# Hamiltonian active particles in an environment

Timo Eichmann  Diego Fieguth  Daniel Brady  James R. Anglin  State Research
Center OPTIMAS and Fachbereich Physik,Technische Univerität
Kaiserslautern,D-67663 Kaiserslautern, Germany

###### Abstract

We examine a Hamiltonian system which represents an active Brownian particle
that can move against an external force by drawing energy from an internal
depot while immersed in a noisy and dissipative environment. The Hamiltonian
consists of two subsystems, one representing the active particle’s motion and
the other its depot of ‘fuel’. We show that although the active particle loses
some of its energy to dissipation from the environment, dissipation can also
help to stabilize the dynamical process that makes the particle active.
Additionally we show how a Hamiltonian active particle can harvest energy from
its environment.

##  I I. Introduction

###  I.1 I.1 Active Brownian particles

Microscopic motors are currently a major subject of investigation in all
natural science: biology, chemistry, and physics. Motor proteins like Kinesin,
Myosin or Prestin seem to play roughly analogous roles in living cells to
automobiles in industrial economies, but the microscopic versions certainly
work differently. They operate out of equilibrium and thermal fluctuations are
not negligible for them Broken ; Noisy . Some proposed theoretical models for
microscopic motors, such as ratchets or Brownian motors, are _passive_
BrownMotor ; Ratchet , in the sense that they obtain their energy from
fluctuations in immediately surrounding reservoirs. Other models are _active_
, either by being driven by systematic external forces from light or sound
Extone ; Exttwo , or by drawing on an internal energy depot Design ; ActStat .

A frequently invoked model for this latter class of active microscopic motors
is the _active Brownian particle_ Complex ; Uphill ; ActiveReview ;
ActiveMotion ; Canon ; DirMot ; ChemFree ; Assym . In this model a particle is
subject to the usual Brownian dissipation and noise, representing the
surrounding medium in which the particle moves, but the particle can also
propel itself through this surrounding medium, by drawing on some form of
internally stored energy (the ‘depot’). Active Brownian particles are intended
to model such real systems as motor proteins or bacteria that swim with
flagella, so the energy depot of an active Brownian particle cannot be as
simple as the tightly coiled spring of a wind-up toy. Precisely because the
microscopic details of how microscopic motors store and use energy are
complex, however, the paradigm of the active Brownian particle represents
energy use and storage phenomenologically. Simple equations of motion are
postulated which involve an abstract sort of internal ‘fuel tank’ and lead to
evolution that sufficiently resembles powered travel; the focus is more on
what active particles can do than on just how they do it.

The use of active Brownian particles as phenomenological models actually goes
well beyond molecular machines, especially when it extends to include
interactions between active particles selfdriven ; ActiveSus ; ActiveReview ;
Information ; Coop . Active particles can serve as simple models for entire
living organisms Active_ComplexEnvironment ; Information  or even as
candidates for a minimal model of cognition Cognition . Within physics, the
concepts of active particles have been helpful in developing stochastic
thermodynamics Seifert ; Limits ; ActiveEntropy ; ActStoch ;
Stochastic_Energetics , and they represent the simplest limit of ‘active
matter’ in general Challenging ; Need ; Opinion ; ActStat .

The phenomenological ‘black box’ treatment of _how_ internally stored energy
is transferred into motion by an active particle has thus been a valuable
feature of the active particle concept, permitting insights into a wide range
of phenomena. Exactly how it is that densely stored energy can enable motion
is still a basic physics question, however, and for many of the systems that
are modeled with active Brownian particles the microscopic mechanisms have yet
to be understood.

###  I.2 I.2 Hamiltonian active particles

When a macroscopic motor operates within a dissipative environment, we can
generally take it for granted that the motor’s strong and massive components
will continue to move as they are designed to move, with only some power loss
to friction; it is not obvious, however, that the much smaller and lighter
mechanism of a microscopic engine will even continue to run under viscous drag
and collisional battering. Since the simple but peculiar “Hamiltonian daemon”
dynamical system of ClassicalDaemon  resembles a combustion engine in its
behavior, yet has only a single non-trivial degree of freedom, it offers a
minimalistic Hamiltonian realization of an active particle. By seeing how the
“daemon” is affected by dissipation and noise, we can identify general issues
that may be involved in the operation of more realistic Hamiltonian active
particles.

Our results will offer two insights for more general active particles. We will
find that dissipation from the environment can actually _enhance_ the
efficiency of the active particle: even quite weak dissipation can allow the
active particle to convert more of its internally stored energy into work, by
helping to control its internal dynamical transitions. And we will confirm
that even a simple Hamiltonian mechanism can harvest energy from its
surroundings, acting as a sort of mindlessly automatic Maxwell’s demon.

###  I.3 I.3 Organization

Our paper is structured as follows. In Section II we briefly review the
Hamiltonian daemon system from Ref. ClassicalDaemon  in the absence of any
environment, to show how this dynamical system can represent an active
particle. We first describe the system’s engine-like behavior and explain the
basic issues which it will illuminate as a toy model for an active Brownian
particle. Then we present the daemon’s Hamiltonian and show how to understand
its behavior in terms of adiabatic orbits in a two-dimensional phase space.

In Section III we then insert our Hamiltonian active particle into an
environment consisting of gas particles that collide with the active particle,
inducing stochastic jumps in its momentum, but without directly affecting its
energy depot.

In Section IV we will discover and quantify our first major result, showing
that although dissipation does lead to some fraction of the particle’s
internal energy being wasted as heat, and excessively strong dissipation can
prevent the particle from being active at all, total energy transfer from
internal fuel into work can actually _increase_ due to moderate dissipation.
The environment is not simply the active particle’s dissipative adversary: the
environment can help to control the dynamical process that powers the active
particle.

In Section V we then replace the dissipative “buffer gas” environment with an
environment that interacts with the active particle’s energy depot. We find a
regime in which this kind of environment acts as a gain medium for the
particle, restoring its internal energy faster than the energy depot is being
drained to power the particle’s motion against friction. This provides a
Hamiltonian realization of the often-modeled scenario in which an active
particle harvests energy from its environment.

We conclude in Section VI with a summary and an outlook toward future work.

##  II II. A Hamiltonian active particle

###  II.1 II.1 Daemon behavior

Before it is coupled to any environment, our Hamiltonian daemon model for an
active particle behaves as shown in Fig. 1(a) and (b). Like a physical
pendulum which can either swing fully around and around or else swing back and
forth, or like a motor which can be running or not, our system has two
dynamical phases. Trajectories which include the “active phase”, in which the
particle draws energy from its depot to move against an external force, are
shown in blue; the trajectories shown in red involve only the “inactive
phase”, in which the depot energy remains untapped and the particle moves
ballistically. In particular Fig. 1(a) shows the particle’s momentum $p$
versus the rescaled time, which decreases steadily under the external force
during inactive phases, but oscillates around a constant positive momentum
$p_{\text{c}}$, moving steadily against the external force, in the active
phase. Figure 1(b) shows correspondingly that the internal energy in the depot
degree of freedom remains essentially constant during inactive phases, but
steadily drops to power the active phase motion.

Figure 1:  Left panels: The evolution of the canonical momentums $p$ and $I$
generated by the Hamiltonian (1) plotted over rescaled time $\lambda(t)$
(defined in (7)). The blue and red trajectories have only slightly different
initial values of the angle coordinate $\alpha$, but these two trajectories
show the two dramatically different dynamical phases of the system. In the red
case the motion is almost the same as if the fuel and motional subsystems were
decoupled, whereas in the blue case most of the energy of the fuel depot is
used to move the particle against the external force. Right panel: The phase
space $\left(\varphi,I\right)$ of the transformed Hamiltonian (3) for the
rescaled time $-\lambda/I_{0}=0.29$. The red diamond and blue square
correspond to the trajectories shown in the left panels, at the time marked
there with diamonds and squares. In (c) the instantaneous energy contours on
which the system is moving at that instant in the two trajectories are marked
in red and blue, respectively. The closed black contour is the instantaneous
separatrix that marks the division in phase space between active and inactive
dynamical phases.

These two dynamical phases, and the possibility of transitions between them,
are the central theme of this paper. Figure 1(c) shows a two-dimensional
reduced phase space, to be constructed below, in which the two kinds of orbit
that represent the two phases can be recognized, divided from each other by a
separatrix, like in the pendulum case former mentioned. Figure 1 was generated
using the Hamiltonian (1), which will be explained in the following, with the
dimensionless parameters $M=\Omega=I_{0}=k=1$, $g=0.001$ and
$\varepsilon=0.02$. The initial values at $-\lambda_{i}/I_{0}=-1.75$ are
$kq_{i}=0$, $p_{i}/p_{c}=2$ and $I_{i}/I_{0}=0.75$. The only difference
between the blue and the red curve in Fig. 2 is the initial value for
$\alpha$, which is $\alpha_{i}=0$ ($\alpha_{i}=-\pi/2$) for the blue (red)
curve. As explained in ClassicalDaemon , the isolated daemon can be understood
in terms of (mostly) adiabatic evolution within this reduced phase space. This
we now briefly review.

###  II.2 II.2 The active particle Hamiltonian

The daemon Hamiltonian of ClassicalDaemon  is a sum of terms representing the
motion of the particle, the depot, and the coupling between them:

$\displaystyle H\\!\left(q,p,\alpha,I\right)$
$\displaystyle=H_{\text{M}}\\!\left(q,p\right)+H_{\text{D}}\\!\left(\alpha,I\right)+\varepsilon
H_{\text{C}}\\!\left(q,\alpha,p,I\right),$ (1)

with a small dimensionless parameter $\varepsilon$ multiplying the coupling
term $H_{C}$, so that the coupling will transfer energy slowly from the depot
into motion and thereby model an active particle rather than a rocket or bomb.
The motion and depot terms in $H$ are ordinary:

$\displaystyle H_{\text{M}}=\frac{p^{2}}{2M}+fq,$ (2) $\displaystyle
H_{\text{D}}=\Omega I\;.$ (3)

$M$ is the mass of the active particle, which can move in the one dimension of
position $q$ with momentum $p$, subject to the constant external opposing
force $-f$. The fuel depot $H_{\text{D}}$ is assumed for simplicity to be
proportional to a single action variable $I$, with the dynamical frequency of
the fuel degree of freedom being $\Omega$. We stipulate that $\Omega$ is by
far the highest frequency in our problem, in order to represent the depot
energy as densely stored (large energy for small mass and size)
ClassicalDaemon .

###  II.3 II.3 Nonlinear coupling

The special coupling $H_{\text{C}}$ allows the daemon to achieve secular
energy transfer from $H_{\text{D}}$ to $H_{\text{M}}$ in spite of the large
frequency gap $\Omega$ and weak coupling $\epsilon$:

$H_{\text{C}}=-\Omega\sqrt{I_{0}^{2}-I^{2}}\cos\\!\left(kq-\alpha\right)\;,$
(4)

where $k$ is a constant with units of inverse length, $\alpha$ is the
canonically conjugate angle variable to the fuel depot action variable $I$,
and the constant $I_{0}$ is a bound on the fuel depot action value such that
$-I_{0}\leq I\leq I_{0}$.

This $H_{\text{C}}$ (4) provides an effectively resonant coupling (a _Chirikov
resonance_ Chirikov ) between the motion of the active particle and its fuel
depot, as long as the particle moves at close to the critical speed
$v_{\text{c}}=\Omega/k$, since then $kq\sim\Omega t$ and $\cos(kq-\alpha)$
becomes slowly changing in time instead of rapidly oscillating. Unlike a
typical Chirikov resonance, in which the long-term effect of the resonant
couple is limited by nonlinear stabilization Chirikov ; ClassicalDaemon , this
resonance can sustain itself over long times: the internally generated force
on the particle from $H_{\text{C}}$ keeps the particle speed near
$v_{\text{c}}$ (momentum near $p_{\text{c}}=M\Omega/k$) while the power to
sustain this speed against the external force $f$ (and as we will see below,
any viscous force) is steadily drawn from the depot energy $H_{\text{D}}$.

Whether the Chirikov resonance actually does sustain active motion in this way
is a dynamical question within the Hamiltonian mechanics of (1). As we will
now explain, the Hamiltonian active motion only persists within the central
“eye” region, bounded by a separatrix, of the effective two-dimensional phase
space shown in Fig. 1(c). In the absence of any environment the isolated
system described by (1) can remain within this separatrix, performing active
motion against the external force, for a certain finite time which is
generally _shorter_ than the limit imposed by energy conservation alone
ClassicalDaemon . Our question in the present paper is how environmental
dissipation and noise may change the duration of active motion by expelling
the system from the active region sooner—or keeping it in the active region
longer.

###  II.4 II.4 Effective 2D phase space

The Hamiltonian (1) can be simplified by a canonical transformation from
$(q,p),(\alpha,I)$ to the new variables $(q,J),(\varphi,I)$ using

$\displaystyle\alpha\to\ $ $\displaystyle\alpha(\varphi,q)=\varphi+kq,$
$\displaystyle p\to\ $ $\displaystyle p(J,I,t)=J-kI-ft\;.$ (5)

This simplifies the system because $J=p+kI+ft$ is an exact constant of the
motion (even though it depends explicitly on time $t$!):
$\frac{\mathrm{d}J}{\mathrm{d}t}=\left\\{J,H\right\\}+\frac{\partial
J}{\partial t}=0$. Taking the time-dependence of the canonical transformation
properly into account, the Hamiltonian in the new coordinates is

$\displaystyle{H^{\prime}}$ $\displaystyle=\frac{[p(J,I,t)]^{2}}{2M}+\Omega
I-\varepsilon\Omega\sqrt{I_{0}^{2}-I^{2}}\cos\\!\left(\varphi\right)$
$\displaystyle\equiv\frac{k^{2}}{2M}\Bigl{[}I-\lambda(t)\Bigr{]}^{2}-\varepsilon\Omega\sqrt{I_{0}^{2}-I^{2}}\cos\\!\left(\varphi\right)$
$\displaystyle\quad+E_{0}(J,t),$ (6)

where

$\displaystyle E_{0}(J,t)$
$\displaystyle=-\frac{M\Omega^{2}}{2k^{2}}-\frac{f\Omega
t}{k}+\frac{\Omega}{k}J,$ $\displaystyle\lambda(t)$
$\displaystyle=\frac{f}{k}(t_{0}-t),$ $\displaystyle t_{0}(J)$
$\displaystyle=\frac{Jk-M\Omega}{fk}\;.$ (7)

The evolution of $(q,J)$ is thus trivial once $\varphi(t)$ and $I(t)$ have
been found from equations of motion that do not involve $q$. For these the
term $E_{0}(J,t)$ is irrelevant and can be ignored, while the effective shift
in the time origin $t_{0}(J)$ is likewise trivial. Our system without
dissipation is thus exactly represented as a time-dependent rotor model, even
though the full Hamiltonian, with its two degrees of freedom, is time-
independent.

###  II.5 II.5 Adiabatic regime

The maximum force which $H_{\text{C}}$ can exert on the particle is
$\varepsilon k\Omega I_{0}$, and so the particle can only make its way
actively against the opposing force if $f<\varepsilon k\Omega I_{0}$; we
assume this from now on. We also note that getting the active particle up to
the critical speed $v_{c}$ requires an initial energy input
$Mv_{\text{c}}^{2}/2$, and so it will only be worthwhile trying to exploit our
system’s energy depot if $\Delta\leq 1$, with

$\displaystyle\Delta^{2}=\frac{M\Omega^{2}/k^{2}}{\Omega I_{0}}.$ (8)

The typical rate at which the Hamiltonian $H^{\prime}$ changes is $f/(\Omega
kI_{0})$, while the frequency of orbits around the instantaneous fixed point
at $\varphi=0$ and $I=-f(t-t_{0})/(\Omega k)$ is $(\varepsilon\Omega
I_{0}k^{2}/M)^{1/2}$. The ratio of these two rates is therefore small in all
regimes we consider, and so the adiabatic theorem of classical mechanics tells
us that our effective rotor system closely follows orbits of constant
instantaneous $H^{\prime}=E$, even as these slowly move and deform while
$H^{\prime}$ changes. Such instantaneous orbits are shown in Fig. 1(c). The
slow (adiabatic) motion and deformation of the orbits depends on time only
through the parameter $\lambda(t)$ defined in (7); the dimensionless ratio
$-\lambda(t)/I_{0}$ can be considered as a conveniently rescaled time
variable.

For our particular $H^{\prime}(\varphi,I,t)$, the eye-shaped separatrix
migrates slowly but steadily downward in $I$, so that on all orbits inside the
separatrix the depot steadily loses energy, sustaining active motion at a
nearly steady speed of $v_{\text{c}}$. Outside the separatrix, in contrast,
orbits merely flatten or bend slightly over time, leaving $I$ nearly constant
and providing no power to move the particle actively.

As well as moving slowly downward in $I$, the separatrix also slowly changes
in shape and size. In particular, as the mean $I$ inside the separatrix
approaches the depot ground state $I=-I_{0}$, the $\sqrt{I_{0}^{2}-I^{2}}$
factor makes the area enclosed by the separatrix shrink to zero. This is
decisive, because under adiabatic conditions the phase space area enclosed by
the system’s actual orbit stays constant (_adiabatic invariance_). The system
can therefore only remain inside the separatrix, keeping the particle active,
as long as the separatrix area is larger than its conserved orbit area. Once
the separatrix shrinks smaller than this, the system exits the separatrix and
the particle goes inactive, even if $H_{\text{D}}$ still holds enough energy
to power more active motion.

We refer readers to ClassicalDaemon  and Liouvillepaper  for further details
on these mechanisms. The question for the present investigation is how the
addition of environmental dissipation and noise, which modify both Liouville’s
theorem and the adiabatic theorem, can alter the behavior of this Hamiltonian
active particle system.

##  III III. Hamiltonian active particle in an environment: Stochastic
collisions

We now consider that our active particle moves through a (one-dimensional)
dilute gas of light particles having mass $m\ll M$. Elastic collisions occur
at random times as the continuous limit of a Poissonian sequence, i.e. with an
exponentially decaying probability distribution of durations $\tau$ between
collisions, $P(\tau)=\bar{\tau}^{-1}e^{-\tau/\bar{\tau}}$ for the average
collision rate $1/\bar{\tau}$. Collisions will alter the active particle’s
momentum instantaneously; between collisions the system evolves under (1) as
in Section II above. The full evolution of the system is thus a piecewise
deterministic process Piecewise_deterministic  which can easily be simulated,
and so we will begin our investigation of Hamiltonian active Brownian
particles by exploring the qualitative effects of a Brownian environment on
our Hamiltonian active particles within this collisional model. After
identifying a potentially dramatic and counter-intuitive effect in which
environmental dissipation can actually help the active particle travel farther
than it otherwise would, we will simplify our model to pure viscous damping in
order to obtain some more general results.

###  III.1 III.1 Including collisions in the model

An elastic collision between the active particle with mass $M$ and an
environment particle with mass $m$ changes $p$ to $p+\delta p$ with

$\delta p=\mu\left(Mv-p\right),$ (9)

where $\mu=\frac{2m}{M+m}\ll 1$. The velocity $v$ of an environment particle
is a Gaussian random variable with a standard deviation $\sigma$, representing
a Maxwell-Boltzmann distribution in 1D.

To see what an environment can do to our Hamiltonian active particle,
therefore, we randomly generate a sequence of collision times at average rate
$1/\bar{\tau}$, at each of which momentum jumps with random $v$ are applied.
Between collisions Hamiltonian evolution under (1)-(4) rules instead. A
typical evolution is shown in Fig. 2, which can be compared directly with Fig.
1—though the comparison is in some ways not trivial. Figure 2 was generated
with the same parameters as Fig. 1 and the additional collision parameters
$\sigma=2\frac{p_{\text{c}}}{M}$, $\mu=\frac{2}{1001}$ and
$\overline{\tau}=\mu\frac{kI_{0}}{f}$.

Figure 2:  Collisions lead to random jumps in the momentum $p$ and as a
consequence to the random behavior of $I$. The coupling $H_{\text{C}}$ manages
nevertheless to keep the momentum near the critical value $p_{c}$ during the
running phase, as can be seen in (a), while $I$ decreases on average linearly
(b). In the reduced phase space a collision leads to a shift of the separatrix
as can be seen in (c). In (a) the red (blue) dot corresponds to before (after)
a collision. In the reduced phase space shown in (c) both dots share the same
location but belong to different energy contours. The horizontal axis variable
$-\lambda(t)/I_{0}$ is now only a monotonic function of $t$ on average: the
inset in (a) shows the small horizontal jitter in the parametric curve
$(\lambda(t),p(t))$, as explained in the text.

A somewhat unusual feature of the horizontal axis variable $-\lambda/I_{0}$ in
Fig. 2 is that it is no longer a simple linear function of time $t$, as it was
in Fig. 1. The definition is still

$\displaystyle-\frac{\lambda(t)}{I_{0}}$
$\displaystyle=\frac{f}{kI_{0}}t+\frac{M\Omega}{I_{0}k^{2}}-\frac{J}{kI_{0}},$
$\displaystyle J$ $\displaystyle=p+kI+ft\;,$ (10)

and $J$ is still exactly constant between collisions. Each instantaneous
collision when it occurs, however, gives a (partly) random kick to $p$; since
$I$ is not directly changed in a collision, the effect on the evolution of
$\varphi$ and $I$ is to give $J$ a random shift in each collision. The
dimensionless ratio $-\lambda(t)/I_{0}$ therefore increases in constant linear
proportion to $t$ between collisions, but makes small a random jump (which can
even be backward!) at each randomly timed collision. These effective small
jitters in “time” (_i.e._ in $-\lambda(t)/I_{0}$) can be seen in the inset of
Fig. 2(a). This quasi-Brownian motion in $\lambda(t)$ provides a Brownian
component in $\dot{\varphi}$ under $H^{\prime}$, so that our active particle
is indeed Brownian, even in the reduced phase $(\varphi,I)$ phase space.

On the larger scale that is easily visible in Fig. 2, nonetheless, the overall
behavior of the Brownian active particle remains quite similar to the case
without collisions with gas particles. An active phase of the dynamics still
exists, in which the particle moves for a certain time with a nearly constant
momentum $p_{c}$, doing work against the external force while the energy in
the internal depot decreases. The self-sustaining Chirikov resonance that is
the minimalistically simple mechanism for this Hamiltonian active particle may
be an exotic piece of nonlinear dynamics but it is not especially fragile. It
can still power an active particle with Brownian noise and dissipation turned
on.

###  III.2 III.2 Viscous drag

One significant effect of the dissipative environment on the active particle
is not apparent in Fig. 2: some of the power being drawn from the depot must
now be used to overcome viscous damping, as the active particle moves through
the gas. This basic effect does not show up in the decrease of $\Omega I$ with
$-\lambda/I_{0}$, however, but rather in making $-\lambda/I_{0}$ increase
faster, on average, with $t$.

This effect can be seen from the fact that between collisions we have
$-\dot{\lambda}(t)/I_{0}=f/(kI_{0})$, while according to (9) the average
change of $-\lambda(t)/I_{0}$ in a collision is $-\mu p_{\text{c}}/(kI_{0})$
in the active phase (since then $p$ stays close to $p_{c}=M\Omega/k$). With
collisions occurring at the average rate $1/\bar{\tau}$, the average increase
rate in $t$ of our rescaled “dimensionless time” is therefore

$\displaystyle\left\langle\frac{d}{dt}\left(-\frac{\lambda(t)}{I_{0}}\right)\right\rangle$
$\displaystyle=\frac{f+\gamma p_{\text{c}}}{kI_{0}},$ (11)

where

$\displaystyle\gamma$ $\displaystyle=\frac{\mu}{\bar{\tau}}$ (12)

can be recognized as the damping coefficient for our one-dimensional ideal gas
dissipation model, inasmuch as for $\varepsilon\to 0$ we would have the
average time dependence $\dot{p}=-f-\gamma p$.

This basic effect of viscous drag can be seen directly by plotting
$H_{\text{M}}$, $H_{\text{D}}$, and their sum; see Fig. 3. In this Figure the
horizontal axis is $-\lambda/I_{0}$ for the case without dissipation, plotted
in blue for comparison; the case without dissipation is simply plotted against
the linearly rescaled time $t$ that coincides with the dissipationless
$-\lambda/I_{0}$. The depot energy is thus depleted sooner because of gas
particle collisions, although the active particle still moves against the same
external force $f$ at the same average $v_{\text{c}}$ that is set by the
Chirikov resonance. Since the collisions are elastic the missing energy has
clearly gone into heating the gas. The collision parameters for the red curve
are $\sigma=0.6\frac{p_{c}}{M}$, $\mu=\frac{2}{101}$ and
$\bar{\tau}=\mu\frac{kI_{0}}{f}$. Both curves start with initially
$p_{i}/p_{\text{c}}=1$. All other parameters are chosen as in Fig. 2.

Figure 3:  Comparison of the energy between the cases with and without
collisions, where $-\lambda/I_{0}$ corresponds to the case without collisions.
The energy in the work term increases at the beginning similar in both cases.
The energy transfer to the work term stops earlier for the case with
collisions. For this case energy gets drained faster from the fuel. The total
energy does not stay constant, since heat is created in the environment. (The
work and total energy have been shifted, by the same constant in all cases,
for visual clarity. The work and total energy with dissipation remain rather
flat for a while after $\lambda=0$, but apart from the visible Brownian
fluctuations these curves do closely follow the motion of a particle
decelerating from $v_{\text{c}}$ and finally attaining terminal velocity
$-f/(\gamma M)$ in the negative direction, under constant force $-f$ and
damping $\gamma$.)

###  III.3 III.3 Thermal fluctuations

With sufficiently mild Brownian kicking the working efficiency of the active
particle is reduced by viscous drag, but the basic operation of the active
particle persists. Stronger thermal fluctuations from gas collisions can
indeed disrupt the Hamiltonian mechanism, however. To show this we present two
different numerical evolutions in Fig. 4, with the gas particle mass $m$
differing by an order of magnitude; the horizontal axis is again the
dimensionless $-\lambda/I_{0}$ rescaled time of the dissipationless case,
which is also shown again for comparison. The vertical axis here is neither
$I$ nor $p$ but rather the action variable $S$, defined as the area enclosed
in the $\varphi,I$ phase-space plane by the contour on which the instantaneous
function $H^{\prime}$ equals the instantaneous value of $H^{\prime}$. The
quantity $A_{0}=4\pi I_{0}$ is the total area of the phase space. The fact
that in Fig. 4 the blue curve of the dissipationless case oscillates around a
constant value represents the well-known adiabatic invariance of the action
Goldstein . All lines correspond to the same initial values
$I_{i}/I_{0}=0.85$, $p_{i}/p_{\text{c}}=1$, and $\alpha_{i}=\frac{5}{8}\pi$.
The collision properties for the red line are $\sigma=0.15\frac{p_{c}}{M}$,
$\mu=\frac{2}{101}$, $\bar{\tau}=\mu\frac{kI_{0}}{f}$, and
$\sigma=0.1\frac{p_{c}}{M}$, $\mu=\frac{2}{1001}$,
$\bar{\tau}=\mu\frac{kI_{0}}{f}$ for the orange line.

Figure 4:  Action for different collision parameters and the separatrix
plotted over the corresponding $-\lambda/I_{0}$. The system changes its state
from running to stalling, if the action of the orbit equals the area of the
separatrix. Friction leads to a decreasing action, and with that to a lower
value of $\lambda$ (orange) compared to the collision less case (blue). Strong
fluctuations might lead to unpredictable transitions (red). On the red curve
collisions are marked with dots.

The black border in Fig. 4 shows the phase space area enclosed by the
instantaneous separatrix within which the active phase occurs. When the
system’s trajectory crosses this border, the active phase ceases. We can see
that for large enough gas particle mass $m$ the larger Brownian fluctuations
in $p$ can include an unlucky run that drives the system outside the
separatrix well before it would have exited without dissipation. For smaller
$m$, in contrast, such large stochastic sorties become extremely unlikely.

The range of possible behaviors of complex Hamiltonian mechanisms like this
one under stochastic forces is too large to explore fully within the scope of
this paper; here we simply note that a low-temperature regime does exist in
which the stochastic effects remain small. Further studies of fluctuations,
perhaps using Fokker-Planck equations, will be left for the future; we focus
now instead upon a significant effect of pure dissipation apart from the
simple drag and heating discussed above.

##  IV IV. Dissipative control

Even though the plots of $p$ and $I$ versus $-\lambda/I_{0}$ in Fig. 2 do not
show the additional energy drain of the viscous drag, they do show a
noticeable trend of steadily _decreasing amplitude_ in the _oscillations_ of
$p$ and $I$ around the constant or linear decrease, respectively, of the
active phase. A similar decrease in oscillation amplitude can also be seen in
Fig. 3; the steadily decreasing action variable in Fig. 4 is the same effect,
since the area enclosed by orbits in phase space is proportional to the
oscillation amplitude squared. The amplitude decrease in these particular
cases is modest, but its potential significance can be seen in the fact that
the orange curve in Fig. 4, with dissipation and weak noise, exits the
separatrix slightly _later_ than the blue curve with no dissipation. This
means that the active phase actually lasts a bit longer in the case with
dissipation than it would have without dissipation.

###  IV.1 IV. 1 Shrinking action

With different parameter values this dissipative effect can become more
dramatic. To investigate the effect systematically we replace our random
collision model with a constant viscous damping, eliminating the random
momentum kicks; this is equivalent to taking the limits $\mu\to 0$ and
$\bar{\tau}\to 0$ in our collision model while keeping $\gamma=\mu/\bar{\tau}$
fixed. We therefore return from our 2D reduced phase space to the full
canonical Hamiltonian equations of motions derived from (1), with the addition
of damping on the particle momentum $p$:

$\displaystyle\dot{p}$ $\displaystyle=-f-\varepsilon
k\Omega\sqrt{I_{0}^{2}-I^{2}}\sin(kq-\alpha)-\gamma p\;,$
$\displaystyle\dot{q}$ $\displaystyle=\frac{p}{M}\;,$ $\displaystyle\dot{I}$
$\displaystyle=\varepsilon\Omega\sqrt{I_{0}^{2}-I^{2}}\sin(kq-\alpha)\;,$
$\displaystyle\dot{\alpha}$ $\displaystyle=\Omega+\varepsilon\frac{\Omega
I}{\sqrt{I_{0}^{2}-I^{2}}}\cos(kq-\alpha)\;.$ (13)

In the active phase of evolution we will have $p\sim M\Omega/k$ as explained
above, and so active motion will only be possible if the total retarding force
$f+\gamma M\Omega/k$ is less than the maximum force $\varepsilon k\Omega
I_{0}$ that $H_{\text{C}}$ can exert.

In Appendix A we use multiple time scale analysis to show that for small
$\sqrt{\varepsilon}$ the adiabatic invariance of the system’s action variable
$S$ is replaced by exponential decay at rate $\gamma$ during the running
phase:

$S(t)=S_{i}e^{-\gamma t}[1+\mathcal{O}(\sqrt{\varepsilon})]\;,$ (14)

where the order $\sqrt{\varepsilon}$ corrections remain small even over long
times. This surprisingly simple but non-trivial result holds even when the
adiabatic orbits are of finite size and the motion in phase space is quite
anharmonic.

This exponential decay of the orbit area in the $(\varphi,I)$ phase space (or
equivalently of the area in the alternative 2D phase space $(q-\alpha/k,p)$)
is an effect of dissipation besides the increased rate of consumption of
$H_{\text{D}}$ due to viscous drag with coefficient $\gamma$ on the particle
moving at steady speed $v_{c}$. The decay of $S(t)$ represents damped motion
of the internal mechanism of the active particle, rather than of the motion of
the particle itself; it is analogous to damping of engine vibration in a car,
rather than to aerodynamic drag on the car’s body. This internal damping
occurs as an indirect effect of the damping that acts directly only on the
active particle’s linear momentum, simply because the particle’s internal
mechanism is coupled to its momentum—as it must be, of course, in order to
power the particle’s motion actively.

###  IV.2 IV.2 Effectiveness of internal damping

The internal damping can potentially have a more significant effect on the
system’s long-term evolution than the viscous drag. The decrease in efficiency
due to drag is simply the proportion of drag force in the total force opposing
the active motion, $\gamma p_{c}/(f+\gamma p_{c})$. The exponential decay of
$S(t)$ continues as long as the active phase lasts, however. Over the total
running time of order

$t_{\text{max}}=\frac{I_{0}\Omega}{(f+\gamma
p_{\text{c}})v_{\text{c}}}=\frac{I_{0}k}{f+\gamma p_{\text{c}}}$ (15)

the e-fold reduction of $S$ will be of order

$\ln\frac{S_{i}}{S(t_{\text{max}})}=\gamma t_{\text{max}}=\frac{\gamma
p_{\text{c}}}{f+\gamma p_{\text{c}}}\frac{I_{0}k^{2}}{M\Omega}=\frac{\gamma
p_{\text{c}}}{f+\gamma p_{\text{c}}}\frac{1}{\Delta}\;,$ (16)

with $\Delta$ defined in (8).

In our numerical evolutions so far we have used $\Delta=1$, in order to avoid
the large ratio of time scales that makes it hard to see all significant
features of time evolution in a single plot. An active particle which can
remain active for a long time because its depot energy is large must have
large $1/\Delta$, however. The longer an active particle remains active, the
more significant the exponentially compounding effects of internal dissipation
become, in comparison to the simple flat rate of efficiency reduction due to
viscous drag.

Such significant internal damping can actually be advantageous, because the
mechanism by which a Hamiltonian active particle converts depot energy into
motional work is not simply a phenomenological _fiat_ constrained only by
energy conservation, but a deterministic dynamical process that stops and
starts under its own equations of motion. The starting and stopping of the
active phase for our particular Hamiltonian active particle model involves
nonlinear dynamics with multiple time scales and a Chirikov resonance, but any
actual active particle will have to control itself, mechanically, somehow.
Dissipation can help with control.

###  IV.3 IV.3 Duration of the active phase

In our particular case the starting and stopping of the active phase is
determined by the adiabatic growing and shrinking of the instantaneous
separatrix in phase space ClassicalDaemon ; Liouvillepaper . As shown in Fig.
4, as $-\lambda/I_{0}$ increases the separatrix first grows and then shrinks.
We consider that the active phase must start from the inactive phase; we do
not consider initial conditions that are fine-tuned, by external forces not
represented in our Hamiltonian, to ensure that the system starts in the active
phase. Exactly how the system enters the separatrix is a non-trivial problem
in post-adiabatic dynamics, because the adiabatic theorem breaks down at the
separatrix ClassicalDaemon ; Liouvillepaper , but whenever the separatrix is
growing there must be some orbits which enter it from outside, because
Liouville’s theorem tells us that the density of orbits inside the separatrix
cannot decrease, and dissipation will only draw in more orbits, because the
active phase is an energy basin.

Our system will therefore always begin its active phase on an orbit just
inside the separatrix, for some initial value of depot fuel level $I/I_{0}$.
The initial value of the system’s action $S_{i}$ is therefore the area
enclosed by the separatrix for $I=I_{i}$. The integral which defines the
separatrix area $S_{\text{sep}}(I/I_{0})$ is given in Appendix A; in the limit
$f+\gamma p_{\text{c}}\to 0$ it approaches

$\displaystyle S_{\text{sep}}\\!\left(\frac{I}{I_{0}}\right)$
$\displaystyle\doteq
16\sqrt{\frac{\varepsilon}{\Delta}}\frac{M\Omega}{k^{2}}\left(1-\left(\frac{I}{I_{0}}\right)^{2}\right)^{1/4}\;.$
(17)

For general $f$ and $\gamma$, $S_{\text{sep}}(I/I_{0})$ remains a well-defined
function which must simply be computed by numerical evaluation of the integral
given in the Appendix. It retains in general the properties that
$S_{\text{sep}}(\pm 1)=0$ and
$S_{\text{sep}}(-I/I_{0})=S_{\text{sep}}(I/I_{0})$.

If the separatrix is expanding the system will remain in the active phase, on
an orbit deeper inside the expanded separatrix, but as fuel is drained the
separatrix must eventually shrink, disappearing completely when $I/I_{0}=-1$.
The system will exit the separatrix even before the depot is thus drained to
its ground state, however, if the system’s orbit action does not decrease fast
enough to remain inside the shrinking separatrix. The depot energy level
$I_{f}$ at the end of the active phase is thus fixed by the condition

$S_{\text{sep}}\left(\frac{I_{f}}{I_{0}}\right)=S_{\text{sep}}\left(\frac{I_{i}}{I_{0}}\right)e^{-\gamma(t_{f}-t_{i})}\;.$
(18)

If we then consider that the steady fuel drain in the active phase implies

$t_{f}-t_{i}=\frac{(I_{i}-I_{f})\Omega}{(f+\gamma
p_{\text{c}})v_{\text{c}}}=\frac{(I_{i}-I_{f})k}{(f+\gamma p_{\text{c}})}\;,$
(19)

we can conclude that the duration of the active phase is fixed by

$S_{\text{sep}}\Big{(}\frac{I_{i}}{I_{0}}\Big{)}e^{-\frac{\gamma
p_{c}}{f+\gamma
p_{c}}\frac{1}{\Delta}\frac{I_{i}}{I_{0}}}=S_{\text{sep}}\Big{(}\frac{I_{f}}{I_{0}}\Big{)}e^{-\frac{\gamma
p_{c}}{f+\gamma p_{c}}\frac{1}{\Delta}\frac{I_{f}}{I_{0}}}\;.$ (20)

The systematic multiple time scale analysis in the Appendix confirms this
result, to leading order in the small parameter $\sqrt{\varepsilon}$.

The general properties that $S_{sep}(I/I_{0})=S_{sep}(-I/I_{0})$ and that
$S_{sep}(\pm 1)=0$ imply that for $I_{i}=I_{0}$ we always have $I_{f}=-I_{0}$,
so that all depot energy is successfully consumed regardless of $\gamma
p_{c}/f$, while for $\gamma=0$ and $I_{i}<I_{0}$ we have $I_{f}=-I_{i}$ (and
the active phase can never begin for $I_{i}<0$ ClassicalDaemon ). For
$I_{i}<1$ the active particle without dissipation is inefficient, in that it
is unable to use all its potentially available fuel. Whenever dissipation is
strong enough that

$\exp\left[-\frac{\gamma p_{c}}{\gamma p_{c}+f}\frac{1}{\Delta}\right]\doteq
0\;,$ (21)

however, we will have $I_{f}\doteq-I_{0}$ regardless of $I_{i}$, meaning that
all fuel is used. When $\Delta$ is small even a very slight amount of
dissipation can dramatically boost the efficiency with which the Hamiltonian
active particle can exploit its energy depot, by extending the duration of the
active phase.

###  IV.4 IV.4 Efficiency

To measure this effect we define the total efficiency $\eta$ of the active
particle as the fraction of depot energy which is successfully consumed, times
the fraction of power which goes into motional work against the external force
$f$ rather than being expended against drag and turned into heat:

$\eta=\frac{I_{i}-I_{f}}{I_{i}+I_{0}}\frac{f}{\gamma p_{c}+f}\;.$ (22)

The analytical predictions for this efficiency are plotted for different
$I_{i}$ in Fig. 5, as a function of $\gamma p_{c}/f$, for a case in which the
multiple time scale theory should be very good because $\varepsilon=10^{-4}$,
and in which the efficiency boost from dissipation can be substantial because
$\Delta=1/4$. Also shown is the same efficiency for a single numerical
solution of the equations of motion (IV.1), from an initial condition just
inside the initial separatrix. The numerically exact evolution generally falls
a few percent short of the analytically predicted efficiency; this is what can
be expected for $\sqrt{\varepsilon}=0.01$.

Figure 5:  Efficiency $\eta$ versus $\gamma p_{c}/f$ for different initial
values $I_{i}$. Dots are from numerical evolutions under the equations of
motion; the curves are the analytical predictions as explained in the text.
Efficiency can be boosted by dissipation significantly.

A less extreme case with the same parameters $\varepsilon=0.2$ and $\Delta=1$,
as in the plots of our previous Sections, shows that dissipation can still
increase efficiency even when conditions are not so ideal, see Fig. 6. At this
considerably larger value of $\sqrt{\varepsilon}$ the post-adiabatic
corrections are larger, and the dependence of efficiency on precise initial
conditions (just where the system enters the separatrix at the start of the
active phase) is significant. The points in Fig. 6 are therefore the average
efficiencies of 100 different numerical evolutions with random initial
conditions for $\alpha$.

Figure 6:  Efficiency $\eta$ versus $\gamma p_{c}/f$ for different initial
values $I_{i}$ with less adiabatic parameters than in Fig. 5.

Even in this much less extreme case the dissipatively enhanced control of the
dynamical process which powers the active particle can more than compensate
for the losses to drag. This enhancement will not matter if the fuel depot can
reliably be filled to $I_{i}=I_{0}$, since then the Chirikov resonance can
sustain itself until all depot fuel is consumed even without dissipation.
Dissipative control can be a major advantage, however, if filling the active
particle’s depot to its maximum capacity cannot be guaranteed. The question of
how the energy depot can be filled leads us to our next Section.

##  V V. Collisions that can change the energy of $H_{\text{D}}$

In Section III we allowed the momentum $p$ of the weight in Hamiltonian (1) to
change through stochastic collisions; the collisions were idealized as
instantaneous momentum transfer between point-particles but our model for
their effects was certainly consistent with Hamiltonian two-body mechanics. In
this complementary section we examine a scenario in which random collisions
instead change the canonical momentum $I$ of the depot subsystem (3). In this
section we will also further explore the relationship between Hamiltonian
mechanics and the phenomenological concept of an active particle, by
experimenting with merging the two kinds of description. Leaving a fully
Hamiltonian description of energy harvesting for future work, we will now
simply interrupt our Hamiltonian time evolution with certain randomly timed
kicks, and then discuss the results and their meaning in our final Section.

###  V.1 V.1 Motivating a model

The kind of kicks which will now affect $I$ are motivated as follows. The
basic idea is that since our active particle has an internal structure it
should be able to exchange with its environment not only the kinetic energy of
its active motion but also the internal energy of its depot. The depot energy
will still be described by the $\left(\alpha,I\right)$ system of (1)-(4), but
we now consider this to be an effective description of a system which is
really composed of multiple subsystems. Our final model will retain the
continuous $I$ variable, but to develop the model we will initially imagine
our depot in discrete terms, as an ensemble of some number $N$ of micro-
depots. We seek a model for how our active particle might be able to gradually
replenish its depot from the environment, by thinking of how it could refill
one micro-depot at a time.

In our discretized motivation for the model, therefore, we imagine an
environment composed of small grains that can each hold enough energy to fill
one empty micro-depot. For simplicity as we motivate the model, we allow each
grain to have only two states: drained or full, denoted by $+$ or $-$. We
propose that a fraction $P_{+}$ of grains are initially in the $+$ state while
$(1-P_{+})$ are in the $-$ state. In an effort to represent the time reversal
symmetry that should be present in a more microscopic description, we will
allow encounters with $+$ grains to fill an empty micro-depot, while
encounters with $-$ grains will deplete the active particle’s depot
analogously, draining one micro-depot.

We further suppose that the chance of actually filling up a micro-depot during
an encounter with a $+$ grain should be proportional to the number of empty
micro-depots that are available to be filled, while likewise the chance of
draining a micro-depot into a $-$ grain is proportional to the number of
currently full micro-depots.

Having motivated our model with a discrete picture, then, we return to the
continuous $I$ with the following stochastic process added to the Hamiltonian
equations of motion for (1)-(4). Collisions occur at random times at the
average rate $1/\bar{\tau}$, as in Section III. In each collision exactly
three abrupt changes in $I$ are possible. With probability $P(I)$ the change
is $I\to I+\delta I$, for a fixed constant $\delta I$ (as long as
$I<I_{0}-\delta I$). With probability $\bar{P}(I)$ the change is instead $I\to
I-\delta I$ (as long as $I>-I_{0}+\delta I$). With the remaining probability
$1-P(I)-\bar{P}(I)$ there is then no change to $I$, because no empty or full
micro-depot happened to connect with the respectively $+$ or $-$ grain. Since
in the actual continuum description of the depot the proportions of full and
empty micro-depots in our motivational picture should correspond to $(I_{0}\pm
I)/(2I_{0})$, the probabilities motivated by our micro-depot picture are
therefore

$\displaystyle P(I)$ $\displaystyle=P_{+}\frac{I_{0}-I}{2I_{0}},$
$\displaystyle\bar{P}(I)$ $\displaystyle=(1-P_{+})\frac{I_{0}+I}{2I_{0}},$
$\displaystyle 1-P(I)-\bar{P}(I)$
$\displaystyle=\frac{1}{2}+\left(P_{+}-\frac{1}{2}\right)\frac{I}{2I_{0}}\;,$
(23)

where it may be noted that the chance of no change in $I$ satisfies $1/4\leq
1-P(I)-\bar{P}(I)\leq 3/4$. The rate of $+\delta I$ collisions in relation to
$-\delta I$ collisions tends to increase as $I$ is lower, as if the active
particle feeds more aggressively when it is hungry.

###  V.2 V.2 Expectations from the $I$ kicks

To anticipate what we may see for small but finite $\delta I$ and
$\bar{\tau}$, it is straightforward to consider the Langevin-like differential
equation which our random kicks would imply in the limit where $\delta I$ and
$\bar{\tau}$ both go to zero while keeping their ratio fixed. In this limit no
stochastic noise survives; we obtain a deterministic set of equations of
motion with a damping-like term in the equation for $I$:

$\displaystyle\dot{I}=-\Gamma\left(I-I_{\text{s}}\right)-\frac{\partial
H}{\partial\alpha}\;,$ (24)

where

$\displaystyle\Gamma$ $\displaystyle=\frac{\delta I}{I_{0}\bar{\tau}},$
$\displaystyle I_{\text{s}}=$ $\displaystyle\left(2P_{+}-1\right)I_{0}.$ (25)

The presence of $I_{\text{s}}$ makes this different from the otherwise similar
case in Section IV.

Since $I_{\text{s}}>-I_{0}$ for any $P_{+}>0$, the fuel depot without coupling
$H_{\text{C}}$ to the active particle motion would always tend to reach
equilibrium at some $I=I_{\text{s}}$ above its ground state $I=-I_{0}$.
Whether $H_{\text{D}}$ increases or decreases then in general depends on the
initial conditions of $I$. If $I$ is initially above $I_{\text{s}}$, it will
decrease and increase otherwise. As opposed to the change in $p$ due to
collisions, which always tends to lower the energy of $H_{\text{M}}$, and with
that the total energy of (1), the change in $I$ can now lead to an energy flow
into the system.

To anticipate what may happen when the fuel depot is put under load to drive
the active phase of our Hamiltonian active particle, we can note that moving
at the operating speed $v_{\text{c}}$ against the external force $f$ implies a
power drain $v_{\text{c}}f=-\Omega\dot{I}$ from the depot. In the active phase
we therefore approximate

$\displaystyle\dot{I}=-\Gamma\left(I-I_{\text{s}}\right)-\frac{f}{k}\;,$ (26)

which drives $I$ towards

$I_{\text{a}}=I_{\text{s}}-\frac{f}{\Gamma k}\;.$ (27)

Depending on the various parameters it can happen that $I_{\text{a}}$ is too
low—or too high!—for the force from $H_{\text{C}}$, which is proportional to
$\sqrt{I_{0}^{2}-I^{2}}$, to overcome $f$ and keep the active particle moving.
It can even be that $I_{\text{a}}<-I_{0}$, so that the depot is driven to
$I=-I_{0}$ where the force from $H_{\text{C}}$ vanishes and the active phase
always ends.

It would seem from this analysis, however, that for sufficiently high $\Gamma$
and $P_{+}$ we will have $I_{\text{a}}$ well within the operating range of the
active phase. The active particle should therefore be able to sustain its
active phase indefinitely, by harvesting energy as it moves through its
environment to continuously replenish its internal energy depot.

###  V.3 V.3 Results with energy-harvesting $I$ kicks

Such a perpetual active state can be seen in Fig. 7, which shows three
representative trajectories for small but finite $\delta I$ and $\bar{\tau}$,
differing only in the probability $P_{+}$, which is $0.05$ for the blue,
$0.20$ for the orange, and $0.40$ for the red curves. The time measure
$t/\frac{kI_{0}}{f}$ would be the same as the $-\lambda/I_{0}$ axis of our
previous plots, if we now had no kicks.

Figure 7:  Momentum measures $p/p_{\text{c}}$ (a) and $I/I_{0}$ (b) for
different probabilities $P_{+}$ of finding an environment particle in a ’+’
state plotted over the time measure $t/\frac{kI_{0}}{f}$. Collisions drain
energy from the fuel system if $I$ is higher than the value $I_{\text{s}}$,
and restore otherwise. If the restoring is high enough the weight might keep
its critical momentum $p_{\text{c}}$ forever (red curve). Hence energy from
the environment is used to lift the weight against a potential.

Without the $I$ kicks the energy transfer would stop at
$t/\frac{kI_{0}}{f}\approx 1$. From the momentum measure $p/p_{c}$ in Fig.
7(a), it can be seen that for the lowest simulated probability $P_{+}=0.05$
the energy transfer stops at $t/\frac{kI_{0}}{f}\approx 0.9$, _before_ it
would have stopped without kicks. For $P_{+}=0.20$ the extended active phase
makes the total amount of energy transferred to the weight increase by a
factor of about $1.5$. And for $P_{+}=0.40$ the active phase goes on forever.
It can be seen in Fig. 7(b) that in the eternal active phase $I$ never falls
to the lower boundary $-I_{0}$, but in the cases where the energy harvesting
only succeeds in extending the duration of the active phase, the active phase
does end when $I$ falls too near the ground state $-I_{0}$. Since this ends
the active phase, the power drain from sustaining the active phase is then
lifted and $I$ settles back to its unloaded asymptotic value $I_{\text{s}}$,
which is $-0.9I_{0}$ in the case $P_{+}=0.05$ and $-0.6I_{0}$ for $P_{+}=0.2$.

Real active particles which are more complicated than this simple model,
including humans, can indeed harvest energy from their environments in order
to sustain active motion for much longer than their internal depots would
allow in isolation. Sufficient consumption efficiency is required, in an
environment that is sufficiently rich in available energy. Our partially
phenomenological model is thus at least qualitatively realistic. We now
conclude by discussing what we have learned from how our models have worked.

##  VI VI. Discussion

Whatever an active particle may actually have for an engine, it must be a
Hamiltonian system of some kind, with dynamical phases at least qualitatively
analogous to those in our model. In such a dynamical system, especially if it
is microscopic, all of the subsystems may well be affected by damping and
noise that act on any of them. So what can happen to dynamical systems of this
unusual but important kind when they are immersed in a dissipative
environment?

###  VI.1 VI.1 Robust mechanisms

In Section II of this paper we reviewed a well-defined microscopic model for a
Hamiltonian active particle, previously presented as an isolated Hamiltonian
system from ClassicalDaemon . In Section III we confirmed the important result
that even though this active particle model relies on the rather exotic
nonlinear mechanism of a Chirikov resonance, it is robust enough to continue
functioning in a dissipative and noisy environment. Too much noise can disrupt
it, and too much dissipation can reduce its efficiency or even stall it
completely, but its tolerances for dissipation and noise are non-zero. The
microscopic mechanisms of real active particles are undoubtedly more
complicated but for the project of nonetheless understanding them from
physical first principles, this result is encouraging.

###  VI.2 VI.2 Dissipative control

In Section IV we then removed noise to focus only on environmental
dissipation, and found that dissipation can actually _enhance_ the efficiency
with which a Hamiltonian active particle exploits its internal depot, by
enhancing its own automatic control of the dynamical process that sustains
active motion.

This is more than just the environment acting as a heat sink in the Carnot
sense. The analogs of cold and hot Carnot reservoirs in the Hamiltonian daemon
are discussed in ClassicalDaemon , where the single rotor degree of freedom
$(\alpha,I)$ of our depot is realized as the Schwinger representation of
angular momentum with two harmonic oscillators of different frequency. The
role of dissipation in enhancing the efficiency of the daemon is more
comparable to the role of cooling a macroscopic engine in preventing excess
micromotion (typically leading to thermal expansion that increases friction
and ultimately makes parts jam together) in the engine’s own moving parts. A
Carnot cycle is an idealized picture of a heat engine in which the engine’s
moving parts are assumed not to deviate from their intended motion in any way,
and not to absorb any heat. In real engines the moving parts, which are
supposed to enforce the thermodynamic cycle mechanically, can in fact overheat
and fail to perform as designed. Approaching closer to the ideal limit of a
Carnot engine requires cooling the real engine’s operating parts, as well as
merely extracting heat from the working fluid into the cold reservoir within
the ideal scheme.

In the daemon the micromotion of the system components, shown in our phase
space plots, is itself the cyclic motion of the active phase engine. It is
nonetheless true that what ultimately limits the efficiency of the Hamiltonian
daemon ClassicalDaemon  is having too much micromotion. The difference from
the overheating problem in macroscopic engines is that in the isolated daemon
micromotion does not increase, but is simply conserved, under the adiabatic
theorem. The efficiency of the daemon can be increased by damping this
undesirable micromotion away. If we consider that dissipation can enhance
engine efficiency by suppressing micromotion in internal components that would
limit efficiency, though, then the role of dissipative control in enhancing
the efficiency of a Hamiltonian active particle can indeed be considered a
microscopic analog of cooling an engine.

Understanding this potential benefit of the dissipative environment for active
Brownian particles has been a major result of our paper:

_Dissipation can increase the total amount of energy transfer from the
internal depot to the motional degree of freedom._

###  VI.3 VI.3 Energy harvesting

In Section V we showed that phenomenological and Hamiltonian approaches to
active particle modelling can be compatible, at least in yielding realistic
results. In particular we showed that it is possible to model the continuous
replenishment of the energy depot of a Hamiltonian active particle, even to
the limit where energy harvesting can sustain active motion indefinitely.

Careful consideration of our procedures in Section V also shows some
limitations of the phenomenological approach, however. Any Hamiltonian system
is intrinsically possible under our most basic understanding of the laws of
nature, but phenomenologically guessed evolution rules may potentially violate
those basic laws, even when this is not obvious. Within the seemingly
reasonable assumptions of our motivational models of micro-depots and grains,
for example, it is not clear that our active particle’s perpetual motion could
not be sustained by harvesting energy from an environment in thermal
equilibrium at a positive temperature. This would be a perpetual motion
machine of the Second Kind, forbidden by the Second Law of Thermodynamics.

The perpetual active motion that we showed in Section V does not have to
violate the Second Law, if it presumes an environment which is far from
thermodynamical equilibrium or has a negative temperature. The problem with
the phenomenological model is not that it is necessarily qualitatively wrong,
but only that it does not reliably indicate when it may be qualitatively
wrong. Identifying such problems can require the kind of careful reasoning
that is typically needed to resolve the apparent paradoxes of Maxwell’s Demon
scenarios, of which this energy harvesting model is potentially one.

Instead of inductively searching for clues to the mistaken assumptions in that
way, the alternative procedure that we mean to suggest with the whole of this
paper is to formulate the entire problem within Hamiltonian mechanics, as at
the start of this paper, and proceed deductively to work out what works, or
does not, when consistency with fundamental mechanics is built-in from the
start.

For many purposes that may not be needed. When enough is known about how
systems in fact behave, phenomenological models can be checked against that
knowledge. To understand fully how active particles work, and know whether
previous limits can in fact be exceeded, may require a deeper understanding,
however. A fully Hamiltonian description of energy harvesting by an active
particle is a project in progress.

The authors acknowledge support from State Research Center OPTIMAS and the
Deutsche Forschungsgemein- schaft (DFG) through SFB/TR185 (OSCAR), Project No.
277625399.

## References

  * (1) F.S. Gnesotto, F. Mura, J. Gladrow, and C.P. Broedersz, Broken detailed balance and non-equilibrium dynamics in living systems: a review, Rep. Prog. Phys. 81 066601 (2018)
  * (2) P. Reimann, Brownian motors: noisy transport far from equilibrium, Phys. Rep. 361 57–265 (2002)
  * (3) B. Lau, O. Kedem, J. Schwabacher, D. Kwasnieskiab and E. A. Weiss, An introduction to ratchets in chemistry and biology, Mater. Horiz. 4, 310-318 (2017)
  * (4) P. Hänggi, F. Marchesoni, and F. Nori, Brownian motors, Ann. Phys. 14(1-3) 51-70 (2005)
  * (5) K. Villa and M. Pumera, Fuel-free light-driven micro/nanomachines: artificial active matter mimicking nature Chem. Soc. Rev. 48, 4966, (2019)
  * (6) T. Xu, L. Xu, and X. Zhang, Ultrasound propulsion of micro-/nanomotors, Appl. Mater. Today 9, 493–503 (2017)
  * (7) P. Pietzonka, É. Fodor, C. Lohrmann, M. E. Cates, and U. Seifert, Autonomous Engines Driven by Active Matter: Energetics and Design Principles Phys. Rev. X 9, 041032 (2019)
  * (8) S. Ramaswamy, The Mechanics and Statistics of Active Matter, Annu. Rev. Condensed Matter Phys. 1, 323-345 (2010)
  * (9) F. Schweitzer, W. Ebeling, and B. Tilch, Complex Motion of Brownian Particles with Energy Depots, Phys. Rev. Lett. 80, 5044 (1998)
  * (10) F. Schweitzer, B. Tilch, and W. Ebeling, Uphill motion of active brownian particles in piecewise linear potentials, Eur. Phys. J. B 14, 157–168 (2000)
  * (11) Y. Zhang, C. Kim, and K.-J.-B. Lee, Active motions of Brownian particles in a generalized energy-depot model, New J. Phys. 10, 103018 (2008)
  * (12) A. Glück, H. Hüffel, and S. Ilijić, Canonical active Brownian motion, Phys. Rev. E 79 021120 (2009)
  * (13) B. Tilch, F. Schweitzer, and W. Ebeling, Directed motion of Brownian particles with internal energy depot, Physica A 273 294-314 (1999)
  * (14) T. Dittrich and N.A. Naranjo, Directed transport in a ratchet with internal and chemical freedoms, Chemical Physics 375 486–491 (2010)
  * (15) P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, and L.Schimansky-Geier, Active Brownian Particles. From Individual to Collective Stochastic Dynamics, Eur. Phys. J. Spec. Top. 202, 1-162 (2012)
  * (16) P.J. Park and K.-J.-B. Lee, A modified active Brownian dynamics model using asymmetric energy conversion and its application to the molecular motor system, J. Biol. Phys. 39, 439–452 (2013)
  * (17) T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel Type of Phase Transition in a System of Self-Driven Particles, Phys. Rev. Lett. 75, 1226 (1995).
  * (18) K. Kanazawa, T. Sano, A. Cairoli, and A. Baule, Loopy Lévy flights enhance tracer diffusion in active suspensions, Nature 579, 364–367 (2020).
  * (19) U. Khadka, V. Holubec, H. Yang, and F. Cichos, Active particles bound by information flows. Nat. Comm. 9, 3864 (2018)
  * (20) C. Touya, T. Schwalger, and B. Lindner, Relation between cooperative molecular motors and active Brownian particles, Phys. Rev. E 83, 051913 (2011)
  * (21) C. Bechinger, R. Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016)
  * (22) Patrick McGivern, Active materials: minimal models of cognition?, Adaptive Behavior (2019)
  * (23) R. Marsland III and J. England, Limits of predictions in thermodynamic systems: a review, Rep. Prog. Phys. 81 016601 (2018)
  * (24) U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75 126001 (2012)
  * (25) C. Ganguly and D. Chaudhuri, Stochastic thermodynamics of active Brownian particles, Phys. Rev. E 88, 032102 (2013)
  * (26) P. Pietzonka and U. Seifert, Entropy production of active particles and for particles in active baths, J. Phys. A: Math. Theor. 51 01LT01 (2018)
  * (27) K. Sekimoto, Stochastic energetics, Lect. Notes Phys. 799 (2010)
  * (28) M.E. Cates, Diffusive transport without detailed balance in motile bacteria: does microbiology need statistical physics?,Rep. Prog. Phys. 75 042601 (2012)
  * (29) A.-S. Smith, Physics challenged by cells, Nat. Phys., 6, 726-729 (2010)
  * (30) N.L. Abbott and O.D. Velev, Active particles propelled into researchers’ focus, Curr. Opin. Colloid Interface Sci. 21, 1–3 (2016).
  * (31) L. Gilz, E. Thesing, and J.R. Anglin, Hamiltonian analogs of combustion engines: A systematic exception to adiabatic decoupling, Phys. Rev. E 94, 042127 (2016)
  * (32) B.V. Chirikov, Research concerning the theory of non-linear resonance and stochasticity, Print N 267 Institute of Nuclear Physics, Novosibirsk, (1969)
  * (33) T. Eichmann, E.P. Thesing, and J.R. Anglin, Engineering separatrix volume as a control technique for dynamical transitions, Phys. Rev. E 98, 052216 (2018)
  * (34) M. Davis and M. Dempster, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models, J. of the Royal Stat. Soc. Series B: Methodological, 46, 353-388 (1984)
  * (35) H.P. Goldstein, P. Charles, and J.L. Safko, Classical mechanics, Addison Wesley (2002)

##  VII Appendix:

###  VII.1 A. Multiple time scale analysis of the dissipative daemon

Here we do not begin with the effective 2D phase space of Section II, but
consider the full canonical Hamiltonian equations of motions derived from (1)
with the addition of a standard viscous damping term on the particle momentum
$p$, with damping coefficient $\gamma$:

$\displaystyle\dot{p}$ $\displaystyle=-f-\varepsilon
k\Omega\sqrt{I_{0}^{2}-I^{2}}\sin(kq-\alpha)-\gamma p\;,$
$\displaystyle\dot{q}$ $\displaystyle=\frac{p}{M}\;,$ $\displaystyle\dot{I}$
$\displaystyle=\varepsilon\Omega\sqrt{I_{0}^{2}-I^{2}}\sin(kq-\alpha)\;,$
$\displaystyle\dot{\alpha}$ $\displaystyle=\Omega+\varepsilon\frac{\Omega
I}{\sqrt{I_{0}^{2}-I^{2}}}\cos(kq-\alpha)\;.$ (28)

In the active phase of evolution we will have $p\sim M\Omega/k$ as explained
above, and so active motion will only be possible if the total retarding force
$f+\gamma M\Omega/k$ is less than the maximum force from $H_{\text{C}}$
$\varepsilon k\Omega I_{0}$.

We now introduce dimensionless variables involving explicit dimensionless
factors $\varepsilon\ll 1$ and $\Delta=\sqrt{M\Omega/(I_{0}k^{2})}\leq 1$:

$\displaystyle t$
$\displaystyle=\frac{\Delta}{\sqrt{\varepsilon}}\frac{\tilde{t}}{\Omega}\;,$
$\displaystyle q$ $\displaystyle=\frac{Q+\alpha}{k}\;,$ $\displaystyle f$
$\displaystyle=\varepsilon I_{0}k\Omega F\;,$ $\displaystyle p$
$\displaystyle=\frac{M\Omega}{k}\Big{(}1+\frac{\sqrt{\varepsilon}}{\Delta}P\Big{)}\;,$
$\displaystyle\gamma$ $\displaystyle=\frac{\varepsilon
I_{0}k^{2}}{M}\Gamma\;,$ $\displaystyle I$ $\displaystyle=XI_{0}\;.$ (29)

With these definitions we expect the active phase to be possible for
$F+\Gamma<1$. This yields the dimensionless equations of motion

$\displaystyle\frac{dQ}{d\tilde{t}}$
$\displaystyle=P-\sqrt{\varepsilon}\Delta\frac{X\cos Q}{\sqrt{1-X^{2}}}\;,$
$\displaystyle\frac{dP}{d\tilde{t}}$
$\displaystyle=-(F+\Gamma)-\sqrt{1-X^{2}}\sin
Q-\frac{\sqrt{\varepsilon}}{\Delta}\Gamma P\;,$
$\displaystyle\frac{dX}{d\tilde{t}}$
$\displaystyle=\sqrt{\varepsilon}\Delta\sqrt{1-X^{2}}\sin Q\;.$ (30)

We do not need the additional equations of motion for $q$ or $\alpha$
separately, given $Q$, since these are trivial; the closed system of equations
(VII.1) determines the full evolution.

We then exploit the assumed smallness of $\varepsilon$ by implementing the
standard multiple time scale perturbation theory. We interpret the
dimensionless time coordinate $\tilde{t}$ as a line of small slope in the
plane of fast and slow times $\tau$ and $T$:

$\displaystyle(\tau,T)$ $\displaystyle=(1,\sqrt{\varepsilon})\tilde{t}\;,$
$\displaystyle Q(\tau,T)$
$\displaystyle=\sum_{n=0}^{\infty}Q_{n}(\tau,T)\varepsilon^{n/2}\;,$
$\displaystyle P(\tau,T)$
$\displaystyle=\sum_{n=0}^{\infty}P_{n}(\tau,T)\varepsilon^{n/2}\;,$
$\displaystyle X(\tau,T)$
$\displaystyle=\sum_{n=0}^{\infty}X_{n}(\tau,T)\varepsilon^{n/2}\;.$ (31)

The evolution of the dynamical variables is only physically meaningful along
the time line $(\tau,T)=(1,\sqrt{\varepsilon})\tilde{t}$; we are free to
invent arbitrary evolution in the rest of the $(\tau,T)$ plane, and as usual
in multiple time scale analysis we use this freedom to force all non-periodic
evolution to occur in $T$ rather than $\tau$. Otherwise we simply expand order
by order in $\sqrt{\varepsilon}$ as in ordinary classical Hamiltonian
perturbation theory.

At order $\varepsilon^{0}$ we find

$\displaystyle\frac{\partial Q_{0}}{\partial\tau}$ $\displaystyle=P_{0}\;,$
$\displaystyle\frac{\partial P_{0}}{\partial\tau}$
$\displaystyle=-(F+\Gamma)-\sqrt{1-X_{0}^{2}}\sin Q_{0}$ (32)

plus the conclusion that $X_{0}=X_{0}(T)$ cannot depend on the fast time
$\tau$. These equations of motion have the first integral

$E_{0}(T)=\frac{P_{0}^{2}}{2}+(F+\Gamma)Q_{0}-\sqrt{1-X_{0}(T)^{2}}\cos
Q_{0}\;,$ (33)

recognizable of course as an energy which is independent of $\tau$ but depends
on $T$ in a manner as yet undetermined.

Identifying this first integral $E_{0}(T)$ may be considered as solving the
dependence of $Q_{0}$ and $P_{0}$ on $\tau$. As $\tau$ increases while $T$ is
held fixed, $Q_{0}(\tau,T)$ and $P_{0}(\tau,T)$ trace out orbits in their 2D
phase space. In particular for any value of $X_{0}(T)=X$ we can identify the
zeroth-order unstable fixed point $(Q_{0},P_{0})=(Q_{S}(X),0)$ for

$Q_{S}(X)=-\sin^{-1}\Big{(}\frac{F+\Gamma}{\sqrt{1-X^{2}}}\Big{)}\;.$ (34)

The energy $E_{0}$ at this fixed point $(Q_{0},P_{0})\to(Q_{S}(X),0)$ is the
energy of the separatrix $E_{S}(X)$ ; the single closed orbit of $Q_{0},P_{0}$
in dependence on $\tau$ with fixed $T$ and $E_{0}(T)=E_{S}(X(T))$ is the
separatrix itself at slow time $T$. All orbits with

$E_{0}<E_{S}(X)=(F+\Gamma)Q_{S}(X)+\sqrt{\frac{1-X^{2}-(F+\Gamma)^{2}}{1-X^{2}}}$
(35)

evolve in $\tau$ as closed orbits in the $Q_{0},P_{0}$ plane, inside the
separatrix. The separatrix as well as the individual orbits all depend on $T$
(through $X\to X_{0}(T)$) as well as on $\tau$; with the $\tau$-dependence
solved we now turn to finding this $T$-dependence—a question analogous to
determine climate change, having solved for the weather.

To determine the $T$-dependence of all our variables it will turn out to be
useful to define the area enclosed in the $(Q_{0},P_{0})$ phase space by a
closed $\tau$-orbit with energy $E_{0}=E\leq E_{S}(X)$, namely the _action_

$\displaystyle S_{0}(E,X):=\oint\\!d\tau P_{0}^{2}=\oint
dQP_{0}(Q,E,X)=2\sqrt{2}\int_{Q_{-}(E,T)}^{Q_{+}(E,T)}\\!\\!\\!\\!\\!\\!dQ\,\sqrt{E-(F+\Gamma)Q+\sqrt{1-X^{2}}\cos
Q}\;,$ (36)

where $\oint\\!d\tau$ denotes the integral over a full period in $\tau$, and
$Q_{\pm}(E,T)$ are the two values of $Q$ (modulo $2\pi$) at which the
integrand vanishes (the turning points where $\partial_{\tau}Q_{0}=P_{0}$
vanishes for $E_{0}=E$). The reason we care about this action is that it will
turn out below to have a simple evolution in $T$. The evolution in $T$ of
$S_{0}(E,T)$ determines implicitly the $T$-dependence of $E_{0}$, and this in
turn fixes the $T$-dependence of $Q_{0}$ and $P_{0}$.

Anticipating its use below, therefore, we compute the derivative of $S_{0}$
with respect to $T$ and find

$\displaystyle\partial_{T}S_{0}\big{(}E_{0}(T),X_{0}(T)\big{)}$
$\displaystyle=2\int_{Q_{-}(E_{0},T)}^{Q_{+}(E_{0},T)}\\!dQ\,\frac{\partial_{T}E_{0}(T)-\big{(}1-X_{0}(T)^{2}\big{)}^{-1/2}\,X_{0}\partial_{T}X_{0}\,\cos
Q}{\sqrt{2\Big{(}E_{0}-(F+\Gamma)Q+\sqrt{1-X_{0}(T)^{2}}\cos Q\Big{)}}}$
$\displaystyle\equiv\oint\\!d\tau\,\Big{(}\partial_{T}E_{0}(T)+\cos\big{(}Q_{0}(\tau,T)\big{)}\partial_{T}\sqrt{1-X_{0}(T)^{2}}\Big{)}$
$\displaystyle\equiv\oint\\!d\tau\,\Big{(}P_{0}\partial_{T}P_{0}+[F+\Gamma+\sqrt{1-X_{0}^{2}}\sin
Q_{0}]\partial_{T}Q_{0}\Big{)}\;,$ (37)

where in the first line there is no contribution from the derivatives of the
integration limits because the integrand vanishes at them, and the $\oint$
symbol in the second and third lines denotes integration over a single period
of the evolution in $\tau$.

To obtain the actual $T$-dependence $S_{0}$ according to the equations of
motion, we proceed to order $\varepsilon^{1/2}$ in our perturbative theory. We
first examine

$\displaystyle\partial_{\tau}X_{1}$
$\displaystyle=-\partial_{T}X_{0}+\Delta\sqrt{1-X_{0}^{2}}\sin
Q_{0}=-X^{\prime}_{0}(T)-\Delta(\partial_{\tau}P_{0}+F+\Gamma)\;.$ (38)

The constraint that all steadily growing or decreasing behavior must appear as
dependence on $T$ rather than on $\tau$ means that our functions must all
depend on $\tau$ periodically (though the period can depend on energy
arbitrarily), and so whatever this periodic evolution in $\tau$ may be, we can
integrate both sides of (38) with respect to $\tau$ over one full period to
obtain

$X^{\prime}_{0}(T)=-\Delta(F+\Gamma)\;.$ (39)

Expressed in the dimensionless variables this is the result described in our
main text, that the fuel depot $I$ steadily decreases, when the particle is
active, to provide the power needed to sustain steady motion against the
external and dissipative force.

Inserting (39) in (38) then yields

$X_{1}(\tau,T)=-\Delta P_{0}(\tau,T)+x_{1}(T)$ (40)

for some function $x_{1}(T)$.

We further find

$\displaystyle\frac{\partial Q_{1}}{\partial\tau}-P_{1}$
$\displaystyle=-\partial_{T}Q_{0}-\Delta\frac{X_{0}\cos
Q_{0}}{\sqrt{1-X_{0}^{2}}}\;,$ (41) $\displaystyle\frac{\partial
P_{1}}{\partial\tau}+\sqrt{1-X_{0}^{2}}\cos(Q_{0})Q_{1}$
$\displaystyle=-\partial_{T}P_{0}-\frac{\Gamma}{\Delta}P_{0}+\frac{X_{0}X_{1}\sin(Q_{0})}{\sqrt{1-X_{0}^{2}}}$
$\displaystyle=-\partial_{T}P_{0}-\frac{\Gamma}{\Delta}P_{0}-\frac{X_{0}(T)[x_{1}(T)-\Delta
P_{0}]}{1-X_{0}(T)^{2}}\left(\partial_{\tau}P_{0}+F+\Gamma\right)\;.$ (42)

We then note that integration by parts and Eqns. (VII.1) imply

$\oint\\!d\tau\,\left[\left(\sqrt{1-X_{0}^{2}}\sin(Q_{0})+F+\Gamma\right)[\partial_{\tau}Q_{1}-P_{1}]+P_{0}[\partial_{\tau}P_{1}+\sqrt{1-X_{0}^{2}}\cos(Q_{0})Q_{1}]\right]=0$
(43)

when the integration is over any full period in $\tau$. We insert (41) and
(42) in (43) and discard total derivatives with respect to $\tau$ from the
full-period integral, then compare the result with (VII.1) to reveal the
surprisingly simple conclusion

$\frac{dS_{0}}{dT}=-\frac{\Gamma}{\Delta}S_{0}(T)\;.$ (44)

This may be recognized as the generalization to include dissipation of the
classical adiabatic theorem, according to which (with $\Gamma=0$) the action
is a so-called _adiabatic invariant_ , meaning that even though the energy and
all other dynamical variables may have slow trends that gradually accumulate
to significant change, these trends will conspire to keep their particular
combination in the action constant even over long times.

With dissipation instead the action decreases exponentially. Restoring
physical time units reveals that up to corrections of order
$\sqrt{\varepsilon}$ (the higher order perturbative terms $n>0$), which remain
that small even over long times, the action defined by $S_{0}$ decreases
exponentially at the rate $\gamma$, _i.e._

$S_{0}(t)=S_{0}(0)e^{-\gamma t}\;.$ (45)

To determine the efficiency with which the active particle uses its depot
energy before the active phase ceases, we use the fact that the active phase
can only begin with the system entering an orbit just inside the separatrix
when $X$ has the initial value $I_{i}/I_{0}$. (We do not consider initial
conditions that have been fine-tuned, through some external process not
described by our Hamiltonian, to start the system in its active phase. So if
the system is deeper inside the separatrix, then it must already have been
inside the separatrix for some time, meaning that the beginning of the active
phase must have been earlier. The active phase therefore always begins with
the system on the separatrix as it exists for $X(T=0)=I_{i}/I_{0}$.)

While the system is in the active phase, the separatrix grows or shrinks as
$X(T)=X(0)-\Delta(\Gamma+F)T$ changes, while the system’s orbit shrinks under
dissipation. The active phase ends when the area enclosed by the system’s
orbit exceeds the area inside the separatrix. The separatrix area for a given
value of $X(T)$ is $S_{sep}(X)=S_{0}(E_{S}(X),X)$, and by identifying the
scaled duration of the active phase as
$T_{f}=[X_{0}(T_{f})-X_{0}(0)]/[(\Gamma+F)\Delta]$ we find the remaining fuel
level $I_{f}=I_{0}X_{0}(T_{f})$ at the end of the active phase from the
condition

$S_{\text{sep}}(I_{i}/I_{0})e^{-\frac{\Gamma}{\Gamma+F}\frac{I_{i}}{I_{0}\Delta^{2}}}=S_{\text{sep}}(I_{f}/I_{0})e^{-\frac{\Gamma}{\Gamma+F}\frac{I_{f}}{I_{0}\Delta^{2}}}\;.$
(46)

To see that (46) is a well-defined condition which can be solved numerically
to determine $X_{f}(X_{i})$, we can note that in the limit of small $\Gamma+F$
we can approximate $Q_{S}(X)=Q_{-}(X)\doteq-\pi\doteq-Q_{+}(X)$ and
$E_{S}(X)\doteq\sqrt{1-X^{2}}$ to obtain the explicit analytical formula

$\displaystyle S_{\text{sep}}(X)$ $\displaystyle\doteq
2\sqrt{2}\left(1-X^{2}\right)^{1/4}\int_{-\pi}^{\pi}\\!\\!\\!\\!\\!\\!dQ\,\sqrt{1+\cos
Q}$ $\displaystyle=16\left(1-X^{2}\right)^{1/4}\;.$ (47)

For general $F$ and $\Gamma$ the numerical task of solving (46) is a bit
harder because $S_{\text{sep}}(X)$ itself requires numerical integration. This
is how we obtained the adiabatically predicted efficiency curves in our main
text.

Regardless of exactly how $S_{\text{sep}}(X)$ behaves it is easy to note that
$S_{\text{sep}}(X)=S_{\text{sep}}(-X)$ and that $S_{\text{sep}}(\pm 1)=0$, so
that $I_{i}=1$ implies $X_{f}=-1$ and all fuel is used regardless of
$\Gamma/F$, while for $\Gamma=0$ and $I_{i}<I_{0}$ we have $I_{f}=-I_{i}$ (and
the active phase can never begin for $I_{i}<0$ ClassicalDaemon ). For
$I_{i}<1$ the active particle without dissipation is inefficient, in that it
is unable to use all its potentially available fuel. Whenever dissipation is
strong enough that

$\exp\left[-\frac{\Gamma}{\Gamma+F}\frac{1}{\Delta^{2}}\right]\doteq 0\;,$
(48)

however, we will have $X_{f}\doteq-1$ regardless of $X_{i}$, meaning that all
fuel is used. When $\Delta^{2}$ is small even a very slight amount of
dissipation can dramatically boost the efficiency with which the Hamiltonian
active particle can exploit its energy depot.