# Simple calculation of the Coulomb-nuclear corrections in $pp$ and $\bar{p}p$
scattering

Loyal Durand<EMAIL_ADDRESS>Present address: 415 Pearl Court, Aspen, CO
81611  Department of Physics University of Wisconsin-Madison, Madison, WI
53706  Phuoc Ha<EMAIL_ADDRESS>Department of Physics, Astronomy, and
Geosciences, Towson University, Towson, MD 21252

###### Abstract

We present a very simple method for calculating the mixed Coulomb-nuclear
effects in the $pp$ and $\bar{p}p$ scattering amplitudes, and illustrate the
method using simple models frequently used to describe their differential
cross sections at small momentum transfers. Combined with the pure Coulomb and
form-factor contributions to the scattering amplitude which are known
analytically from prior work, and the unmixed nuclear or strong-interaction
scattering amplitude, the results give a much simpler approach to fitting the
measured $pp$ and $\bar{p}p$ cross sections and extracting information on the
real part of the forward scattering amplitudes than methods now in use.

##  I Introduction

The effect of the Coulomb interaction in high-energy proton-proton and
antiproton-proton scattering has been studied by many authors; see [1, 2, 3,
4, 5, 6, 7, 8, 9] and the many further references therein. A primary objective
has been the use of Coulomb-nuclear interference effects to determine the
ratios $\rho$ of the real to imaginary parts of the $pp$ and $\bar{p}p$
scattering amplitudes in the forward direction. The only other direct
information on the real parts of the amplitudes is that obtained at much
larger angles very near the observed dips in the differential cross sections.
These dips are associated with diffraction zeros in the imaginary parts of the
amplitudes where the scattering is dominated by the real part [10].

The most commonly used method for calculating the Coulomb-nuclear effects in
data analyses appears to be that of Cahn [3] as later modified by Kundrát and
M. Lokajiček [5]. This method is based on the use of the Fourier-Bessel
convolution theorem to calculate the corrections that involve Coulomb and
nuclear interactions simultaneously, and to include the effects of the nucleon
charge form factors. The results, which involve delicate manipulations in
their derivation to avoid singularities associated with the infinite range of
the Coulomb interaction, involve further complications in the subsequent
evaluation of convolutions, and are not transparent.

We show here that the mixed Coulomb-nuclear correction term can in fact be
calculated very simply as the rapidly-convergent double integral

$f_{N,c}^{\,\rm
Corr}(s,q^{2})=\int_{0}^{\infty}dbb\left[\left(e^{2i\delta^{\prime}_{c}(b,s)+2i\delta_{c}^{FF}(b,s)}-1\right)\times\int_{0}^{\infty}dq^{\prime}q^{\prime}f_{N}(s,q^{\prime
2})J_{0}(q^{\prime}b)\right]J_{0}(qb),$ (1)

where the inner integration on $q^{\prime}$ must be performed first, as
indicated. Here $\delta^{\prime}_{c}(b,s)$ is the pure Coulomb phase shift or
eikonal function after an impact-parameter independent term $i\eta\ln{(2p/q)}$
is extracted, and $\delta_{c}^{FF}(b,s)$ is the phase shift associated with
the effects of the charge form factors of the nucleons. $f_{N}(b,s)$ is the
nuclear scattering amplitude. This also appears as a separate term in the full
amplitude as shown below. The remaining Coulomb and form-factor-associated
terms in the amplitude are known analytically [8].

In a typical analysis such as that in [11], the sum of a model nuclear
amplitude $f_{N}$ and the Coulomb amplitude modified by form factors is used
to fit the data with the model $f_{N}$ also used to calculate the rather
complicated mixed Coulomb, nuclear, and form-factor terms following Cahn [3]
or Kundrát and M. Lokajiček [5]. The fit is then adjusted as necessary to
account for those corrections. Our approach is similar, but with the important
differences that we use analytically known expressions [8] for the combined
Coulomb and form-factor contribution, write the amplitude in a form that
isolates a pure nuclear term, and write the remaining mixed Coulomb-nuclear
term as in Eq. (1). As we show by example, that correction is small and easily
calculated, significantly simplifying the analysis.

##  II Theoretical background

In the absence of significant spin effects, generally thought to be very small
at high energies, the spin-averaged differential cross section for proton-
proton scattering can be written in terms of a single spin-independent
amplitude

$f(s,q^{2})=i\int_{0}^{\infty}db\,b\left(1-e^{2i\delta_{tot}(b,s)}\right)J_{0}(qb).$
(2)

The total phase shift $\delta_{tot}$ the sum of terms $\delta_{c}$ for pure
Coulomb scattering, $\delta_{c}^{FF}$ for the effects of the charge form
factors of the proton, and $\delta_{N}$ for the strong-interaction or
“nuclear” scattering,

$\delta_{tot}(b,s)=\delta_{c}(b,s)+\delta_{c}^{FF}(b,s)+\delta_{N}(b,s).$ (3)

Here

$\delta_{c}(b,s)=\eta(\ln{pb}+\gamma)$ (4)

where $\gamma=0.5772\ldots$ is Euler’s constant,
$\eta=z_{1}z_{2}\alpha=\alpha$ $(-\alpha)$ for $pp$ $(\bar{p}p)$ scattering,
and [12]

$\delta_{c}^{FF}(b,s)=\sum_{m=0}^{3}\frac{\eta}{2^{m}\Gamma(m+1)}(\mu
b)^{m}K_{m}(\mu b)$ (5)

for the standard proton charge form factor

$F_{Q}(q^{2})=\frac{\mu^{4}}{\left(q^{2}+\mu^{2}\right)^{2}}$ (6)

with $\mu^{2}=0.71$ GeV2.

With our normalization, the differential scattering cross section is

$\frac{d\sigma}{dq^{2}}=\pi\lvert f(s,q^{2})\rvert^{2},$ (7)

where $q^{2}=-t$ is the square of the invariant momentum transfer and
$W=\sqrt{s}$ the total energy in the center-of-mass system.

As shown in [10], an overall momentum-dependent phase factor
$(4p^{2}/q^{2})^{i\eta}$ which does not affect the differential cross section
can be extracted from the scattering amplitude, and the remaining amplitude
written in the form

$f(s,q^{2})=f^{\prime}_{c}(s,q^{2})+f_{c}^{FF}(s,q^{2})+f_{N}(s,q^{2})+f_{N,c}^{\,\rm
Corr}(s,q^{2}),$ (8)

with $\delta_{c}\rightarrow\delta^{\prime}_{c}$ now given in Eq. (3) by

$\delta^{\prime}_{c}(b,s)=\eta(\ln{(qb/2)}+\gamma).$ (9)

The Coulomb and form-factor terms combine as shown in [8], Sec. IIC, to give

$f_{c}(s,q^{2})+f_{c}^{FF}(s,q^{2})=-\frac{2\eta}{q^{2}}\left[1-\left(\frac{q^{2}}{q^{2}+\mu^{2}}\right)^{i\eta}\left(1-\frac{\mu^{8}}{(q^{2}+\mu^{2})^{4}}\right)+O(\eta)\right],$
(10)

with the overall phase $\left(4p^{2}/q^{2}\right)^{i\eta}$ which appears in
Eq. (21) of [8] removed.

The purely nuclear amplitude, which is to be determined from fits to
scattering data, is

$f_{N}(s,q^{2})=i\int_{0}^{\infty}dbb\left(1-e^{2i\delta_{N}(b,s)}\right)J_{0}(qb).$
(11)

Finally,

$f_{N,c}^{\,\rm
Corr}(s,q^{2})=\int_{0}^{\infty}dbb\left(e^{2i\delta^{\prime}_{c}(b,s)+2i\delta_{c}^{FF}(b,s)}-1\right)\times
i\left(1-e^{2i\delta_{N}(b,s)}\right)J_{0}(qb)$ (12)

is the mixed Coulomb-nuclear term.

We note that the last factor in Eq. (12) is just the integrand for $f_{N}$ in
Eq. (11), so may be evaluated as the inverse Fourier-Bessel transform
$\tilde{f}_{N}(b,s)$ of the nuclear amplitude $f_{N}$ [13],

$\tilde{f}_{N}(b,s)=\int_{0}^{\infty}dqqf_{N}(s,q^{2})J_{0}(qb)=i\left(1-e^{2i\delta_{N}(b,s)}\right).$
(13)

Thus,

$f_{N,c}^{\,\rm
Corr}(s,q^{2})=\int_{0}^{\infty}dbb\left(e^{2i\delta^{\prime}_{c}(b,s)+2i\delta_{c}^{FF}(b,s)}-1\right)\tilde{f}_{N}(b,s).$
(14)

The key observation is that $\tilde{f}_{N}(b,s)$ can be determined for any
successful phenomenological model for $f_{N}(s,q^{2})$ by performing the
inverse transform in Eq. (13). This can be calculated analytically for the
exponential-type models in $q^{2}$ commonly used in fitting the $pp$ and
$\bar{p}p$ data at high energies and small momentum transfers, and some other
models as well, giving simple expressions that make the calculation of the
Coulomb-nuclear correction straightforward by numerical evaluation of the
remaining rapidly-convergent integral. We will consider some examples in Sec.
III.

The mixed Coulomb-nuclear correction can also be evaluated efficiently
numerically for models in which $\tilde{f}_{N}(b,s)$ cannot be calculated
analytically. In that case,

$f_{N,c}^{\,\rm
Corr}(s,q^{2})=\int_{0}^{\infty}dbb\left[\left(e^{2i\delta^{\prime}_{c}(b,s)+2i\delta_{c}^{FF}(b,s)}-1\right)\times\int_{0}^{\infty}dq^{\prime}q^{\prime}f_{N}(s,q^{\prime
2})J_{0}(q^{\prime}b)\right]J_{0}(qb)$ (15)

as stated in the Introduction. The key is to first evaluate the integral over
$q^{\prime}$ in Eq. (15). This integral converges rapidly for any reasonable
model for $f_{N}$ that describes the rapid, nearly exponential, fall of the
differential cross sections with increasing $q^{2}$ observed at high energies,
and gives a result that vanishes rapidly for large $b$ as expected from the
long-range behavior of strong interactions. The second integral over $b$ is
therefore also expected to converge rapidly. This will be seen explicitly in
the examples in Sec. III.

The order of the integrations is crucial: the Coulomb plus form factor term in
parentheses in Eq. (15) does not provide convergence at large $b$ if one tries
to integrate in the opposite order, and one encounters the singularities that
caused trouble in Cahn’s approach [3] and its later modifications. The double
integral in Eq. (15) converges well overall when performed in the order
specified, and can easily be evaluated numerically.

In our approach, as in that of Cahn [3] or Kundrát and M. Lokajiček [5], or
the various modifications of it, one has to start with a model for $f_{N}$
fitted to the data, then calculate the Coulomb-nuclear correction, which will
be small, and then refit to get the final $f_{N}$.

##  III Examples

Consider as an example the simple exponential model

$f_{N}^{\,\rm exp}(s,q^{2})=(i+\rho)\sqrt{A/\pi}e^{-\frac{1}{2}Bq^{2}}$ (16)

with $A,\,B$ and $\rho$ functions of $s$ but independent of $b$. This has been
used over a wide range of energies to fit experimental data on the $pp$ and
$\bar{p}p$ differential cross sections at small $q^{2}$ to determine the
forward slope parameters $B=-d(\ln{d\sigma/dq^{2}})/dq^{2}$, the total cross
sections $\sigma_{\rm tot}=4\pi\Im f_{N}(s,0)$, and to determine the ratios
$\rho(s)=\Re f_{N}(s,0)/\Im f_{N}(s,0)$ of the real to the imaginary parts of
the forward amplitudes from Coulomb-nuclear interference effects. See, for
example, [15] and [11] for examples at 52.8 GeV and 8 TeV. For this model

$\tilde{f}_{N}^{\,\rm
exp}(b,s)=(i+\rho)\int_{0}^{\infty}dqq\sqrt{\frac{A}{\pi}}e^{-\frac{1}{2}Bq^{2}}J_{0}(qb)=(i+\rho)\sqrt{\frac{A}{\pi}}\frac{1}{B}e^{-b^{2}/2B}.$
(17)

The remaining integral over $b$ in Eq. (15) converges exponentially and is
easily evaluated numerically to get the mixed Coulomb-nuclear correction term
$f_{N,c}^{\,\rm Corr}(s,q^{2})$.

The model can be extended to

$f_{N}^{\,\rm
exp^{\prime}}(s,q^{2})=(i+\rho)\sqrt{A/\pi}e^{-\frac{1}{2}B^{\prime}q^{2}}(1-cq^{2})$
(18)

to include the diffraction structure caused by the expected zero in $\Im
f_{N}$. Here $c=1/q_{0}^{2}$, $q_{0}^{2}$ is the location of the observed
diffraction minimum, and $B^{\prime}=B-2c$ is fixed to reproduce the observed
forward slope $B$. In this case,

$\tilde{f}_{N}^{\,\rm
exp^{\prime}}(b,s)=(i+\rho)\sqrt{\frac{A}{\pi}}\frac{1}{B^{\prime}}\left(1-\frac{2c}{B^{\prime}}+\frac{cb^{2}}{B^{\prime
2}}\right)e^{-b^{2}/2B^{\prime}}.$ (19)

We have not found this extension to be necessary in calculating the
corrections for $q^{2}\ll q_{0}^{2}$.

The inner integral in Eq. (15) can also be evaluated analytically for some
other models, for example, that of Ferreira, Kohara, and Kodama [16], which
fits the $pp$ data quite well over a wide range in $q^{2}$. However, for most
models, the integrals must be evaluated numerically. As shown by the simple
exponential model, the inner and outer integrals may be expected to converge
very rapidly as functions of $q^{\prime}$ and $b$ for realistic $f_{N}$.

In Fig. 1 we show the ratios of the real and imaginary parts of the mixed
Coulomb-nuclear corrections $f_{N,c}^{\,\rm Corr}(s,q^{2})$ calculated using
the eikonal model of Block et al. [17] (red solid curves), and the exponential
model of Eq. (16) (blue dashed curves), to the real and imaginary parts of the
simple exponential model $f_{N}^{\,\rm exp}(s,q^{2})$. The full eikonal model
satisfies the constraints of unitarity, analyticity, and crossing symmetry,
fits the data on $\sigma_{\rm tot}$, $\sigma_{\rm elas}$, $B$, and $\rho$ for
$pp$ and $\bar{p}p$ scattering from 5 GeV to 57 TeV, and gives a good
description of the differential scattering cross sections and dip structure
even though the data on $d\sigma/dq^{2}$ other than B were not used in the
fit. It is taken here as representing the experimental data. The simple model
uses the values of $\sigma_{\rm tot}$, $B$, and $\rho$ obtained in the eikonal
fit to obtain $A,\,B$ and $\rho$ in Eq. (16) as would be obtained in a fit to
the data.

As seen in the top row in Fig. 1, the real parts of the corrections calculated
using the simple exponential model and Eq. (14) agree remarkably well at small
$q^{2}$ with those calculated in the eikonal model using the expression in Eq.
(12) with the eikonal phase shift. This agreement would be expected. The real
part of the correction is associated mainly with the imaginary part of the
nuclear scattering amplitude as may be seen by expanding the exponential in
the factor in parentheses in Eq. (14) to first order in the small quantity
$\eta$. Since $\Im f_{N}\gg\lvert\Re f_{N}\rvert$, the very good fit of the
exponential model to $d\sigma/dq^{2}$ over a range of small $q^{2}$ over which
the cross section decreases rapidly as seen, for example, in [15] and [11],
implies a correspondingly good fit to $\Im f_{N}$ over a region in which the
integral in Eq. (14) converges rapidly.

The imaginary parts of the mixed Coulomb-nuclear correction found using the
exponential model are considerably less accurate, but are quite small as seen
in the bottom row in Fig. 1. They arise mainly from the real part of $f_{N}$,
small compared to the imaginary part, and give very small corrections to the
latter which are not involved in the Coulomb-nuclear interference, hence, in
the determination of $\rho$.

The relative inaccuracy of these corrections seen in Fig. 1 results from the
poor description of $\Re f_{N}$ given by the exponential model. As expected
from a theorem of Martin [18] and seen in the eikonal model, there is a
diffraction zero in $\Re f_{N}$ between $q^{2}=0$ and the first diffraction
zero in $\Im f_{N}$. This is not evident in the differential cross section
because of the small size of $\Re f_{N}$ relative to $\Im f_{N}$ for $q^{2}$
below the dip region, but still leads to a much more rapid decrease of $\Re
f_{N}$ than $\Im f_{N}$ as $q^{2}$ increases from 0. The Martin zero is not
included in $f_{N}^{\rm exp}$, Eq. (16), and can only be incorporated in a
model like that in Eq. (18) using information on the position of the zero that
is not available from experiment. We have found that simply including a
separate exponential term for the real part of the model amplitude with a
magnitude $\rho$ relative to the imaginary part and a slope parameter $B_{R}$
matched to that in the eikonal model eliminates most of the errors in the
comparisons in the bottom row in Fig. 1.

Figure 1: Top row: ratios of the real parts of the mixed Coulomb-nuclear
corrections $f_{N,c}^{\,\rm Corr}(s,q^{2})$ to the $pp$ elastic scattering
amplitude calculated using the eikonal model of Block et al. [17] for the
nuclear amplitude $f_{N}(s,q^{2})$ (solid red lines) and the simple
exponential model $f_{N}^{\rm exp}(s,q^{2})$ of Eq. (16) (dashed blue line),
to the real parts of $f_{N}^{\rm exp}(s,q^{2})$. Bottom row: ratios of the
imaginary parts of $f_{N,c}^{\,\rm Corr}(s,q^{2})$ calculated using the
eikonal model (solid red line) and the exponential model (dashed blue line) to
the imaginary part of the exponential model.

In Fig. 2 we show ratios of the real and imaginary parts of the mixed Coulomb-
nuclear corrections $f_{N,c}^{\,\rm Corr}(s,q^{2})$ to the real and imaginary
parts of $f_{N}^{\rm exp}$ for the eikonal and exponential models at very
small $q^{2}$, the region of the observed Coulomb-nuclear interference. The
agreement of the results for the real parts is excellent. The corrections to
the real part of the amplitude are substantial and diverge logarithmically at
small $q^{2}$ because of the term $\ln{(qb/2)}$ in $\delta^{\prime}_{c}$, Eq.
(9). While small compared to the Coulomb term itself, the corrections in $\Re
f_{N}^{\,\rm Corr}$ cannot be neglected in analyses of Coulomb-nuclear
interference.

Figure 2: Left-hand column: ratios of the real parts of the mixed Coulomb-
nuclear corrections $f_{N,c}^{\,\rm Corr}(s,q^{2})$ to the $pp$ elastic
scattering amplitude calculated using the eikonal model (solid red lines) and
the simple exponential model of Eq. (16) (dashed blue line), to the real parts
of $f_{N}^{\rm exp}$ in the region of Coulomb-nuclear interference. Right-hand
column: ratios of the imaginary parts of $f_{N,c}^{\,\rm Corr}(s,q^{2})$
calculated using the eikonal model (solid red line) and the exponential model
(dashed blue line) to the imaginary part of the exponential model.

In Fig. 3, we show the real and imaginary parts of $f_{N}$ in the neighborhood
of the diffraction zero in $\Im f_{N}$ for $pp$ scattering at 8 TeV as
calculated in the eikonal model. The real part of the amplitude is dominant
near the zero, and the Coulomb term as modified by the charge form factors of
the protons leads to an approximately 20% difference between the $pp$ and
$\bar{p}p$ cross sections even with no significant crossing-odd hadronic term
present in $f_{N}$ [10]. The mixed Coulomb-nuclear corrections to $\Re f_{N}$
are again small but significant. The magnitude of the differential cross
section at the minimum provides a measure of $\Re f_{N}$ at that point, where
it is known to be negative since $\rho$ is positive as measured for
$q^{2}\approx 0$ and $\Re f_{N}$ changes sign at the Martin zero [18].

Figure 3: Plots of the nuclear amplitude and corrections in the neighbor hood
of the diffraction zero in $\Im f_{N}$ in the eikonal model of Block et al.
[17] at 8000 GeV. Solid black line: $\Im f_{N}(s,q^{2})$. Solid blue line:
$\Re f_{N}(s,q^{2})$. Dash-dot blue line: sum of $\Re f_{N}$ and the Coulomb
plus form-factor term in Eq. (10). Dashed red line: the real part of the full
$pp$ scattering amplitude including the Coulomb and form-factor terms and the
mixed Coulomb-nuclear corrections.

##  IV Conclusions

We conclude that the very simple calculation of the Coiulomb-nuclear
corrections to the $pp$ and $\bar{p}p$ scattering amplitudes through Eq. (14)
or Eq. (15) using a reasonable model for $f_{N}$ is quite accurate at high
energies, and represents substantial improvement in simplicity and clarity
over the methods most commonly used at present [3, 5, 7]. The combination of
the analytic expression for the pure Coulomb and form-factor terms in Eq.
(10), the model nuclear amplitude $f_{N}$, and this small correction term
gives our expression for the full $pp$ or $\bar{p}p$ scattering amplitude
$f(s,q^{2})$, Eq. (8).

We have illustrated the calculation of the mixed Coulomb-nuclear term for $pp$
scattering using the simple exponential model for the nuclear scattering
amplitude in Eq. (16) which is frequently used to fit cross sections at high
energies and small $q^{2}$, and shown that the results agree with those
obtained in the comprehensive eikonal model of Block et al. [17] at energies
from $W=52.8$ GeV to 13 TeV. A useful parametrization of the correction term
is given in [10], Sec. II. We have not investigated the corrections at low
energies, but the method still applies for appropriate models for $f_{N}$.

## References

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††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: These authors contributed equally.

# Optimal closed-loop control of active particles and a minimal information
engine

Rosalba Garcia-Millan  Department of Mathematics, King’s College London,
London WC2R 2LS, United Kingdom  DAMTP, Centre for Mathematical Sciences,
University of Cambridge, Cambridge CB3 0WA, UK  St John’s College, Cambridge
CB2 1TP, UK  Janik Schüttler  DAMTP, Centre for Mathematical Sciences,
University of Cambridge, Cambridge CB3 0WA, UK  Michael E. Cates  DAMTP,
Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA,
UK  Sarah A. M. Loos<EMAIL_ADDRESS>DAMTP, Centre for Mathematical
Sciences, University of Cambridge, Cambridge CB3 0WA, UK

###### Abstract

To establish general principles of optimal control of active matter, we study
the elementary problem of moving an active particle by a trap with minimum
work input. We show analytically that (open-loop) optimal protocols are not
affected by activity, but work fluctuations are always increased. For closed-
loop protocols, which rely on initial measurements of the self-propulsion, the
average work has a minimum for a finite persistence time. Using these
insights, we derive an optimal cyclic active information engine, which is
found to have a higher precision and information efficiency when operated with
a run-and-tumble particle than for an active Ornstein-Uhlenbeck particle and,
we argue, than for any other type of active particle.

Exploring strategies to control the individual or collective motion of self-
propelled particles Schneider and Stark (2019); Falk  _et al._ (2021);
Liebchen and Löwen (2019); Baldovin  _et al._ (2023); Shankar  _et al._ (2022)
enhances our understanding of active dynamics and represents a first step
towards technological applications; ranging from the design of active
metamaterials to the construction of nanosized robots. Recent research has
explored various means of control, e.g. adjusting the potential landscape
Davis  _et al._ (2024); Gupta  _et al._ (2023); Aurell  _et al._ (2011); Ekeh
_et al._ (2020); Majumdar  _et al._ (2022); Blaber and Sivak (2023); Lee  _et
al._ (2023), the confinement Malgaretti and Stark (2022); Cocconi  _et al._
(2023), or the level of intrinsic activity Falk  _et al._ (2021). In the realm
of small-scale organisms and synthetic machines, high frictional losses and
the prevalence of thermal and non-thermal fluctuations become particularly
important challenges. In this context, “thermodynamically optimized” control
schemes aim to minimize the associated heat dissipation, the work input, or
the generated entropy, while transitioning a system from one state to another.
Even for passive systems, thermodynamically optimal control can give rise to
non-intuitive solutions such as discontinuously time-dependent driving
Schmiedl and Seifert (2007); Engel  _et al._ (2023); Whitelam (2023); Loos
_et al._ (2024). To explore the basic principles of thermodynamically
optimized control of active systems, which are characterized by noise, memory,
(non-)Gaussian statistics and intrinsic driving, it is thus vital to study
simplified paradigmatic toy models.

In this spirit, we investigate the elementary problem of moving a harmonic
trap containing a single active particle to a target position in a given
finite time through some fluid environment that forms a heat bath, in one
spatial dimension. We optimize the dragging protocol to minimize the
experimenter’s work input, a directly measurable quantity Loos  _et al._
(2024) that bounds below the controller’s total energy expenditure. The
problem is complex enough to show key features of the optimal control of
active matter, yet can be solved exactly. Different from passive particles
(PPs), for which this problem was solved in the seminal paper Schmiedl and
Seifert (2007), self-propelled particles can actively swim towards the target,
which could suggest that less external work input is needed. However, this
internal self-propulsion can just as well increase the resistance against the
translation. We explore under which conditions the intrinsic activity
facilitates or hinders the transport. To gain more general insights, we
consider two different active models: a run-and-tumble particle (RTP) and an
active Ornstein-Uhlenbeck particle (AOUP).

From the viewpoint of control theory, the dragging problem defined above
corresponds to a so-called open-loop control, because the protocol is
predefined and fixed for every stochastic realization. More refined approaches
are closed-loop or feedback control schemes, which dynamically adapt the
protocol taking into account real-time knowledge of the system state,
resembling a mesoscopic “Maxwell demon.” As a consequence, closed-loop control
can generally yield a more precise manipulation; but it necessarily relies on
measurements which, according to Laundauer’s principle, are associated with
thermodynamic costs Parrondo  _et al._ (2015). As an example for a simple
closed-loop control problem, we study how an initial measurement of the system
state affects the optimal driving protocol, generalising previous studies on
PPs Abreu and Seifert (2011); Whitelam (2023) to the active case. Remarkably,
we find that if the particle is initially already moving towards the target
position, the optimal protocol can involve an initial jump of the trap _away_
from the target, challenging naive intuition.

As an application, we theoretically construct a minimal, thermodynamically
optimized cycle that uses measurement information and returns extractable
work, creating an “information engine” Saha  _et al._ (2021); Parrondo  _et
al._ (2015). Here, the harvested work however stems from the in-built activity
Malgaretti and Stark (2022); Cocconi  _et al._ (2023); Cocconi and Chen
(2024), like in an active heat engine Krishnamurthy  _et al._ (2016); Holubec
_et al._ (2020); Fodor and Cates (2021); Speck (2022); Zakine  _et al._
(2017). To access the efficiency of information-to-work conversion, we
calculate the fundamental thermodynamic cost associated with the required
real-time measurements Malgaretti and Stark (2022); Sagawa and Ueda (2010);
Kullback  _et al._ (2013); Cao and Feito (2009); Parrondo  _et al._ (2015),
and show that work extraction from RTPs is more efficient and more precise
than from AOUPs.

Figure 1: Optimal transport protocol for the closed-loop control problem,
where a demon measures of the particle’s initial self-propulsion of $v_{0}>0$:
(a) protocol stages ($1\to 2$ initial jump, $2\to 3$ continuous driving
regime, $3\to 4$ final jump to target), (b) protocol (blue), average particle
position (grey) and self-propulsion (arrows). The thin blue line shows the
corresponding protocol without measurement for reference. (c) shows the
cumulative average work $\left\langle
W(t)\right\rangle_{v_{0}}=\int_{0}^{t}\mathrm{d}{t^{\prime}}\
\dot{\lambda}k(\lambda-\langle x\rangle_{v_{0}})$ with initial measurement
(thick line) and without initial measurement (thin line). Here, $\tau=0.2$,
$k=1$, $t_{\text{f}}=1$, and $v_{0}=1$.

Model.—We consider a one-dimensional model for a self-propelled particle at
position $x$ in a harmonic potential $V(x,\lambda)=k(x-\lambda)^{2}/2$, where
$k$ is the stiffness and $\lambda$ is the center of the trap. The particle’s
motion follows the overdamped Langevin equation

$\dot{x}(t)=-\partial_{x}V(x,\lambda)+v(t)+\sqrt{2D}\ \xi(t)\ ,$ (1)

where $v$ denotes the self-propulsion, $D$ is a thermal diffusion constant,
and $\xi$ is a unit-variance Gaussian white thermal noise,
$\left\langle\xi(t)\right\rangle=0$,
$\left\langle\xi(t)\xi(t^{\prime})\right\rangle=\delta(t-t^{\prime})$. We have
absorbed the friction constant and $k_{\mathrm{B}}$ into other parameters. We
consider two standard models of active motility: (i) an AOUP, where
$\tau\dot{v}(t)=-v(t)+\sqrt{2D_{v}}\ \xi_{v}(t),$ with self-propulsion
“diffusion” constant $D_{v}$ and unit-variance Gaussian white noise $\xi_{v}$
independent of $\xi$; and (ii) an RTP, where the self-propulsion is a
dichotomous, or telegraphic, Poissonian noise that randomly reassigns values
$\pm{\omega}$ with tumbling rate of $1/\tau$ Schnitzer  _et al._ (1990);
Schnitzer (1993); Schneider and Stark (2019); Cates (2012); Garcia-Millan and
Pruessner (2021). A direct comparison between the two active models can be
established up to second-order correlators by fixing
${D_{v}}/{\tau}\equiv{\omega}^{2}$, such that both motility models give rise
to the exponentially decaying correlator

$\left\langle
v(t)v(t^{\prime})\right\rangle={\omega}^{2}\mathchoice{\mathrm{e}^{-\frac{|t-t^{\prime}|}{\tau}}}{\operatorname{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)}{\operatorname{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)}{\operatorname{exp}\left(-\frac{|t-t^{\prime}|}{\tau}\right)}$
(2)

at stationarity, with correlation time $\tau$. Considering AOUPs versus RTPs
confirm robustness of our main results and to isolate the effect of
introducing non-Gaussianity in the form of a Poissonian noise (RTPs) Lee and
Park (2022). We compare both active particle models against the case of a PP,
where $v(t)\equiv 0$. Both active models reduce to PPs under the limits
$\omega\to 0$ or $\tau\to 0$. In the limit $\tau\to\infty$, the active models
yield a ballistic motion with a constant velocity randomly set by the initial
condition.

As an example of an optimal control problem, we consider moving the trap from
the origin $\lambda_{0}=0$ at initial time $t=0$ (where the system is in a
steady state) to a target position $\lambda(t_{\text{f}})=\lambda_{\text{f}}$
in a finite time $t_{\text{f}}$. Due to the frictional resistance of the
surrounding fluid and changes in the potential energy of the particle, such a
process generally requires an external work input by the controller of
Jarzynski (1997); Crooks (1998)

$W=\int_{0}^{t_{\text{f}}}\mathrm{d}{t}\ \dot{\lambda}\frac{\partial
V(x,\lambda)}{\partial\lambda}=\int_{0}^{t_{\text{f}}}\mathrm{d}{t}\
\dot{\lambda}k(\lambda-x)\,.$ (3)

We optimise the time-dependent protocol, denoted by $\lambda(t)$, such that
the required average work input is minimised. We deliberately do not account
for the dissipation associated with the self-propulsion, as the internal
mechanism is unresolved at RTP/AOUP level, nonuniversal, and hard to measure
experimentally (in order to quantify the dissipation of the active swimmer,
one could explicitly model the self-propulsion mechanism, as done in Gaspard
and Kapral (2017); Pietzonka and Seifert (2017); Speck (2018, 2022)).

Open-loop control.—First we discuss the open-loop control case where no
measurements are taken. Using the Euler-Langrange formalism, we minimise the
noise-averaged work functional given in Eq. (3) with the non-equilibrium
steady state as initial condition, such that $\langle x(0)\rangle=0$ and
$\langle v(0)\rangle=0$. The optimization, which is derived in Schüttler  _et
al._ , reveals that the optimal protocol $\lambda(t)$ is independent of the
self-propulsion, and therefore, identical for AOUPs, RTPs and PPs Schmiedl and
Seifert (2007). For passive and active particles, the optimal protocol has two
symmetric jumps at $t=0$ and $t=t_{\text{f}}$ of length
$\lambda_{\text{f}}/(kt_{\text{f}}+2)$ connected by a linear dragging regime,
shown as the thin line in Fig. 1(b), and its energetic cost is $\left\langle
W\right\rangle=\lambda_{\text{f}}^{2}/(t_{\text{f}}+2/k)\geq 0$. Thus, the
activity does not decrease the required work. These findings are important in
experimental settings, where the underlying particle dynamics might be
unknown. They also show that in this context, activity cannot be harnessed
unless it is monitored in some fashion.

However, the activity influences the _fluctuations_ of the work done by the
controller. As we derive in Schüttler  _et al._ , the variance of the work for
PPs is
${\mathrm{Var}}W_{\text{passive}}=2D\lambda_{\text{f}}^{2}/(t_{\text{f}}+2/k)$,
while for both active models, there is the same additional contribution of

$\displaystyle{\mathrm{Var}}W_{\text{active}}-{\mathrm{Var}}W_{\text{passive}}=$
$\displaystyle\\!\\!\\!=\frac{2\lambda_{\text{f}}^{2}\omega^{2}}{(\frac{t_{\text{f}}}{\tau}+\frac{2}{k\tau})^{2}}\left[\frac{t_{\text{f}}}{\tau}+\frac{\frac{2}{k\tau}+(1-k\tau)(1-e^{-t_{\text{f}}/\tau})}{1+k\tau}\right]\\!\\!\geq
0\,.$ (4)

Thus, the variance of the work is always increased by the activity, consistent
with the picture of an increased effective temperature. The additional
variance only vanishes in the trivial limits: $\lambda_{\text{f}}\to 0$;
$\omega\to 0$, where active particles become passive; and in the quasistatic
limit, $t_{\text{f}}\to\infty$, where ${\mathrm{Var}}W$ vanishes for all
models.

Taken together, our findings suggest that activity does not pose an advantage
in the open-loop case: while the minimum mean work and optimal protocol remain
identical to the passive case, the stochastic work entails a larger
uncertainty. Furthermore, at the level of first and second moments, the
(non-)Gaussianity is irrelevant for the open-loop control, as we show in
Schüttler  _et al._ , though a distinction arises in the higher moments. For
example, while the positional distribution remains Gaussian for PPs and AOUPs
throughout the protocol, the non-Gaussian distribution of RTPs is generally
changed by the dragging Garcia-Millan and Pruessner (2021).

Closed-loop control (initial measurement).—We turn to the case where a
feedback controller or mesoscopic “demon,” is able to take an initial real-
time measurement before deciding and executing the protocol. The optimal
closed-loop protocol can be obtained by the same optimization procedure as
before, now incorporating the (measured) states $x(0)=x_{0},v(0)=v_{0}$ as
initial condition. We here focus on the case where only $v$ is measured, which
is obtained by subsequently averaging over $x_{0}$ while taking into account
that a measurement of $v$ also yields information about $x$. Concretely, we
use that $\left\langle x(0)\right\rangle_{v_{0}}=v_{0}/(k+\tau^{-1})$. Here,
$\langle\bullet\rangle_{v_{0}}=\langle\bullet|v(0)\\!=\\!v_{0}\rangle$ denotes
the conditional average over trajectories with $v(0)=v_{0}$. As detailed in
Schüttler  _et al._ , the optimal protocol after only a $v$-measurement, which
again coincides for AOUPs and RTPs, reads

$\lambda_{v_{0}}(t)=\left\\{\begin{array}[]{l l}0,&t=0\\\ \left\langle
x(t)\right\rangle_{v_{0}}+\frac{d}{kt_{\text{f}}+2}-\frac{v_{0}}{2k}e^{-t/\tau}&0<t<t_{\text{f}}\\\
\lambda_{\text{f}},&t=t_{\text{f}}\end{array}\right.\ ,$ (5)

with the corresponding mean particle trajectory,

$\left\langle
x(t)\right\rangle_{v_{0}}=\frac{v_{0}}{k+\tau^{-1}}+\frac{d}{t_{\text{f}}+2/k}t+\frac{\tau
v_{0}}{2}(1-e^{-t/\tau})\ ,$ (6)

where $d$ is an effective distance defined as

$d=\lambda_{\text{f}}-\frac{v_{0}}{k+\tau^{-1}}-\frac{\tau
v_{0}}{2}\left(1-\mathchoice{\mathrm{e}^{-t_{\text{f}}/\tau}}{\operatorname{exp}\left(-t_{\text{f}}/\tau\right)}{\operatorname{exp}\left(-t_{\text{f}}/\tau\right)}{\operatorname{exp}\left(-t_{\text{f}}/\tau\right)}\right)\,.$
(7)

The quantity $d$ corresponds to the distance between the expected value of the
particle’s initial position after measurement of $v$ and the target position
of the trap (first two terms) minus the distance covered “for free” by the
particle’s self-propulsion until its orientation decorrelates. In Schüttler
_et al._ , we also derive the solutions for control with both $x$ and $v$
measurements, and with only $x$-measurement, the latter giving consistent
results to the closed-loop control of PPs studied in Abreu and Seifert (2011),
with no additional terms due to activity.

Figures 1(a) and (b) show a typical optimal protocol (5) for some $v_{0}>0$,
$\lambda_{\text{f}}>0$. Different from the open-loop case, the jumps at $t=0$
and $t=t_{\text{f}}$ can be asymmetric and, at intermediate times, the
protocol is generally nonlinear in time. Furthermore, the initial jump

$\lambda_{v_{0}}(0^{+})-\lambda_{v_{0}}(0)=\frac{v_{0}}{k+\tau^{-1}}-\frac{{v_{0}}}{2k}+\frac{d}{kt_{\text{f}}+2}\,,$
(8)

can be in the opposite direction to the target position $\lambda_{\text{f}}$.
What seems at first glance even more counterintuitive is that these reversed
jumps occur when the initial self-propulsion already pushes the particle
towards the target trap position $\lambda_{\text{f}}$, as illustrated in Figs.
1(a) and (b). The expectation value for the corresponding cumulative work
$\int_{0}^{t}\mathrm{d}{t^{\prime}}\ \dot{\lambda}(\lambda-\left\langle
x\right\rangle)$, shown in Fig. 1(c), reveals the underlying reason. While the
initial reversed jump is costly, it brings the system into a configuration
which allows better subsequent extraction of net energy from the self-
propulsion, overcompensating the initial cost.

The average total work associated with $\lambda_{v_{0}}(t)$ is

$\displaystyle\left\langle W\right\rangle_{v_{0}}$
$\displaystyle=-\frac{kv_{0}^{2}}{2(k+\tau^{-1})^{2}}-{\frac{\tau
v_{0}^{2}}{8}}\left(1-\mathchoice{\mathrm{e}^{-2t_{\text{f}}/\tau}}{\operatorname{exp}\left(-2t_{\text{f}}/\tau\right)}{\operatorname{exp}\left(-2t_{\text{f}}/\tau\right)}{\operatorname{exp}\left(-2t_{\text{f}}/\tau\right)}\right)+\frac{d^{2}}{t_{\text{f}}+2/k}\
,$ (9)

which implies that $v$ measurements contribute towards work extraction through
two different physical mechanisms. First, measuring $v$ gives average
positional information that makes it possible to extract some of the potential
energy stored in the initial configuration, concretely $(k/2)\left\langle
x(0)\right\rangle_{v_{0}}^{2}\leq\left\langle V\right\rangle_{v_{0}}$, by the
Cauchy-Schwarz inequality. Second, energy can be extracted directly from the
self-propulsion. After measuring $v_{0}\not=0$, setting the trap to a position
where the particle actively climbs up the potential and then moving the trap
along the direction of self-propulsion one can extract net work from the
activity until the orientation decorrelates. In contrast to the work extracted
from the potential energy, this operation requires a finite process duration,
$t_{\text{f}}>0$. Neither mechanism requires an ambient temperature, $D>0$; in
contrast with work extraction from potential energy through positional
measurements Abreu and Seifert (2011).

The average work with respect to $v_{0}$ measurements is obtained by averaging
Eq. (9) over the steady state distribution of $v$, leading to

$\displaystyle\begin{split}\mathcal{W}=&-\frac{k\omega^{2}}{2(k+\tau^{-1})^{2}}-{\frac{\tau\omega^{2}}{8}}\left(1-\mathchoice{\mathrm{e}^{-2t_{\text{f}}/\tau}}{\operatorname{exp}\left(-2t_{\text{f}}/\tau\right)}{\operatorname{exp}\left(-2t_{\text{f}}/\tau\right)}{\operatorname{exp}\left(-2t_{\text{f}}/\tau\right)}\right)\\\
&+\frac{\lambda_{\text{f}}^{2}+\left(\frac{2}{k+\tau^{-1}}+\tau(1-e^{-t_{\text{f}}/\tau})\right)^{2}\omega^{2}/4}{t_{\text{f}}+2/k}\;.\end{split}$
(10)

$\mathcal{W}$ has a pronounced minimum at finite $\tau$, Fig. 2. Thus, the
cost to translate an actively self-propelled particle is lower than for a
passive particle ($\tau\to 0$), and, remarkably, also a ballistic particle
($\tau\to\infty$). In both these limits the work $\mathcal{W}$ reduces to that
of the passive case, $\frac{\lambda_{\text{f}}^{2}}{t_{\text{f}}+2/k}$, on
average. Therefore, a finite persistence time, i.e., the presence of non-
equilibrium fluctuations, poses an advantage in the closed-loop protocol.

Figure 2: The work of the closed-loop protocol averaged over the measured
values $v_{0}$ shows a pronounced minimum for a finite persistence time
$\tau$; here shown for $k=1$ and $t_{\text{f}}\in[10^{-2},10^{0.5}]$. The
$y$-axis shows the reduction in average work compared to the passive limit
($\tau\to 0$). Inset shows the kink in the minimum $\tau$ (black line) as a
function of $t_{\text{f}}$.  Figure 3: Active information engine: (a) cyclic
protocol $\lambda^{\mathrm{engine}}$ determined upon an initial self-
propulsion measurement $v_{0}$ in each cycle, (b) single realisation of cyclic
engine protocol (thick blue), corresponding noise-averaged particle position
(grey), and cumulative work (thick red). The arrows indicate the velocity
measurements. Thin lines show trajectories for other sequences of self-
propulsion measurements ($\omega=1.0,\tau=0.5,t_{\text{f}}=10,k=0.5$). (c)
Work distribution per cycle for AOUP and RTP from simulations
($\omega=0.2,\tau=1,t_{\text{f}}=10,k=2$). While the mean ($\times$) is
identical for AOUP and RTP and matches the theory (dashed line), AOUPs have a
larger standard deviation (vertical bars); indicating that a motor run with
RTPs is more precise. Moreover, for RTPs, the unique mode of the distribution
is below the mean value, while it is close to zero for AOUPs.

Active information engine.—A notable application of optimal closed-loop
protocols are optimized cyclic machines. Based on our results so far, one can
readily construct a minimal, optimal information engine that harvests energy
from the activity using self-propulsion measurements, as follows. Starting
from Eqs. (5)-(9), we perform a secondary optimization of $\left\langle
W\right\rangle_{v_{0}}$ with respect to $\lambda_{\text{f}}$, obtaining that
the optimal target distance after measuring $v_{0}$ is
$\lambda_{\text{f}}=\frac{1}{k+\tau^{-1}}+\frac{1}{2}\tau
v_{0}(1-e^{-t_{\text{f}}/\tau})$, so that $d=0$. Thus, $\left\langle
W\right\rangle_{v_{0}}\leq 0$ for all $v_{0}$. Setting $d=0$ in Eq. (5) gives
the corresponding optimal protocol $\lambda^{\mathrm{engine}}$,

$\lambda^{\mathrm{engine}}=\left\\{\begin{array}[]{l l}0,&t=0\\\ \frac{\tau
v_{0}}{2}\left(\frac{3+k\tau}{1+k\tau}-\frac{1+k\tau}{k\tau}e^{-t/\tau}\right)&0<t<t_{\text{f}}\\\
\frac{\tau
v_{0}}{2}\left(\frac{3+k\tau}{1+k\tau}-e^{-t_{\text{f}}/\tau}\right)&t=t_{\text{f}}\end{array}\right.\
.$ (11)

Iterating $\lambda^{\mathrm{engine}}$ yields a cyclic engine with period
$t_{\text{f}}$, as visualized in Fig. 3. The motor builds up nontrivial
correlations between the cycles. However, we can nevertheless access the
average work extraction per cycle, by reasoning as follows. First, the
orientational dynamics is not affected by the trap, so that all $v_{0}$
measurements are simply drawn from its steady state. Second, the final trap
position now automatically coincides with the average particle position at the
end of the protocol, so that the protocol extracts any average potential
energy build-up during the process. Therewith, we can readily obtain from Eq.
(9) the average work extraction per cycle
$-\frac{k}{2(k+\tau^{-1})^{2}}\omega^{2}-\frac{1}{8}\tau\omega^{2}\left(1-e^{-2t_{\text{f}}/\tau}\right),$
both for AOUPs and RTPs 111We can assume that the particle has equilibrated by
the end of the cycle if its duration $t_{\text{f}}$ is greater than the
timescales $\tau$ and $k^{-1}$, so that the particle position has effectively
reached its steady state, and we can thus apply Eq. (9). , which has no
optimum value of $\tau$, and reaches its maximum in the quasistatic regime. We
find that, while the average work is identical for AOUPs and RTPs, the work
distributions differ significantly, Fig. 3(c). Most significantly, the maximum
value of the distribution is at a negative value below the average for RTPs,
whereas it is around zero for AOUPs 222At long timescales, the center of the
trap performs a diffusive motion. This diffusive delocalization can be
prevented at the cost of lowering the work extraction by modifying the
protocol, for instance, by resetting the engine to the origin at the end of
each cycle. This process is described by Eqs. (5)-(9) with
$\lambda_{\text{f}}=0$. If $t_{\text{f}}$ is sufficiently long, the engine
still produces net work..

Finally, we address the effect of measurement uncertainty on the extractable
work, and the thermodynamic cost of the information acquisition Sagawa and
Ueda (2010); Kullback  _et al._ (2013); Cao and Feito (2009); Parrondo  _et
al._ (2015), which has not been included in previous studies of active
information engines Cocconi  _et al._ (2023). Since the closed-loop protocol
relies on a measurement, its associated uncertainty prevents the controller
from applying the optimal protocol accurately and thus from extracting the
maximum available work. Assuming that the $v$ measurement has a Gaussian-
distributed error $\epsilon$, we show in Schüttler  _et al._ that the net
extracted work has an additional cost term
$\Delta\mathcal{W}(\epsilon)\propto\epsilon^{2}$, leading to a total average
work extraction per cycle of

$\mathcal{W}^{\mathrm{engine}}=-\frac{k\omega^{2}}{2(k+\tau^{-1})^{2}}-\frac{\tau\omega^{2}}{8}\left(1-e^{-2t_{\text{f}}/\tau}\right)+\Delta\mathcal{W}(\epsilon)\,.$
(12)

The cost term $\Delta\mathcal{W}(\epsilon)\geq 0$ reaches its highest value of
$\tau\epsilon^{2}/2$ in the quasi-static limit and is identical for RTPs and
AOUPs, despite the different $v$ distributions.

The information-efficiency of the engine is obtained by dividing the mean
extracted work per cycle by the information acquisition cost,

$\displaystyle\eta=\frac{-\mathcal{W}^{\mathrm{engine}}}{D_{v}\,\Delta\mathcal{I}}\,.$
(13)

For an individual measurement, the information acquisition costs are given by
the relative entropy, and when averaged over multiple cycles, by the mutual
information $\Delta\mathcal{I}$ between the true value $v_{0}^{\mathrm{r}}$
and the measurement outcome $v_{0}^{\mathrm{m}}$ Sagawa and Ueda (2010, 2008);
Cao and Feito (2009); Abreu and Seifert (2011). Here, the information-
theoretical contribution is scaled by $D_{v}$, which plays the role of
“temperature” of the random variable.

As we explicitly show in Schüttler  _et al._ , the mutual information for
AOUPs is

$\displaystyle\Delta\mathcal{I}_{\mathrm{AOUP}}[v_{0}^{\mathrm{m}}:v_{0}^{\mathrm{r}}]=\frac{1}{2}\ln\left(1+\frac{{\omega}^{2}}{\epsilon^{2}}\right).$
(14)

As expected, it vanishes for $\epsilon\to\infty$, where measurements become
useless, and diverges for $\epsilon\to 0$. This divergence is not surprising,
in view of the infinite information content in a continuous variable,
$v_{0}\in\mathbb{R}$ Polyanskiy and Wu (2014); Gray (2011). For RTPs, we find
that in the regime of small errors $\epsilon\ll\omega$,

$\displaystyle\Delta\mathcal{I}_{\mathrm{RTP}}[v_{0}^{\mathrm{m}}:v_{0}^{\mathrm{r}}]=\ln(2)\,.$
(15)

Thus, the measurement yields exactly one bit of information. Comparing (15)
and (14) indicates that, at least for modest measurement uncertainty, which is
the regime of interest, the cost of information acquisition is always smaller
for RTPs than for AOUPs. In fact, RTPs have the smallest information cost (one
bit) of any possible active particle model. Recalling that the extractable
work is independent of the model of self-propulsion, we conclude that the
information efficiency $\eta$ is always highest for RTPs.

Together with Eqs. (12) and (15), this gives a universal upper bound to the
information-efficiency,

$\displaystyle\eta\leq\eta_{\mathrm{max}}={1}/{[4\ln(2)]}\,.$ (16)

The engine is most efficient when run with an RTP at low frequency
$1/t_{\text{f}}$, with low error $\epsilon$, and $k\tau=1$ 333Taking into
account the underlying thermodynamic cost of the self-propulsion as a
background dissipation term results in a maximum efficiency at some finite
frequency $1/t_{\text{f}}$. Specifically, assuming a constant rate
$\mathcal{A}$ of ATP depletion of the self-propulsion mechanism, the
efficiency
$\eta\propto\frac{\mathcal{W}^{\mathrm{engine}}}{D_{v}\,\Delta\mathcal{I}+\mathcal{A}\,t_{\text{f}}}$
peaks at a finite $t_{\text{f}}$.. Finally, we note that even for finite
$t_{\text{f}}$, the efficiency $\eta$ admits a maximum at a finite persistence
time $\tau$.

Conclusions.—We have generalized a canonical optimization problem to the case
of active particles. The open-loop protocol is not affected by activity, but
flucutations of the work are generally increased. In constrast, closed-loop
protocols can extract energy from self-propulsion measurements by jumps away
from the target position. Surprisingly, a finite persistence time is
beneficial as compared to “infinitely persistent” (ballisitic) particles. We
have further derived an optimal engine to harvest work from the self-
propulsion measurements by simple shifts of the trap. Such engine is most
efficient and most precise for RTPs, i.e., it benefits from non-Gaussianity.
We found a universal upper efficiency bound for this design of engine of
$1/[4\ln(2)]\approx 0.36$. Higher efficiencies can presumably be reached with
more complicated machines, for instance by dynamically adjusting the trap
stiffness.

To find optimal solutions for more complicated processes, one may apply recent
machine-learning algorithms Whitelam (2023); Engel  _et al._ (2023); Loos  _et
al._ (2024). With such, also collective active systems could be studied. Our
results for AOUPs readily generalize to higher dimensions, and it would be
interesting to explore RTPs and other active matter models in higher
dimensions as well.

Our insights are complementary to recent progress on optimal control of active
matter using a perturbative framework based on response theory Davis  _et al._
(2024); Gupta  _et al._ (2023). While this framework provides a general
approach applicable to more complicated scenarios, it applies only to the
regime of slow and weak driving. Instead, we have focused on the fast driving
regime, allowing us to explicitly study the emergence of driving
discontinuities, which play a crucial role in all our optimal protocols. As
opposed to an active ratchet, the information-driven work extraction minimises
dissipation from unprofitable non-equilibrium fluctuations during the initial
correlation time.

We thank Robert Jack and Édgar Roldán for insightful discussions. R.G.-M.
acknowledges support from a St John’s College Research Fellowship, Cambridge.
J.S. acknowledges funding through the UK Engineering and Physical Sciences
Research Council (Grant number 2602536). S.L. acknowledges funding through the
postdoctoral fellowship of the Marie Skłodowska-Curie Actions (Grant Ref.
EP/X031926/1) undertaken by the UKRI, through the Walter Benjamin Stipendium
(Project No. 498288081) from the Deutsche Forschungsgemeinschaft (DFG), and
from Corpus Christi College Cambridge.

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# Cultivating Online: Question Routing in a Question and Answering Community
for Agriculture

1st Xiaoxue Shen  China Agricultural University
college of Information and
Electrical Engineering
Beijing, China
<EMAIL_ADDRESS>2nd Liyang Gu  China Agricultural University
college of Information and
Electrical Engineering
Beijing, China
<EMAIL_ADDRESS>3rd Adele Lu Jia  China Agricultural University
college of Information and
Electrical Engineering
Beijing, China
<EMAIL_ADDRESS>
###### Abstract

Community-based Question and Answering (CQA) platforms are nowadays
enlightening over a billion people with crowdsourced knowledge. A key design
issue in CQA platforms is how to find the potential answerers and to provide
the askers timely and suitable answers, i.e., the so-called question routing
problem. State-of-art approaches often rely on extracting topics from the
question texts. In this work, we analyze the question routing problem in a CQA
system named Farm-Doctor that is exclusive for agricultural knowledge. The
major challenge is that its questions contain limited textual information.

To this end, we conduct an extensive measurement and obtain the whole
knowledge repository of Farm-Doctor that consists of over 690 thousand
questions and over 3 million answers. To remedy the text deficiency, we model
Farm-Doctor as a heterogeneous information network that incorporates rich side
information and based on network representation learning models we accurately
recommend for each question the users that are highly likely to answer it.
With an average income of fewer than 6 dollars a day, over 300 thousands
farmers in China seek online in Farm-Doctor for agricultural advices. Our
method helps these less eloquent farmers with their cultivation and hopefully
provides a way to improve their lives.

###### Index Terms:

question and answering, question routing, network representation learning

111Extension of this work is currently under review and details will be
updated soon.

##  I INTRODUCTION

Community-based Question and Answering (CQA) systems have become popular
knowledge sharing platforms where users get answers for the questions they
raised. They have received great attention both in industry and in academia
[1, 2]. One of the most important goals of CQA systems is to provide an asker
with a suitable answer in the shortest possible time, i.e., the so-called
question routing problem. In contrast to previous works [3, 4, 2] that focus
on general CQA websites such as Quora and Yahoo! Answers, in this work, we
analyze the question routing problem in a CQA platform named Farm-Doctor that
is exclusive for agricultural knowledge.

In China, over 300 thousand rural resident, although with limited income (on
average less than 6 dollars a day), managed to connect to the internet and
seek online in Farm-Doctor for agricultural knowledge. Accurate question
routing will provide timely advices for their cultivation and potentially
improve their lives. However, question routing in Farm-Doctor faces a major
challenge, i.e., the limited textual information problem.

In general CQA platforms, most questions are described in natural languages
and question routing is often performed through extracting topics from the
rich textual information[3, 2]. In contrast, users in Farm-Doctor raise their
questions mostly through pictures, along with simple questions like which is
the problem? and what should I do? Due to the lack of textual information,
topic models that are widely used in CQA platforms are not applicable to the
case of Farm-Doctor. On the other hand, although image recognition has
received great attention and success both in industry and in academia,
efficient tools on identifying crop diseases and even crops are still missing.
As a consequence, it is difficult to infer the topics from the questions
(texts or images) alone. To solve this problem, in this work we incorporate
rich side information and model CQA platforms as a heterogeneous information
network (HIN), and based on network representation learning (NRL) models, we
conduct, to the best of our knowledge, the first analysis of the question
routing problem without using textual information.

Our analysis of Farm-Doctor mainly consists of three parts:

Measuring Farm-Doctor. In this work, we have obtained the whole knowledge base
of Farm-Doctor. Our dataset covers all the 697,695 questions raised before
April 21st, 2018, along with the information on the associated 3,179,333
answers, 438 crops and 305,359 users. The information we obtained include not
only the basic question characteristics but also user activities such as who
raised/answered which question at what time, which crop is tagged in which
question and is interested to which user. The dataset is publicly available
through requests to the first author.

Characterizing question and answer dynamics. We first provide an analysis on
the scale and the characteristics of the question repository in Farm-Doctor.
We examine the number of answers received by each question and the timeliness
of the answers. We find that questions in Farm-Doctor normally attract a few
answers shortly after they are raised, but as time passes by they no longer
receive any attention. Then, we analyze the user activities in terms of the
number questions they raised and answered. We find a highly skewed activity
level of the users, with a small number of users raising and answering a large
number of questions. These results all indicate the need of a proper question
routing method in Farm-Doctor, so that questions will get more answers and
hopefully that less active users will be encouraged to answer the questions
personalized recommended to them.

Question routing. To tackle the limited textual information problem in Farm-
Doctor, we build a HIN model based on a variety of relationships and we adopt
heterogeneous NRL models to learn the low-dimensional embeddings of the
questions, the users, and the crops. Taking the learned representations as
input features, we build machine-learned classifiers to recommend timely and
accurately, for the newly posted questions, the potential users that most
likely will answer the questions.

The main contributions of this work are as follows:

  * •

We obtain the whole knowledge repository of FarmDoctor (until April 2018) that
contains information of 305,359 users, 697,695 questions, and 3,179,333
answers (Section 2). Our dataset is publicly available upon request to the
first author.

  * •

We analyze the basic characteristics of the question repository and the user
activities in Farm-Doctor (Section3).

  * •

We propose a heterogeneous graph model that incorporates a variety of
relationships among users, questions and crops. Based on this model, we adopt
NRL methods to, for each question, accurately predict the potential answerers
(Section 4).

## References

  * [1] Z. Chen, C. Zhang, Z. Zhao, and D. Cai, “Question retrieval for community-based question answering via heterogeneous network integration learning,” 2016.
  * [2] L. Yang, M. Qiu, S. Gottipati, F. Zhu, J. Jiang, H. Sun, and Z. Chen, “Cqarank: jointly model topics and expertise in community question answering,” in _ACM International Conference on Conference on Information and Knowledge Management_ , 2013, pp. 99–108.
  * [3] B. Li and I. King, “Routing questions to appropriate answerers in community question answering services,” in _ACM International Conference on Information and Knowledge Management_ , 2010, pp. 1585–1588.
  * [4] B. Li, I. King, and M. R. Lyu, “Question routing in community question answering: putting category in its place,” in _ACM International Conference on Information and Knowledge Management_ , 2011, pp. 2041–2044.