}(w(t))=-w\nu/2$, in
which case $\dot{W}_{\rm max}\equiv\langle\dot{W}[F^{*}_{\rm
ext}]\rangle=\nu^{2}/4$ [23]. The naïve protocol is thus defined by $F_{\rm
ext}^{\rm(n)}(s(t))=-{s\nu}/{2}$, resulting in the average power output

$\langle\dot{W}[F_{\rm ext}^{\rm(n)}]\rangle=\frac{\nu^{2}}{4}\left[\langle
ws\rangle-\frac{1}{2}\langle s^{2}\rangle\right]=\dot{W}_{\rm
max}\left[\frac{m\left(\frac{1}{2}-\epsilon\right)-\alpha}{2\alpha+m}\right]\leavevmode\nobreak\
,$ (7)

where we have used

$\displaystyle\langle ws\rangle$
$\displaystyle=\pi_{1,1}+\pi_{-1,-1}-\pi_{1,-1}-\pi_{-1,1}=\frac{m(1-2\epsilon)}{2\alpha+m}$
(8)

and $s^{2}(t)=1$. The average power output (7) is a monotonically decreasing
function of $\epsilon$ and, for any finite switching rate $\alpha$, there is a
critical error probability $\epsilon_{c}=(m-2\alpha)/2m$ above which the power
output (7) becomes negative. Furthermore, even in the ideal case $\epsilon=0$,
no positive power can be extracted by the application of the naïve protocol
when $m<2\alpha$. The requirement for positive power extraction is more
stringent than demanding that the rate of measurement exceeds that of
tumbling, reflecting the asymmetry between positive and negative work
extracted upon correct and incorrect guessing, respectively.

Combining the results of Sec. II.1 with Eq. (7) for the average power output,
we find the following expression for the “informational” efficiency of the
conversion from entropy to work of the naïve protocol,

$\displaystyle\eta^{\rm(n)}\equiv\frac{\langle\dot{W}[F_{\rm
ext}^{\rm(n)}]\rangle}{\beta^{-1}\dot{\mathcal{S}}_{\rm exc}}$
$\displaystyle=\beta\dot{W}_{\rm
max}\frac{m\left(\frac{1}{2}-\epsilon\right)-\alpha}{\alpha
m(1-2\epsilon)\log\frac{1-\epsilon}{\epsilon}}$
$\displaystyle=\frac{\beta\dot{W}_{\rm
max}}{\log\frac{1-\epsilon}{\epsilon}}\left[\frac{1}{2\alpha}-\frac{1}{m(1-2\epsilon)}\right]$
(9)

where $\beta$ is introduced to make the efficiency dimensionless. Note that
the definition of the efficiency $\eta$ used in Eq. (9) and henceforth differs
from that of Ref. [19], which we denote $\tilde{\eta}$, in that the
dissipation of the free active particle is not included in the operational
cost. However, they can be related straightforwardly via
$\tilde{\eta}(1+\eta^{-1}+\beta^{-1}\dot{S}_{\rm
RnT}/\langle\dot{W}\rangle)=1$. We observe that $\eta^{\rm(n)}$ is maximal at
an intermediate value of $\epsilon$, i.e. at finite power, see Fig. 2, and
becomes negative for $\epsilon>\epsilon_{c}$. Since the self-propulsion speed
$\nu$ only features via $\dot{W}_{\rm max}\sim\nu^{2}$, the efficiency can be
made arbitrarily large at fixed temperature (constant $\beta$) by increasing
$\nu$, in a clear breach of Landauer’s bound for equilibrium information
engines.

###  II.3 Bayesian protocol

Within the context of direct coupling of the auxiliary process $s$ to the
self-propulsion state $w$, let us now consider a more accurate protocol, which
we dub the Bayesian protocol. It was shown in Ref. [23] that maximal average
power output from an active particle with hidden states is achieved for a
protocol proportional to the posterior expectation of the self-propulsion
velocity given the observable degrees of freedom. Denoting
$\pi_{w|s}=\pi_{w,s}/\pi_{s}$ the steady-state conditional probabilities,
which can be computed from Eq. (3), the Bayesian protocol is accordingly
defined by

$F^{\rm(b)}_{\rm
ext}(s(t))=-\frac{\nu}{2}(\pi_{1|s}-\pi_{-1|s})=-\frac{s\nu}{2}\frac{m(1-2\epsilon)}{2\alpha+m}\leavevmode\nobreak\
.$ (10)

Unlike the naïve protocol, Eq. (10) depends explicitly on the error
probability $\epsilon$, vanishing at $\epsilon=1/2$ and saturating to
$\pm\nu/2$ in the simultaneous limit $\epsilon\to 0$ , $m\to\infty$, at fixed
$\alpha$. The associated average power output is

$\displaystyle\langle\dot{W}[F^{\rm(b)}_{\rm ext}]\rangle$
$\displaystyle=\frac{\nu^{2}}{2}\left[\frac{m(1-2\epsilon)}{2\alpha+m}\langle
ws\rangle-\frac{m^{2}(1-2\epsilon)^{2}}{2(2\alpha+m)^{2}}\langle
s^{2}\rangle\right]$ $\displaystyle=\dot{W}_{\rm
max}\frac{m^{2}(1-2\epsilon)^{2}}{(2\alpha+m)^{2}}\leavevmode\nobreak\ ,$ (11)

where we used Eq. (8) to substitute for $\langle ws\rangle$. Once again,
$\dot{W}_{\rm max}$ denotes the power that could have been extracted were $w$
directly accessible to the controller, such that the fraction in the right-
hand side of Eq. (11) corresponds to the reduction in performance due to the
presence of the measurement device as an intermediary.

Using Eq. (5) for the excess entropy production, the efficiency of the
conversion from entropy to work of the Bayesian protocol is thus

$\eta^{\rm(b)}\equiv\frac{\langle\dot{W}[F^{\rm(b)}_{\rm
ext}]\rangle}{\beta^{-1}\dot{\mathcal{S}}_{\rm exc}}=\frac{\beta\dot{W}_{\rm
max}m(1-2\epsilon)}{\alpha(m+4\alpha)\log\frac{1-\epsilon}{\epsilon}}$ (12)

which is a monotonically increasing function of $\epsilon$, meaning that in
this instance efficiency is maximal in the limit $\epsilon\to 1/2$ and thus at
zero power, see Fig. 2. In particular, using

$\lim_{\epsilon\to\frac{1}{2}}(1-2\epsilon)\left[\log\left(\frac{1-\epsilon}{\epsilon}\right)\right]^{-1}=\frac{1}{2}$
(13)

we have the value of the max efficiency $\eta^{\rm(b)}_{\rm
max}/\beta\dot{W}_{\rm max}=m/[2\alpha(m+4\alpha)]$. Both work and efficiency
vanish in the limit $\alpha\to\infty$, consistent with the idea that the
auxiliary system can’t “keep up” with the $w$ dynamics when the latter’s
correlation time scale becomes too small. Similarly to the naïve protocol
studied in the previous Section, the efficiency (12) can be made arbitrarily
large at fixed temperature by increasing $\nu$.

Figure 2: Average power output as a fraction of maximum achievable power
(left) and efficiency (right) as a function of the error probability
$\epsilon\in[0,1/2]$ for protocols relying on direct coupling to the RnT
motility state $w$. Dotted and solid lines refer to results for the naïve and
Bayesian protocols, as introduced in Sec. II.2 and II.3, respectively. The
power output is a monotonically decreasing function of $\epsilon$ in both
cases, with the Bayesian protocol generically outperforming the naïve one. The
efficiency, on the other hand, displays a maximum at finite power for the
naïve protocol and at zero power for the Bayesian one. Here, we set
$\dot{W}_{\rm max}=1$ and $m=1$.

##  III Indirect measurement

In many realistic scenarios, establishing a direct coupling between a
measurement device and the internal degree of freedom controlling the self-
propulsion direction is unfeasible. For instance, while the position of a
molecular motor might be easily tracked, neither the chemical states of its
motor heads nor the polarisation of the cytoscheletal filament to which the
motor is bound are typically accessible to direct observation [31, 32]. In
light of this consideration, we now consider a scenario where only the history
of the position of the RnT particle is observable. Remarkably, it was shown in
Ref. [23] that it is still possible to design an auxiliary system dynamic,
such that unidirectional coupling of the latter to the instantaneous particle
velocity allows for positive power extraction, the intermediate step being the
inference of the hidden state.

Let us introduce the notation $\\{x(t)\\}_{-\infty}^{\tau}$ for the positional
trajectory of a RnT particle ending at time $t=\tau$, as well as an auxiliary
continuous process $Q[\\{x(t)\\}_{-\infty}^{\tau}]$ playing the role of a
measurement device, which couples unidirectionally to the co-transported
particle position. We take, in particular,

$\displaystyle\dot{Q}(t)$ $\displaystyle=\frac{\nu}{D_{x}}[\dot{x}(t)-F_{\rm
ext}(t)]-2\alpha Q+\tilde{\xi}_{Q}(t)$
$\displaystyle=\frac{\nu^{2}}{D_{x}}w(t)-2\alpha Q(t)+\xi_{Q}(t)$ (14)

where $\tilde{\xi}_{Q}(t)$ and $\xi_{Q}(t)$ are Gaussian white noises with
covariance
$\langle\tilde{\xi}_{Q}(t)\tilde{\xi}_{Q}(t^{\prime})\rangle=2D_{Q}\delta(t-t^{\prime})$
and
$\langle\xi_{Q}(t)\xi_{Q}(t^{\prime})\rangle=2(D_{Q}+\nu^{2}/D_{x})\delta(t-t^{\prime})$,
while $\dot{x}(t)$ was replaced in accordance with Eq. (1). Thus, $Q(t)$
amounts to a weighted time average of past velocities, with a memory kernel
that decays exponentially over a characteristic time scale $1/(2\alpha)$,
capturing the intuition that RnT velocities tend to be persistent and of the
same sign as $w$ over this time scale. The non-zero noise correlation
$\langle\xi_{Q}(t)\xi_{x}(t^{\prime})\rangle=2\nu\delta(t-t^{\prime})$ means
that this cannot be thought of as a bipartite system [33]. In fact, the limit
$D_{Q}\rightarrow 0$ of Eq. (14) can be obtained as the exact low-Péclet
number asymptote [here ${\rm Pe}\equiv\nu^{2}/(\alpha D_{x})$] of the dynamics
of the confidence parameter

$\mathfrak{Q}[\\{x\\}_{-\infty}^{\tau}]=\ln\frac{P(w(\tau)=+1|\\{x\\}_{-\infty}^{\tau})}{P(w(T)=-1|\\{x\\}_{-\infty}^{\tau})}\leavevmode\nobreak\
,$ (15)

where $P(v|\\{x\\}_{-\infty}^{\tau})$ denotes the posterior probability that
the internal self-propulsion state is currently $v\in\\{1,-1\\}$ given the
observed positional trajectory [23]. Rearranging Eq. (15) as

$P(w(\tau)=\pm
1|\\{x\\}_{-\infty}^{\tau})=\frac{1}{2}\pm\frac{e^{\mathfrak{Q}}-1}{2(1+e^{\mathfrak{Q}})}$
(16)

clarifies that there exists a one-to-one correspondence between these
complementary probabilities and the confidence parameter $\mathfrak{Q}$. The
confidence parameter in turn defines the optimal protocol via the relation

$F_{\rm
ext}^{*}(\mathfrak{Q}(t))=-\frac{\nu\tanh(\mathfrak{Q}/2)}{2}=-\frac{\nu\mathfrak{Q}}{4}+\mathcal{O}(\mathfrak{Q}^{2})\leavevmode\nobreak\
,$ (17)

which draws on the result, already mentioned in Sec. II, that maximal average
power output from an active particle with hidden states is achieved for a
protocol proportional to the posterior expectation of the self-propulsion
velocity given the observable degrees of freedom, which is in this case $x(t)$
[23].

While remaining a sensible intermediary observable to infer $w$, Eq. (14) for
$Q$ stops being an exact description of the dynamics of the real confidence
parameter $\mathfrak{Q}$ for ${\rm Pe}\gtrapprox 1$ and in general when
$D_{Q}>0$. In the following, we will nonetheless define the auxiliary dynamics
via Eq. (14) at all ${\rm Pe}$, on the basis that no obvious alternative is
available. Similarly, the need for a finite measurement noise $D_{Q}$ in order
to establish a well-defined thermodynamic picture of the joint dynamics will
become apparent shortly.

###  III.1 Excess entropy production

Following Sec. II.1 and in order to precisely quantify the efficiency of this
dynamic Szilard engine, we define the operational cost of measurement as the
excess entropy production rate induced by coupling the auxiliary degree of
freedom $Q$ to the RnT dynamics,

$\displaystyle\dot{\mathcal{S}}$
$\displaystyle\equiv\lim_{T\to\infty}\frac{1}{T}\left\langle\ln\frac{\mathbb{P}_{F}[x,w,Q]}{\mathbb{P}_{R}[x,w,Q]}\right\rangle$
$\displaystyle=\lim_{T\to\infty}\frac{1}{T}\left\langle\ln\frac{\mathbb{P}_{F}[x,Q|w]}{\mathbb{P}_{R}[x,Q|w]}\right\rangle\leavevmode\nobreak\
,$ (18)

where we have used the fact that a symmetric dichotomous noise $w$ is time-
reversal symmetric. We write the Onsager-Machlup path probability functional
[29] for the forward trajectories in the Stratonovich convention [34, 27] as

$\mathbb{P}_{F}[x,Q|w]\propto{\rm exp}\left[-\frac{1}{2}\int_{0}^{T}dt\
\mathbf{U}^{T}(t)\mathsf{C}^{-1}\mathbf{U}(t)\right]$ (19)

with

$\mathbf{U}(t)=\begin{pmatrix}\dot{x}-F_{\rm ext}(Q)-\nu w(t)\\\
\dot{Q}-\frac{\nu^{2}}{D_{x}}w(t)+2\alpha Q(t)\end{pmatrix}$ (20)

and

$\mathsf{C}^{-1}=\frac{1}{2D_{x}D_{Q}}\begin{pmatrix}D_{Q}+\frac{\nu^{2}}{D_{x}}&-\nu\\\
-\nu&D_{x}\end{pmatrix}$ (21)

the inverse covariance matrix. Similarly, we have for the time-reversed path
probability

$\mathbb{P}_{R}[x,Q|w]\propto{\rm exp}\left[-\frac{1}{2}\int_{0}^{T}dt\
\mathbf{U}_{R}^{T}\mathsf{C}^{-1}\mathbf{U}_{R}\right]$ (22)

with

$\mathbf{U}_{R}=\begin{pmatrix}\dot{x}+F^{\dagger}_{\rm ext}(Q)+\nu w(t)\\\
\dot{Q}+\frac{\nu^{2}}{D_{x}}w(t)-2\alpha
Q(t)\end{pmatrix}\leavevmode\nobreak\ .$ (23)

Using the dual-reversed convention [30, 27] for the time-reversed protocol,
$F_{\rm ext}^{\dagger}(Q)=-F_{\rm ext}(Q)$, and after some algebra, we find

$\dot{S}=\frac{\nu}{D_{x}}\langle(\dot{x}-F_{\rm
ext})w\rangle+\frac{2\nu\alpha}{D_{x}D_{Q}}\langle Q(\dot{x}-F_{\rm
ext})\rangle-\frac{2\alpha}{D_{Q}}\langle\dot{Q}Q\rangle\leavevmode\nobreak\
.$ (24)

The first term can be identified as the entropy production rate of the free
run-and-tumble particle, $\dot{S}_{\rm RnT}$. The remaining two contributions
quantify instead the excess entropy originating purely from the coupling to,
and nonequilibrium dynamics of, the auxiliary system. Using the steady-state
correlations $\langle wQ\rangle={\rm Pe}/4$ and $\langle
Q^{2}\rangle=D_{Q}/(2\alpha)+{\rm Pe}/2+{\rm Pe}^{2}/8$, which are derived in
Appendix A based on Ref. [35], we finally arrive at the simple expression for
the excess entropy production

$\dot{S}_{\rm exc}=\frac{\alpha^{2}{\rm Pe}}{D_{Q}}\left(2+\frac{\rm
Pe}{2}\right)+2\alpha\leavevmode\nobreak\ .$ (25)

Note that the excess entropy diverges as $D_{Q}\to 0$, similarly to the limit
$\epsilon\to 0$ of the direct measurement setup, Eq. (5), indicating a
vanishing efficiency at finite temperature for the optimal protocol studied in
Ref. [23]. Unexpectedly, Eq. (25) does not vanish in the limit
$D_{Q}\to\infty$ of infinite measurement noise. We rationalise this result by
observing that the correlator $\langle wQ\rangle$ remains finite in this
limit, indicating the persistence of a nonequilibrium information flow
generated by the coupling between $x$ and $Q$. Finally, unlike Eq. (5), the
excess entropy for the case of indirect measurement depends on $\nu$ via the
Péclet number.

###  III.2 Naïve protocol

As was done in Sec. II.2 for the case of direct measurement, we start with a
naïve protocol, meaning in this case one that treats the auxiliary system $Q$
as an exact copy of the confidence parameter $\mathfrak{Q}$ of Eq. (15),
ignoring the measurement noise $D_{Q}$. Accordingly, the naïve protocol is
defined via Eq. (17) as $F^{(n)}_{\rm ext}(Q(t))=F^{*}_{\rm
ext}(\mathfrak{Q}=Q(t))$. The associated average power output is

$\displaystyle\langle\dot{W}[F_{\rm ext}^{(n)}]\rangle$
$\displaystyle=\left\langle\dot{x}(t)\frac{\nu\tanh[Q(t)/2]}{2}\right\rangle\leavevmode\nobreak\
.$ (26)

The expectation in the right-hand side of Eq. (26) can be computed in closed
form in the low $\rm Pe$, low $D_{Q}$ asymptote (specifically
$D_{Q}/\alpha\ll{\rm Pe}\ll 1$). Indeed, we show in Appendix A that $\langle
Q^{2}\rangle=(D_{Q}/\alpha+{\rm Pe})/2+\mathcal{O}({\rm Pe}^{2})$, thus in
this limit $Q\sim{\rm Pe}$ is typically small and the protocol can be
linearised to give

$\displaystyle\langle\dot{W}[F_{\rm ext,lin}^{(n)}]\rangle$
$\displaystyle=\left\langle\dot{x}\frac{\nu Q}{4}\right\rangle$
$\displaystyle=\dot{W}_{\rm max}\left[\frac{\rm Pe}{8}\left(1-\frac{\rm
Pe}{4}\right)-\frac{D_{Q}}{8\alpha}\right]\leavevmode\nobreak\ .$ (27)

This result, which is exact for all $\rm Pe$ and $D_{Q}$ but approximates (26)
only when $D_{Q}/\alpha\ll{\rm Pe}\ll 1$, shows that the fraction of the
available power that is actually extracted upon application of the linearised
naïve protocol is a monotonically decreasing function of the measurement noise
$D_{Q}$, vanishing at a finite critical value $D_{Q,c}=\alpha{\rm Pe}(1-{\rm
Pe}/4)$ of the latter. This is also the case for the average power output of
the full nonlinear protocol (26), as shown in Fig. 3 (left column).
Furthermore, the power extracted upon application of the linearised protocol
eventually becomes negative as $\rm Pe$ is increased. This is not the case for
the power output of the full nonlinear protocol (26), which instead approaches
from above a finite positive value at large $\rm Pe$, see Fig. 3 (left
column).

Combining Eqs. (25) and (26) we finally obtain the efficiency $\eta^{(b)}$ of
the conversion from entropy to work of the naïve protocol, plotted in Fig. 3
(right column). For the linearised protocol, the dimensionless efficiency can
be written in closed form,

$\eta^{(n)}_{\rm lin}=\frac{\beta\dot{W}_{\rm max}\left[\frac{\rm
Pe}{8}\left(1-\frac{\rm
Pe}{4}\right)-\frac{D_{Q}}{8\alpha}\right]}{2\alpha\left[\frac{\alpha{\rm
Pe}}{D_{Q}}\left(1+\frac{\rm Pe}{4}\right)+1\right]}\leavevmode\nobreak\ ,$
(28)

and retains a nonmonotonic dependence on $\rm Pe$ and $D_{Q}$. Unlike the case
of direct measurement studied in Sec. II, here the self-propulsion speed $\nu$
enters both via $\dot{W}_{\rm max}\sim\nu^{2}$ and ${\rm Pe}\sim\nu^{2}$ and
it is thus not immediately obvious whether $\eta^{(n)}_{\rm lin}$ is bounded
above. Assuming that the Stokes-Einstein relation applies,
$D_{x}=k_{B}T/\gamma$, we can rewrite the efficiency (28) as a function of the
dimensionless parameters $\rm Pe$ and $D_{Q}/\alpha$ only. The resulting
expression is plotted in Fig. 4 and shows a maximum at $\rm Pe=2.159...$ and
$D_{Q}/\alpha=2.797...$, where $\eta^{(n)}_{\rm lin}\simeq 8\times 10^{-2}$,
indicating the existence of a nonequilibrium upper bound analogous to that
established by Landauer for equilibrium information engines.

Figure 3: Average power output (left column) and efficiency (right column) as
a function of Péclet number ${\rm Pe}\equiv\nu^{2}/(\alpha D_{x})$ and
measurement error $D_{Q}\in\mathbb{R}^{+}$ for the naïve protocol relying on
inference of the RnT motility state $w$ introduced in Sec. III.2, setting
$\alpha=\beta=1$. Solid lines denote results for the full nonlinear protocol,
Eq. (26) and are obtained by numerical integration. Dashed lines are closed-
form analytical results for the linearised protocol, Eqs. (27) and (28). The
linearised protocol performs similarly to the full protocol in the regime
$D_{Q}/\alpha<{\rm Pe}\ll 1$, as expected. In both cases, we observe a non-
monotonic dependence of the efficiency on both ${\rm Pe}$ and $D_{Q}$. The
black dashed line in the top-right panel indicates the maximum extractable
power under the constraint of a hidden self-propulsion state obtained in Ref.
[23].  Figure 4: Color map of the efficiency $\eta^{(n)}_{\rm lin}$ of Eq.
(28) as a function of the dimensionless parameters $\rm Pe$ and $D_{Q}/\alpha$
for the linearised naïve protocol with indirect measurement discussed in Sec.
III.2. The dashed line indicates the contour $\eta^{(b)}_{\rm lin}=0$. The
white cross indicates the location of the global maximum, for $\rm
Pe=2.159...$ and $D_{Q}/\alpha=2.797...$, where $\eta^{(n)}_{\rm lin}\simeq
0.0794$.

While no closed-form expression is available for the efficiency of the fully
nonlinear protocol, it is nevertheless possible to argue that $\eta^{(b)}$ can
be made arbitrarily large at fixed temperature. In particular, it is clear
that Eq. (26) for the average power output is bounded above by $\dot{W}_{\rm
max}$ and we further expect $\langle\dot{W}[F_{\rm ext}^{(n)}]\rangle$ to
approach a finite fraction of $\dot{W}_{\rm max}$ which is independent of
$D_{Q}/\alpha$ as $\rm Pe\to\infty$, consistently with numerical results in
Fig. 3. Thus, for ${\rm Pe}\gg 1$,

$\eta^{(n)}\propto\left[\frac{\alpha}{D_{Q}}\left(1+\frac{\rm
Pe}{4}\right)+{\rm Pe}^{-1}\right]^{-1}\leavevmode\nobreak\ .$ (29)

Taking the limits ${\rm Pe}\to\infty$ and $\alpha{\rm Pe}/D_{Q}\to 0$
simultaneously produces a divergent efficiency, demonstrating that
$\eta^{(n)}$ is unbounded in $({\rm Pe},D_{Q}/\alpha)\in\mathbb{R}^{2}$.

Figure 5: Average power output (left column) and efficiency (right column) as
a function of Péclet number ${\rm Pe}\equiv\nu^{2}/(\alpha D_{x})$ and
measurement error $D_{Q}\in\mathbb{R}^{+}$ for the Bayesian protocol relying
on inference of the RnT motility state $w$ introduced in Sec. III.3, Eq. (32).
The results are obtained by numerical integration of exact expressions,
setting $\alpha=\beta=1$, for which no closed form is available. Similarly to
the naïve protocol, we observe a non-monotonic dependence of the efficiency on
both ${\rm Pe}$ and $D_{Q}$. The black dashed line in the top-left panel
indicates the maximum extractable power under the constraint of a hidden self-
propulsion state (data from Ref. [23]).

###  III.3 Bayesian protocol

Completing the parallel with Sec. II, we now consider a more accurate
“Bayesian” protocol designed to account and partially compensate for the
measurement noise parametrised by $D_{Q}$. Rather than treating the auxiliary
variable $Q$ as a copy of the confidence parameter $\mathfrak{Q}$ of Eq. (15),
which is related to the posterior probability $P(w(\tau)|Q(\tau))$ of the
self-propulsion mode via the simple relation (16), we compute the true self-
propulsion probabilities conditioned on $Q$ explicitly. In particular, we
write

$\displaystyle P(w(\tau)=\pm 1|Q(\tau))$ $\displaystyle=\int
d\mathfrak{Q}P(w(\tau)=\pm 1|\mathfrak{Q})P(\mathfrak{Q}|Q)$
$\displaystyle=\int
d\mathfrak{Q}\left[\frac{1}{2}\pm\frac{e^{\mathfrak{Q}}-1}{2(1+e^{\mathfrak{Q}})}\right]\frac{P(Q|\mathfrak{Q})P(\mathfrak{Q})}{P(Q)}$
(30)

where we have used Bayes’ theorem together with Eq. (16). By inspection of the
Langevin equation (14), it is clear that the steady state probability $P(Q)$
is identical to that of the position of a RnT particle confined in a harmonic
potential of stiffness $2\alpha$, which is known in closed form from Refs.
[35, 36], see Appendix A. On the other hand, neither $P(Q|\mathfrak{Q})$ nor
$P(\mathfrak{Q})$ are known analytically. Nevertheless, we can draw on the
result of Ref. [23] that the dynamics of $\mathfrak{Q}$ are well approximated
at low $\rm Pe$ by Eq. (14) with $D_{Q}=0$ to write $P(\mathfrak{Q})\simeq
P(\tilde{Q})$ and $P(Q|\mathfrak{Q})\simeq P(Q|\tilde{Q})$, where

$Q(t)=\tilde{Q}(t)+\int_{-\infty}^{\tau}dt\
\tilde{\xi}_{Q}(t)e^{-2\alpha(\tau-t)}\leavevmode\nobreak\ .$ (31)

The second term in the right-hand side of Eq. (31) is simply an Ornstein-
Uhlenbeck process [37] with zero mean and variance $D_{Q}/2\alpha$, implying
$P(Q|\tilde{Q})=\mathcal{N}(Q;\tilde{Q},D_{Q}/2\alpha)$, where
$\mathcal{N}(\bullet;m,v)$ denotes a Gaussian distribution of mean $m$ and
variance $v$. Our Bayesian protocol is then defined as

$\displaystyle F_{\rm ext}^{\rm(b)}(Q(t))$
$\displaystyle=-\frac{\nu}{2}[P(w(t)=+1|Q(t))-P(w(t)=-1|Q(t))]$
$\displaystyle=-\frac{\nu}{2}\int
d\tilde{Q}\left(\frac{e^{\tilde{Q}}-1}{e^{\tilde{Q}}+1}\right)\mathcal{N}(Q;\tilde{Q},D_{Q}/2\alpha)\frac{P(\tilde{Q})}{P(Q)}\leavevmode\nobreak\
.$ (32)

The average power output associated with this protocol can be computed by
numerical evaluation of the exact integral expressions, giving the curves
shown in Fig. 5 (left column). These results can then be combined with Eq.
(25) for the excess entropy production to compute the efficiency
$\eta^{\rm(b)}$, also shown in Fig. 5 (right column). The non-monotonic
dependence of $\eta^{\rm(b)}$ on ${\rm Pe}$, which reflects the onset of a
regime where measurement expenditure exceeds power output, is reminiscent of
recent results in quantum Szilard engine with finite-time measurement [38] and
partially observable information engines [39]. While no closed-form expression
can realistically be obtained for the average power output associated with the
protocol (32), a similar logic to the one presented in the previous section
can be used to argue for the corresponding efficiency to be unbounded in
$({\rm Pe},D_{Q}/\alpha)\in\mathbb{R}^{2}$. Indeed, such an argument relies
solely on the assumption, consistent with our numerical results, that the
average power output should tend to a finite constant independent of
$D_{Q}/\alpha$ as $\rm Pe\to\infty$.

##  IV Conclusion

We have addressed the thermodynamic consistency of an autonomous, dynamic
Szilard engine [19] operating on a RnT particle in one dimension, a canonical
model in active matter physics reminiscent of the motility strategy of
_E.Coli_ and _Salmonella_ bacteria [40, 36, 41, 42, 43]. We have considered
two realistic, i.e. error-prone, measurement scenarios: in the first, binary
measurements are performed directly on the internal self-propulsion state of
the active particle, while in the second only the particle trajectory in space
is accessible and the self-propulsion state is estimated by means of inference
from weighted time-averages of past velocities [23]. In each scenario, we
contrast a naïve version of the protocol, where measurement errors are
neglected, with a Bayesian version, where knowledge of the error statistics is
drawn upon to generically enhance efficiency. The thermodynamic cost of
operating the measurement device is quantified formally as the excess entropy
production rate induced by the coupling of an auxiliary process playing the
role of a measurement device, which we compute analytically and in closed
form. We showed that, in both cases, the precision of measurements has a
nontrivial impact on the efficiency of the engine. With the exception of the
linearised naïve protocol studied in Sec. III.3, we observed that in all
instances the efficiency of our dynamic Szilard engine can be made arbitrarily
large at constant temperature by increasing the dissipation of the free RnT
particle, consistent with recent results [19] where operational costs were
estimated in a more heuristic way. We understand this breach of Landauer’s
equilibrium bound as a consequence of information being used to redirect part
of the intrinsic internal dissipation originating from activity into useful
work, rather than to extract the latter directly from a heat bath with an
infinitesimal correlation time scale.

The analysis presented here could be refined by considering the efficiency of
a similarly designed autonomous information engine operating on a
thermodynamically consistent active particle model, such as those introduced
in Refs. [44, 45, 46, 47], which typically involve a weak coupling between
configurational/chemical and mechanical degrees of freedom, with multiple
dissipative currents, rather than describing self-propulsion via an effective
external “active force” [48, 49].

Both LCs acknowledge Connor Roberts, Yu-Han Ma and Henry Alston for useful
feedback on the manuscript.

##  Appendix A Statistics of RnT motion in a harmonic trap

The authors of Ref. [35] have obtained a simple integral expression for the
steady-state probability density of a generic active particle in a harmonic
potential, on which we draw in various places of this work for the particular
case of RnT motion and thus report here for convenience. In particular, they
considered an Ornstein-Uhlenbeck process with time-dependent potential
described by the Langevin equation

$\dot{x}(t)=-\frac{k}{\gamma}\left[x(t)-c(t)\right]+\xi(t)\leavevmode\nobreak\
$ (33)

where $k$ is the stiffness of the harmonic potential $V(x,c)=k(x-c)^{2}/2$
centered at $c(t)$, $\gamma$ denotes the effective friction coefficient and
$\xi(t)$ is a Gaussian white noise with zero mean and covariance
$\langle\xi(t_{1})\xi(t_{2})\rangle=2D\delta(t_{1}-t_{2})$. By the Stokes-
Einstein relation, the effective diffusion coefficient satisfies
$D=k_{B}T/\gamma$. RnT motion is captured by Eq. (33) when centre $c(t)$ is a
symmetric dichotomous process taking the values $\pm c_{0}$ with switching
rate $r$. With these definitions and setting $\gamma=1$ without loss of
generality, the stationary conditional distributions of $x$, given
$\sigma=c/c_{0}=\pm 1$, reads

$\displaystyle P(x|\sigma)$ $\displaystyle=\frac{N}{2}\int_{-1}^{+1}dz\
\mathcal{N}\left(x;c_{0}z,\frac{D}{\nu}\right)(1-\sigma z)^{r/\nu-1}(1+\sigma
z)^{r/\nu}\leavevmode\nobreak\ ,$ (34)

where $\mathcal{N}(\bullet;m,v)$ once again denotes a Gaussian distribution of
mean $m$ and variance $v$ and

$\displaystyle N^{-1}$
$\displaystyle=\frac{2\nu}{r}{}_{2}F_{1}\left(1,1-\frac{r}{\nu},1+\frac{r}{\nu},-1\right)$
(35)

is a normalisation factor. Here, $\nu={k}/{\gamma}$, while ${}_{2}F_{1}$
denotes the hypergeometric function.

For the $Q$ dynamics of Eq. (14), we thus have, by direct comparison with (33)
and incorporating results of Eqs. (34) and (35), the steady-state conditional
probability density

$\displaystyle P(Q|w)=\frac{N_{Q}}{8}\int_{-1}^{+1}dz\ \mathcal{N}$
$\displaystyle\left(Q;\frac{\nu^{2}}{2\alpha
D_{x}}z,\frac{1}{2\alpha}\left(\frac{\nu^{2}}{D_{x}}+D_{Q}\right)\right)$
$\displaystyle\qquad\times(1-wz)^{-\frac{1}{2}}(1+wz)^{\frac{1}{2}}$ (36)

with $N_{Q}^{-1}={}_{2}F_{1}(1,1/2,3/2,-1)$. Using Eq. (36) is is
straightforward to derive the expectations

$\langle Q^{2}\rangle=\frac{D_{Q}}{2\alpha}+\frac{\rm Pe}{2}\left(1+\frac{\rm
Pe}{4}\right),\quad\langle wQ\rangle=\frac{\rm Pe}{4}\leavevmode\nobreak\ .$
(37)

The corresponding statistics of $\tilde{Q}$, as introduced in Eq. (31), are
obtained from the above by simply setting $D_{Q}=0$. Equivalent results have
been obtained in Ref. [36], albeit in a different representation.

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