ARTICLES
PUBLISHED ONLINE: 30 MARCH 2014 | DOI: 10.1038/NNANO.2014.40
Dynamic relaxation of a levitated nanoparticle
from a non-equilibrium steady state
Jan Gieseler1, Romain Quidant1,2, Christoph Dellago3 * and Lukas Novotny4 *
Fluctuation theorems are a generalization of thermodynamics on small scales and provide the tools to characterize the
fluctuations of thermodynamic quantities in non-equilibrium nanoscale systems. They are particularly important for
understanding irreversibility and the second law in fundamental chemical and biological processes that are actively driven,
thus operating far from thermal equilibrium. Here, we apply the framework of fluctuation theorems to investigate the
important case of a system relaxing from a non-equilibrium state towards equilibrium. Using a vacuum-trapped
nanoparticle, we demonstrate experimentally the validity of a fluctuation theorem for the relative entropy change occurring
during relaxation from a non-equilibrium steady state. The platform established here allows non-equilibrium fluctuation
theorems to be studied experimentally for arbitrary steady states and can be extended to investigate quantum fluctuation
theorems as well as systems that do not obey detailed balance.
O
ne of the tenets of statistical physics is the central limit
theorem. It allows systems with many microscopic degrees
of freedom to be reduced to only a few macroscopic thermodynamic variables. The central limit theorem states that, independently of the distribution of microscopic variables, a macroscopic
extensive quantity U, such as the total energy of a system with
N degrees of freedom, follows a Gaussian distribution with mean
kUl / N and variance s2U / N. Consequently, for large N, the relative fluctuations sU /kUl vanish and the macroscopic quantity
becomes sharp. With the advance of nanotechnology it is now possible to study experimentally systems small enough that the relative
fluctuations become comparable to the mean value. This gives rise
to new physics where transient fluctuations may run counter to
the expectations of the second law of thermodynamics1.
The statistical properties of the fluctuations of thermodynamic
quantities like heat, work and entropy production are described
by exact relations known as fluctuation theorems2–5, which allow
us to express the inequalities familiar from macroscopic thermodynamics as equalities6,7. Fluctuation relations are particularly important for understanding fundamental chemical and biological
processes, which occur on the mesoscale where the dynamics are
dominated by thermal fluctuations8. For example, they allow us to
relate the work along non-equilibrium trajectories to thermodynamic free-energy differences9,10. Fluctuation theorems have been
tested experimentally on a variety of systems, including pendulums11, trapped microspheres1, electric circuits12, electron tunnelling13,14, two-level systems15 and single molecules16,17. Most of
these experiments are described by an overdamped Langevin
equation. However, systems in the underdamped regime18, or in
quantum systems19 where the concept of a classical trajectory
loses its meaning, are less explored.
Here, we study the thermal relaxation of a highly underdamped
nanomechanical oscillator from a non-equilibrium steady state
towards equilibrium. Because of the low damping of our system,
the dynamics can be precisely controlled, even at the quantum
level20–22. This high level of control allows us to produce nonthermal steady states and makes nanomechanical oscillators ideal
candidates for investigating non-equilibrium fluctuations for transitions between arbitrary steady states. Although for the initial
steady state, detailed balance is violated, the relaxation dynamics
are described by a microscopically reversible Langevin equation
that satisfies detailed balance23. Under these conditions, a transient
fluctuation relation holds7,24 for the relative entropy change characterizing the irreversibility of the relaxation process. Similar relations hold
also for relaxation processes in ageing systems as studied both theoretically25 and experimentally26–28 in gels and glasses. For the initial
non-equilibrium steady state generated in our experiment we derive
an analytical expression for the phase-space distribution, which is
y
z
x
ton
+
toff
Feedback
Σ ← Δϕ ← 2Ω0
Parametric drive
Ωmod, ε
Figure 1 | Experimental set-up. A nanoparticle is trapped by a tightly
focused laser beam in high vacuum. In a first experiment, the nanoparticle is
initially cooled by parametric feedback. At time t ¼ toff , the feedback is
switched off and the nanoparticle trajectory is followed as it relaxes to
equilibrium. After relaxation, the feedback is switched on again and the
experiment is repeated. In a second experiment, the nanoparticle is initially
excited by an external modulation of frequency Vmod in addition to feedback
cooling. Again at a time t ¼ toff, both the feedback and the external
modulation are switched off and the nanoparticle is monitored as it relaxes.
1
ICFO–Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain, 2 ICREA–Institució Catalana de Recerca i
Estudis Avançats, 08010 Barcelona, Spain, 3 University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria, 4 ETH Zürich, Photonics
Laboratory, 8093 Zürich, Switzerland. * e-mail: [email protected]; [email protected]
358
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40
toff
Fit
Experimental
0.5
0
−40
Tfb
0
0
0.2
0.4
0.6
0
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0.10
Time (ms)
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75
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150
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Position (nm)
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Position (nm)
Theory
Experimental
Thermal
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10−5
Time (s)
c
×10−3
10−1
ρ(x) (nm−1)
E / kBT0
−150
8
6
5
0s
10−5
0.8
Time (s)
b
50
p(x) (nm–1)
1.0
x (nm)
E / kBT0
a
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DOI: 10.1038/NNANO.2014.40
ρ(x) (nm−1)
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0
0.9 s
10−3
75
10−5
−150
−150
0
0.2
0.4
0.6
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0
75
150
Position (nm)
0.8
Time (s)
Figure 2 | Relaxation from a non-equilibrium steady state generated by parametric feedback cooling. The initial non-equilibrium temperature is Tfb.
At time toff, the feedback is switched off and the particle energy relaxes to the equilibrium energy kBT0. a, Time evolution of the average energy evaluated
from 104 individual experiments. The red dashed line is a fit according to equation (10). Inset: The particle oscillates with constant amplitude on short
timescales. b, Four different realizations of the relaxation experiment. Each run yields a different trajectory and the time it takes for the particle to acquire an
energy of kBT0 deviates considerably from the ensemble average shown by the blue curve in a. c, Time evolution of the position distribution, shown as a
density plot. d, Position distributions evaluated at three different times. Distributions correspond to vertical cross-sections in c. Superimposed red curves in
top and bottom panels are theoretical distributions. The initial distribution deviates notably from a thermal equilibrium distribution with the same average
energy (grey dashed line in top panel). The inset in d (top panel) shows a zoom-in of the top region of the distribution r(x) highlighting the deviation from a
thermal distribution.
in excellent agreement with the experimental data and directly validates the fluctuation theorem. Our experimental framework can be
extended to study transitions between arbitrary steady states and, furthermore, lends itself to the experimental investigation of quantum
fluctuation theorems29 for nanomechanical oscillators20–22.
The experimental set-up is shown in Fig. 1. We consider a silica
nanoparticle of radius r ≈ 75 nm and mass m ≈ 3 × 10218 kg that is
trapped in vacuum by the gradient force of a focused laser beam.
Within the trap, the nanoparticle oscillates in all three spatial directions. To a first approximation, the three motional degrees of
freedom are well decoupled. Hence, the time evolution of the
particle position x is described by the one-dimensional
Langevin equation
1
F fluct + Fext
(1)
ẍ + G0 ẋ + V20 x =
m
where V0/2p ≈ 125 kHz is the particle’s frequency along the direction
of interest, G0 is the friction coefficient and Fext is an externally applied
force. The random nature of the collisions does not only provide deterministic damping G0 , but also a stochastic force Ffluct , which thermalizes the energy of the nanoparticle. The fluctuation–
dissipation theorem links the damping
rate intimately to the strength
of the stochastic force, F fluct (t) = 2mG0 kB T0 j(t), where T0 , kB and
j (t) are the bath temperature, Boltzmann’s constant and white noise
corresponding to kj (t)l ¼ 0 and kj (t)j (t′ )l ¼ d(t 2 t′ ).
The total energy of the harmonically oscillating nanoparticle is
given by
1
p2
1
= mV20 x(t)2
E(x, p) = mV20 x2 +
2m 2
2
(2)
where x is the displacement from the trap centre and p is the
momentum. The second equality in the above equation follows
from
the
slowly
varying
amplitude
approximation,
x(t) = x sin(V0 t), x˙ ≪ V0 x. This approximation is well satisfied
in our experiments because it takes many oscillation periods for
the oscillation amplitude to change appreciably (Fig. 2a, inset).
Applying a time-dependent external force Fext for a sufficiently
long time, the system is initially prepared in a non-equilibrium
steady state with distribution rss(u, a), which, in general, is not
known analytically. Here, u specifies the state of the system and a
denotes one or several parameters that determine the initial
steady-state distribution, such as the strength of the external force.
At time t ¼ toff the external force is switched off and we follow
the evolution of the undisturbed system. In this relaxation phase
(external force Fext off ) the dynamics satisfies detailed balance
with respect to the equilibrium distribution req / exp(2b0E(u))
at reciprocal temperature b0 ¼ 1/kBT0. As shown be Evans and
Searles24,30 for thermostatted dynamics and by Seifert7 for stochastic
dynamics, the time reversibility of the underlying dynamics implies
the transient fluctuation theorem
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359
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a
b
100
20 ms
50 ms
100 ms
10−1
10−2
p(Δ )
ρfb(E)
100
Experimental
Fit
Thermal
10−2
0.1
0.2
0.3
150 ms
300 ms
10−3
−4
10−4
0
0.4
−2
E/kBT0
c
5
d
0
Δ
2
4
8
6
∑( p(Δ ) t)
∑(Δ )
4
3
2
4
Experimental
Theory
2
1
0
DOI: 10.1038/NNANO.2014.40
0
1
2
3
4
5
Δ
0
0
2
4
6
8
Δ
Figure 3 | Fluctuation theorem for the relaxation experiment in Fig. 2. a, Energy distribution with feedback on (red circles). The black solid curve is a fit
according to equation (11). Large-amplitude oscillations experience stronger damping and are therefore suppressed relative to an equilibrium distribution
(grey dashed line). b, Probability density p(DS) evaluated for different times after switching off the feedback. c, Function S(DS) evaluated for the
distributions shown in b. d, Function S evaluated for the time-averaged distributions kp(DS)lt. The data are in good agreement with the fluctuation theorem
of equation (3) (black dashed line).
p(−DS)/p(DS) = e−DS
(3)
such that the average relative entropy change is non-negative. The
average relative entropy change is related to the total entropy
change of the oscillator and bath together by7
(4)
kDSl = DS tot + D(rt rss )
holding for the relative entropy change
DS = b0 Q + Df
Here, Q is the heat absorbed by the bath at reciprocal temperature
b0. Because no work is done on the system, the heat Q exchanged
along a trajectory of length t starting at u0 and ending at ut equals
the energy lost by the system, Q ¼ 2[E(ut) 2 E(u0)]. The quantity
Df ¼ f(ut) 2 f(u0) is the difference of the trajectory-dependent
entropy f(u) ¼ 2ln rss(u, a) (ref. 31) between the initial and
final states of the trajectory. Thus, DS is the change in relative
entropy32, or Kullback–Leibler divergence, between the initial
steady-state distribution and the equilibrium distribution observed
along a particular trajectory. Note that the fluctuation theorem (3)
holds for any time t at which DS is evaluated and it is not required
that the system reaches the equilibrium distribution at time t. The
relative entropy change, which equals the dissipation function introduced by Evans and Searles for thermostatted dynamics24,30,33, is the
logarithmic ratio of the probability to observe a particular trajectory
and the probability of the corresponding time-reversed
trajectory7,34,35. As such, DS can be viewed as a measure of the
irreversibility occurring during the relaxation process.
From the detailed fluctuation theorem of equation (3), the
integral fluctuation theorem
ke−DS l = 1
(5)
directly follows. Through Jensen’s inequality, the convexity of the
exponential function implies the second law-like inequality
kDSl ≥ 0
360
(6)
(7)
where D(rtrss) is the relative entropy of the statistical state of the
system at time t with respect to the initial steady-state distribution.
Slightly modifying the definition of DS, one can also derive a different but related integral fluctuation theorem7,31,36, from which the
non-negativity of the total entropy change follows, DStot ≥ 0, providing a direct link to the second law of thermodynamics.
However, no detailed fluctuation theorem holds for this case.
Analogous fluctuation relations for the total entropy production
have also been verified for two coupled systems kept in a nonequilibrium steady state by holding each system at a different
temperature37,38. For further discussion of the fluctuation theorem
and the significance of DS, see Supplementary Information, Section 2.
If the initial steady-state distribution is an equilibrium distribution, rss(u, a) ¼ e 2b[E(u)2F(b)], corresponding to a temperature
T ¼ 1/kBb and with free energy F(b) ¼ 2kBT ln du e2bE(u),
the expressions become particularly simple and the fluctuation
theorem for DS acquires a physically very transparent meaning.
In this case, f(u) ¼ b[E(u) 2 F(b)], such that DS ¼ (b0 – b)Q
and the fluctuation theorem simplifies to p(2Q)/p(Q) ¼
exp{2(b0 2 b)Q}. Note that this particular fluctuation expression
for the special case of transitions between equilibrium states has
been obtained earlier39 and was shown experimentally to hold
also in the case of an ageing bath27. As a consequence of this fluctuation relation for the heat, the probability of observing energy
flowing from the colder system to the hotter bath is exponentially
small compared with the probability of observing energy transfer
in the other direction. Because Q is an extensive quantity, irreversibility for macroscopic systems is a direct consequence of the
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Tmod
E / kBT0
3
d 10−2
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Experimental
2
toff
ρ(x) (nm−1)
a
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DOI: 10.1038/NNANO.2014.40
1
0
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75
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b 8
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6
Run 3
Run 4
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4
ρ(x) (nm−1)
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Harmonic
Experimental
10−4
10−5
0.8
Time (s)
2
0
0
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0.1 s
10−3
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75
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75
10−2
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0s
10−3
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0s
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0.9 s
−150
0
0.2
0.4
0.6
0.8
0.9 s
10−3
10−4
10−5
−150
Time (s)
−75
0
75
Position (nm)
150
Figure 4 | Relaxation from a non-equilibrium steady state generated by external parametric modulation. The initial effective non-equilibrium temperature is
kBTeff. At time toff, the feedback is switched off and the particle energy relaxes to the equilibrium energy kBT0. a, Time evolution of the average energy evaluated
from repeated individual experiments. The red dashed line is a fit according to equation (10). b, Four different realizations of the relaxation experiment.
Each run yields a different trajectory and the time it takes for the particle to acquire an energy of kBT0 deviates considerably from the ensemble average
shown by the blue curve in a. c, Time evolution of the position distribution shown as a density plot. d, Position distributions evaluated at three different times.
Distributions correspond to vertical cross-sections in c. Superimposed red curves in top and bottom panels are theoretical distributions. The initial distribution
features a sharply peaked double-lobe distribution, characteristic for a harmonic oscillator at constant energy. As the system evolves, the two peaks smear
out and merge into a single Gaussian distribution.
fluctuation theorem. The integral fluctuation theorem for the relative entropy change further implies that (b0 2 b)kQl ≥ 0, such
that heat flows from hot to cold on average, in line with the
second law of thermodynamics.
In the following, we experimentally investigate the fluctuation
theorem (3) for two different initial non-equilibrium steady-state distributions. The first steady state is generated by parametric feedback
cooling (ss ¼ fb) and the second one by external modulation (ss ¼
mod) in addition to feedback cooling. In the case of parametric feedback cooling we enforce a non-equilibrium state by applying a force
Fext ¼ Ffb to the oscillating particle through a parametric feedback
scheme (cf. Fig. 1)40. The feedback Ffb = −hmV0 x2 ẋ adds a cold
damping Gfb to the natural damping G0. Here, the parameter h
defines the strength of the feedback. Note that parametric feedback
is different from thermal damping, where an increased damping is
accompanied by an increase in fluctuations. Because parametric feedback adds an amplitude-dependent damping Gfb / x 2, oscillations
with a large amplitude experience a stronger damping than oscillations with a small amplitude. As a consequence, the position distribution is non-Gaussian and assumes the form (see Supplementary
Information, equation 69)
b0 (4+amV20 x2 )2
32a
b0 mV20 (4 + amV20 x2 ) exp −
rfb (x, a) =
8p3
erfc b0 /a
(8)
b0 (4 + amV20 x2 )2
× K1/4
32a
where a ¼ h/mG0V0 , and erfc and K1/4 are the complementary
error function and a generalized Bessel function of the second
kind, respectively. In analogy to the thermal equilibrium temperature of the harmonic oscillator, we define an effective temperature
Tfb ¼ kElfb/kB of the system. Here kElfb denotes the average energy
with feedback on. Using distribution (8) to calculate the average
energy we find the effective temperature
b0
e−b0 /a
b0
4mG0 V0 T0
− 2
(9)
≈
Tfb = T0 2
√
a p erfc b0 /a
a
pkB h
where the approximation holds for Tfb/T0 ≪ 1.
At time t ¼ toff , the feedback is switched off and the system
relaxes back to the thermal equilibrium distribution at temperature
T0. The experimental data for this relaxation process are shown in
Fig. 2c,d. Without the feedback, the collisions with the surrounding
molecules are no longer compensated and the oscillator energy
increases. Exploiting that at low friction the oscillator energy
changes slowly, one finds from equations (1) and (2) that the time
evolution
of the energy is governed by Ė = −G0 (E − kB T0 ) +
2EG0 kB T0 j(t). An average over noise then yields the differential
equation kĖl = −G0 (kEl − kB T0 ), which implies that the average
energy of the oscillator relaxes exponentially to the equilibrium
value kBT0 ,
kE(t)l = kB T0 + kB (Tss − T0 )e−G0 t
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(10)
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a
b
10−1
10−1
p(Δ )
ρfb(E)
100
Experimental
Thermal
10−2
10−3
2.0
3.0
E/kBT0
3.5
10−3
−5.0
4.0
−2.5
0
2.5
5.0
Δ
d
5
20 ms 150 ms
50 ms 300 ms
100 ms
4
∑(Δ )
2.5
10−2
8
6
∑( p(Δ ) t)
c
DOI: 10.1038/NNANO.2014.40
3
2
4
Experimental
Theory
2
1
0
0
0
1
2
3
4
5
Δ
0
2
4
6
8
Δ
Figure 5 | Fluctuation theorem for the relaxation experiment of Fig. 4. a, The energy distribution with external modulation on (red circles) differs
significantly from an equilibrium distribution with identical average energy (grey dashed line). b, Probability density p(DS) evaluated for different times after
switching off the modulation. c, Function S(DS) evaluated for the distributions shown in b. d, Function S evaluated for the time-averaged distributions
kp(DS)lt. The data are in good agreement with the fluctuation theorem of equation (3) (black dashed line).
where Tss denotes an arbitrary initial steady-state temperature, for
example Tfb.
To verify this equation, we repeated the relaxation experiment
104 times. Each time, the same initial distribution rfb(u0 , a) was
established by parametric feedback and, after switching off the feedback, the system was followed as it evolved from u0 to ut within time
t. Along each 1 s trajectory we sampled the particle position at a
rate of 625 kHz and, from integration over 64 successive position
measurements, we obtained the energy at a rate of 9.8 kHz. In
Fig. 2a we show the average over the individual time traces together
with a fit to equation (10). Equilibrium is reached after a time of
order t0 ¼ 1/G0 ¼ 0.17 s. According to equation (10) and the data
shown in Fig. 2, the average energy of the particle increases monotonically. However,
due to the small size of the particle, the fluctu
ating part, 2EG0 kB T0 j(t), is comparable to the deterministic part
2G0(E 2 kBT0), so an individual trajectory can be quite different
from the ensemble average of equation (10). Figure 2b shows four
realizations of the relaxation experiment. Each particle trajectory
x(t) results from switching off the feedback at initial time t ¼ toff.
The 104 trajectories allow us to evaluate the distributions
pfb (DS) = kd[DS − DS(ut )]lfb for different times t. Here, the subscript ‘fb’ denotes the average over the initial distributions obtained
under the action of feedback. For this initial non-equilibrium steady
state, the energy distribution is calculated analytically as (see
Supplementary Information, equation 64)
rfb (E, a) =
ab0 exp −b0 /a
a exp −b0 E + E2
p erfc b0 /a
4
(11)
This distribution has the form of a Boltzmann–Gibbs distribution
for the generalized energy E þ aE 2/4, where the term aE 2/4
arises from the feedback and strongly penalizes high energy
states. It is consistent with the phonon number distribution of an
optomechanical system with a quadratic coupling term41.
362
Inserting the above distribution into equation (4) we find that, for
the relaxation
from rfb , the relative entropy change is given by
DS = b0 a Et2 − E02 /4. In this case, the integral fluctuation
theorem implies that kDE 2l ≥ 0; that is, the average of the squared
energy does not decrease during the relaxation process. Figure 3a
shows the measured steady-state distribution of the energy to be
in excellent agreement with the prediction of equation (11). For
small energies, the measured distribution features a small dip
caused by measurement noise. For comparison, we also show the
corresponding equilibrium distribution with the same average
energy (grey dashed line). It is evident that it deviates strikingly
from the true distribution rfb(E, a). In Fig. 3b we plot the distributions rfb(DS) for different times t. They become increasingly
asymmetric for long times, with higher probabilities for positive
DS and lower probabilities for negative DS. To test fluctuation
theorem (3) for our measurements we define
p(DS)
= DS
(12)
S(DS) = ln
p(−DS)
where S(DS) is predicted to be time-independent. Using the
distributions for DS shown in Fig. 3b, we compute S(DS) and
show the resulting data in Fig. 3c. Because the fluctuation
theorem (3) is time-independent, we evaluate the time-average for
each r(DS) in Fig. 3c and render it in the plot shown in Fig. 2d.
The averaging improves the statistics and leads to excellent agreement with the fluctuation theorem for DS. The offset for small
DS results from measurement noise.
The experimental scheme introduced here allows us to study
non-equilibrium processes for arbitrary initial states and for arbitrary transitions between states. To demonstrate that the fluctuation
theorem holds for arbitrary non-equilibrium initial states, we apply
an external harmonic drive signal in addition to the parametric
feedback as illustrated in Fig. 1. The harmonic drive generates a
force Fmod ¼ e mV20 cos(Vmodt)x acting on the nanoparticle, with
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DOI: 10.1038/NNANO.2014.40
modulation frequency Vmod/2p ¼ 249 kHz and modulation depth
e ¼ 0.03. Modulation at Vmod brings the particle into oscillation
at frequency 124.5 kHz and amplitude x. The resulting steadystate position distribution rmod(x) deviates strongly from an equilibrium Gaussian distribution and resembles the characteristic
double-lobe function
p−1
rmod (x) = √
x2 − x2
(13)
of a harmonic oscillator with constant energy. As in the previous
experiment, at t ¼ toff the modulation and the feedback are switched
off, and the nanoparticle dynamics is measured during relaxation.
Figure 4 shows the relaxation of the particle’s average energy and
the evolution of the position distribution.
Due to the additional driving, the average initial energy is larger
than the thermal energy kBT0. After the driving is switched off, the
average energy relaxes exponentially to the equilibrium value
according to equation (10). As in the previous experiment, individual realizations of the switching experiment differ significantly from
the average (Fig. 4b). As the system relaxes, the two lobes of the
initial position distribution broaden until they merge into a single
Gaussian peak corresponding to temperature T0.
In the case of parametric modulation, the form of the initial
energy distribution rmod(E) is not known analytically and therefore
needs to be determined experimentally. Using the measured initial
distribution together with the energies E0 and Et evaluated at
times 0 and t, respectively, we calculate DS ¼ b0Q þ Df.
Figure 5a shows the initial energy distribution rmod(E), which has
a narrow spread around a non-zero value and therefore differs
significantly from a thermal distribution with identical effective
temperature (grey dashed line). The measured distributions of DS
evaluated at different times after switching off the modulation are
shown in Fig. 5b. As before, we use the distributions p(DS) to
evaluate S(DS) and plot it in Fig. 5c. To reduce the variance we
time-average the distributions p(DS) and plot the corresponding
S function in Fig. 5d. As in the previous experiment, we find excellent agreement with theory (black dashed line), providing solid
experimental validation for the fluctuation theorem (3) being
valid for initial steady states that are out of equilibrium.
In conclusion, we have experimentally demonstrated the validity
of a fluctuation theorem for the relaxation from a non-equilbrium
state towards equilibrium. The theorem holds for the relative
entropy change DS, which is related (but not identical) to the
total entropy production. Using a levitated nanoparticle in high
vacuum we have verified the fluctuation theorem for different
initial non-equilibrium states, demonstrating that this theoretical
framework can be used to understand fluctuations in nanoscale
systems. Our experimental approach allows us to measure the
dynamics of a nanoparticle during relaxation from an arbitrary
initial state and to study its statistical properties. We succeeded in
deriving an analytic expression for the non-equilibrium steady
state under the action of a feedback force and demonstrated excellent agreement with experimental data. The presented experimental
framework naturally extends to the study of transitions between
arbitrary steady states and to quantum fluctuation theorems,
similar to recent proposals for trapped ions19,29. We envision that
our approach of using highly controllable nanomechanical oscillators will open up experimental and theoretical studies of fluctuation theorems in complex settings, which arise, for instance,
from the interplay of thermal fluctuations and nonlinearities42
where detailed balance does not hold43,44. Furthermore, it serves
as an experimental simulator platform in analogy to quantum
simulators based on ultracold gases, superconducting circuits or
trapped ions45.
ARTICLES
Received 5 November 2013; accepted 6 February 2014;
published online 30 March 2014
References
1. Wang, G. M., Sevick, E. M., Mittag, E., Searles, D. J. & Evans, D. J. Experimental
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Acknowledgements
This research was supported by ETH Zürich, ERC-QMES (no. 338763), ERC-Plasmolight
(no. 259196), Fundació Privada CELLEX and the Austrian Science Fund (FWF) within the
SFB ViCoM (grant F41). The authors acknowledge support from the ESF Network
Exploring the Physics of Small Devices.
Author contributions
L.N. and J.G. conceived and designed the experiments. J.G. performed the experiments.
J.G., C.D. and L.N. analysed the data. C.D. developed the theoretical framework. R.Q.
contributed materials/analysis tools. J.G., C.D. and L.N. co-wrote the paper.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints. Correspondence and
requests for materials should be addressed to L.N. and C.D.
Competing financial interests
The authors declare no competing financial interests.
NATURE NANOTECHNOLOGY | VOL 9 | MAY 2014 | www.nature.com/naturenanotechnology
© 2014 Macmillan Publishers Limited. All rights reserved.
SUPPLEMENTARY INFORMATION
DOI: 10.1038/NNANO.2014.40
Supplementary Information for
Dynamic relaxation
of a levitated nanoparticle
from
a non-equilibrium
steady state
Dynamic
Relaxation of a Levitated
Nanoparticle from a
Non-Equilibrium Steady State
Jan Gieseler1 , Romain Quidant1,2 , Christoph Dellago3 and Lukas Novotny4
1
2
ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860
Castelldefels (Barcelona), Spain
ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain
3
University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria
4
ETH Zürich, Photonics Laboratory, 8093 Zürich, Switzerland
Abstract
The supplementary information provides the theory of parametric feedback cooling
and a discussion of the fluctuation theorem, which is demonstrated experimentally in
the main text.
Contents
1 Theory of parametric feedback cooling
S2
1.1
Stochastic differential equation for the energy . . . . . . . . . . . . . . .
S3
1.2
Energy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S7
1.3
Effective temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S9
1.4
Relaxation of average energy
1.5
Phase space distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . S12
. . . . . . . . . . . . . . . . . . . . . . . . S11
2 Fluctuation theorem for ∆S
S16
2.1
Relative entropy change ∆S . . . . . . . . . . . . . . . . . . . . . . . . . S16
2.2
Detailed fluctuation theorem
2.3
Integral fluctuation theorem . . . . . . . . . . . . . . . . . . . . . . . . . S20
2.4
Relaxation from an initial equilibrium state . . . . . . . . . . . . . . . . S21
. . . . . . . . . . . . . . . . . . . . . . . . S19
S1
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© 2014 Macmillan Publishers Limited. All rights reserved.
1
1
2.5
Relaxation from a steady state generated by parametric feedback . . . . S22
2.6
Distributions of ∆S for t → ∞ . . . . . . . . . . . . . . . . . . . . . . . S22
Theory of parametric feedback cooling
The trapped particle experiences both the trap force generated by the laser as well as a
viscous force (Stokes force) due to the random impact of gas molecules. For small oscillation amplitudes, the trapping potential is harmonic and the three spatial dimensions
are decoupled. Each direction can be characterised by a frequency Ω0 , which is defined
p
by the particle mass m and the trap stiffness k as Ω0 = k/m. The equation of motion
for the particle’s motion along q = {x, y, z} is therefore
q̈(t) + Γ0 q̇(t) + Ω20 q(t) =
1
[Ffluct (t) + Ffb (t)] ,
m
(1)
where Γ0 is the friction coefficient and Ffluct is a random Langevin force that satisfies the
fluctuation dissipation theorem hFfluct (t) Ffluct (t′ )i = 2mΓ0 kB T0 δ(t − t′ ). In the above
equation, Fopt (t) = Ω0 ηq 2 p is a time-varying, non-conservative optical force introduced
by parametric feedback of strength η and p = mq̇ is the momentum.
Starting from Equ. (1), we now derive a stochastic differential equation for the energy
E(q, p) =
p2
1
mΩ20 q 2 +
2
2m
(2)
in the limit of a highly underdamped system (Q = Ω0 /Γ0 ≫ 1). As shown below, in
the low friction limit, the stochastic equation of motion for the energy (or rather, for
the square root of the energy) can be written in a form which resembles over-damped
Brownian motion in energy space. As a result, we can consider the energy as the only
relevant variable1 . As a side product, we also obtain the energy and the position distributions in the non-equilibrium steady state generated by the application of the feedback
loop. In fact, in energy space the dynamics of the system with feedback can be viewed
as an equilibrium dynamics occurring in a system with an additional force term.
1
note that this works only in the low friction limit
S2
© 2014 Macmillan Publishers Limited. All rights reserved.
1.1
Stochastic differential equation for the energy
For the developments below it is more convenient to write (1) as a stochastic differential
equation (SDE),
p
dt,
m
p
dp = (−mΩ20 q − Γ0 p − Ω0 ηq 2 p)dt + 2mΓ0 kB T0 dW,
dq =
(3a)
(3b)
where W (t) is a Wiener process with
hW (t)i = 0,
hW (t)W (t′ )i = t′ − t.
(4a)
(4b)
Note that in particular hW 2 (t)i = t for any time t ≥ 0. Accordingly, for a short
(infinitesimal) time interval dt we have
hdW i = 0,
h(dW)2 i = dt.
(5)
(6)
The time derivative of the Wiener process, ξ(t) = dW (t)/dt, is white noise and it is
√
related to the random force by Ffluct (t) = 2mΓ0 kB T0 ξ(t).
We determine the energy change dE that occurs during the short time interval dt
during which position and momentum change by dq and dp as specified by the equations
of motion (3a) and (3b). To lowest order, the energy change is given by
∂E
∂E
1 ∂2E
dE =
dq +
dp +
(dp)2 .
∂q
∂p
2 ∂p2
(7)
Note that this equation differs from the usual chain rule because we have to keep the
term proportional to (dp)2 . The reason is that according to Equ. (3b), dp depends on
√
dW which is of order dt. Hence, if we want to keep all terms at last up to order dt,
we cannot neglect the second order term in the above equation because (dp)2 is of order
dt. In contrast, we can safely neglect the terms proprtional to (dq)2 and dqdp, because
they are of order dt2 and (dt)3/2 , respectively.
Computing the derivatives of the energy with respect to q and p we obtain
dE = mΩ20 qdq +
p
1
dp +
(dp)2 .
m
2m
S3
© 2014 Macmillan Publishers Limited. All rights reserved.
(8)
Inserting dq and dp from Eqs. (3a) and (3b) and neglecting all terms of order (dt)3/2
or higher yields
dE = −m(Γ0 + Ω0 ηq 2 )
p 2
pp
dt +
2mΓ0 kB T0 dW + Γ0 kB T0 dW 2 .
m
m
To avoid the multiplicative noise of Equ. (9) we consider the variable ǫ =
(9)
√
E
instead of the energy E. The change dǫ due to the changes dq and dp occurring during
an infinitesimal time interval dt is given by
∂ǫ
∂ǫ
1 ∂2ǫ
dǫ =
dq +
dp +
(dp)2
∂q
∂p
2 ∂p2
(10)
as all other terms are of order (dt)3/2 or higher. Evaluation of the partial derivatives
yields
dǫ =
mΩ20
q
1 p
1
dq +
dp +
2ǫ
2ǫ m
2
1
1 p2
− 3 2
2mǫ 4ǫ m
(dp)2 .
(11)
Using the equations of motion (3a) and (3b) and exploiting that (dp)2 = 2mΓ0 kB T0 (dW )2
up to order dt we obtain
Γ0 + Ω0 ηq 2 p2
dǫ = −
dt +
2ǫ
m
√
2mΓ0 kB T0 p
Γ0 kB T0
dW +
2ǫ
m
2ǫ
p2
1−
(dW )2 .
2mǫ2
(12)
We now integrate this equation over an oscillation period τ = 2π/Ω0 to obtain the
Rτ
change ∆ǫ = 0 dǫ over one oscillation period,
Z
Z
Z τ
p
Γ0 τ p 2
Ω 0 η τ q 2 p2
p
∆ǫ = −
dt −
dt + 2mΓ0 kB T0
dW
(13)
2 0 mǫ
2 0 mǫ
0 2mǫ
Z τ
1
p2
+Γ0 kB T0
1−
(dW )2 .
(14)
2
2ǫ
2mǫ
0
To compute the integrals on the right hand side of the above equation, we assume
that in the low-friction limit the energy E, and hence also ǫ remains essentially constant
over one oscillation period. We also assume that the feedback mechanism changes the
energy of the system slowly and that the motion of the system during one oscillation
period is practically not affected by the feedback either. In the low friction regime, where
the coupling to the bath is weak, a small feedback strength (i.e., a small η) is sufficient
for considerable cooling. Accordingly, during one oscillation period the position q and
the momentum p are assumed to evolve freely:
s
2
q(t) = ǫ
sin Ω0 t,
mΩ20
√
p(t) = mq̇(t) = ǫ 2m cos Ω0 t,
S4
© 2014 Macmillan Publishers Limited. All rights reserved.
(15)
(16)
where we have selected the phase of the oscillation such that the position q = 0 at time
0. Hence, the first two integrals of (14) are given by
Z τ 2
Z τ
p
dt = 2ǫ
cos2 Ω0 t dt = ǫτ
0 mǫ
0
and
Z
τ
0
q 2 p2
4ǫ3
dt =
mǫ
mΩ20
Z
τ
sin2 Ω0 t cos2 Ω0 t dt =
0
(17)
ǫ3 τ
.
2Ω20 m
(18)
Insertion of these results and of the harmonic expressions for q(t) and p(t) from above
into Equ. (14) gives
∆ǫ = −
p
ηǫ3
Γ0 kB T0
Γ0 ǫ
τ−
τ + Γ0 kB T0 ∆R1 +
∆R2 .
2
4mΩ0
2ǫ
(19)
where ∆R1 and ∆R2 are given by
∆R1 =
τ
Z
cos Ω0 t dW
(20)
sin2 Ω0 t (dW )2 .
(21)
0
and
∆R2 =
Z
τ
0
Since W (t) is a Wiener process, ∆R1 and ∆R2 are random numbers. Next we will
determine the statistical properties of ∆R1 and ∆R2 .
As ∆R1 is the result of a (weighted) sum of Gaussian random numbers, it will be a
Gaussian random number, too. The mean of ∆R1 is given by
Z τ
Z τ
h∆R1 i = h
cos Ω0 t dW i =
cos Ω0 thdW i = 0,
0
(22)
0
where the angular brackets imply an average over all noise histories. The variance of
∆R1 is given by
Z τ
Z τ
h(∆R1 ) i = h
cos Ω0 t dW
cos Ω0 t′ dW ′ i
0
0
Z τZ τ
=
cos Ω0 t cos Ω0 t′ hdW dW ′ i
0
0
Z τ
τ
=
cos2 Ω0 t dt = ,
2
0
2
(23)
where we have exploited that hdW dW ′i = δ(t′ − t)dt. Hence, the random variable ∆R1
can be written as
∆R1 =
r
1
W (τ ),
2
S5
© 2014 Macmillan Publishers Limited. All rights reserved.
(24)
where W (τ ) is a Wiener process at τ , i.e., a Gaussian random variable with variance τ .
In a similar way, we can show that the mean of ∆R2 is given by
Z τ
h∆R2 i = h
sin2 Ω0 t (dW )2 i
Z 0τ
=
sin2 Ω0 th(dW )2 i
0
Z τ
τ
=
sin2 Ω0 tdt = ,
2
0
because of h(dW )2 i = dt. For the second moment of ∆R2 we obtain
Z τ
Z τ
h(∆R2 )2 i = h
sin2 Ω0 t (dW )2
sin Ω0 t′ (dW ′ )2 i
0
Z 0τ Z τ
2
2
=
sin Ω0 t sin Ω0 t′ h(dW )2 (dW ′ )2 i
Z0 τ Z0 τ
=
sin2 Ω0 t sin2 Ω0 t′ dtdt′
0
0
Z τ
2
τ2
2
=
sin Ω0 tdt = ,
4
0
(25)
(26)
where we have used that (dW )2 and (dW ′ )2 are uncorrelated and that h(dW )2 i = dt.
Thus, the variance of ∆R2 vanishes,
h(∆R2 )2 i − h∆R2 i2 =
τ2 τ2
−
= 0.
4
4
(27)
This result implies that the random variable ∆R2 is sharp such that it can be replaced
by its average, ∆R2 = τ /2.
Putting everything together we obtain
∆ǫ =
Γ0 ǫ
ηǫ3
Γ0 kB T0
−
−
+
2
4mΩ0
4ǫ
τ+
r
Γ0 kB T0
W (τ ).
2
(28)
Since the oscillation period τ is assumed to be short compared to the dissipation time
scale 1/Γ0 , we can finally write the stochastic differential equation for the variable ǫ,
r
Γ0 ǫ
ηǫ3
Γ0 kB T0
Γ0 kB T0
dǫ = −
−
+
dt +
dW.
(29)
2
4mΩ0
4ǫ
2
This equation, in which ǫ is the only variable, is the main result of this section. It
implies that the relaxation process can be understood as a Brownian motion of ǫ (or,
equivalently, of the energy) under the influence of an external “force”.
S6
© 2014 Macmillan Publishers Limited. All rights reserved.
Similarly, we can derive the corresponding stochastic differential equation for the
energy E = ǫ2 :
dE =
p
ηE 2
+ Γ0 kB T0 dt + 2EΓ0 kB T0 dW.
−Γ0 E −
2mΩ0
(30)
Note that, in contrast to ǫ, the energy is subject to multiplicative noise.
1.2
Energy distribution
Equation (29) derived in the previous section resembles the Langevin equation of a
variable ǫ evolving at temperature kB T0 under the influence of an external force f (ǫ) at
high friction ν:
1
dǫ = f (ǫ)dt +
ν
r
2kB T0
dW.
ν
(31)
The isomorphism is established by setting ν = 4/Γ0 and
f (ǫ) = −2ǫ −
kB T0
ηǫ3
+
.
mΩ0 Γ0
ǫ
(32)
Interestingly, a low friction Γ0 , which determines the magnitude of the frictional force
acting on the particle, corresponds to a high friction ν for the time evolution of ǫ and,
thus, of the energy E. The Langevin equation (31) is known to sample the BoltzmannGibbs distribution
ρ(ǫ) ∝ exp {−β0 U (ǫ)} ,
(33)
where β0 = 1 /kB T0 is the reciprocal temperature and U (ǫ) is the potential corresponding to the force f (ǫ) = −dU (ǫ)/dǫ. In our case, integration of the force f (ǫ) of Equ.
(32) yields the potential
U (ǫ) = ǫ2 +
α 4
ǫ − kB T0 ln ǫ,
4
(34)
η
mΩ0 Γ0
(35)
where we have introduced
α=
to simplify the notation. Hence, Equ. (29) generates the distribution
n
α o
ρ(ǫ, α) ∝ ǫ exp −β0 ǫ2 + ǫ4 ,
4
(36)
which can be viewed as the equilibrium distribution of the potential U (ǫ). In the above
equation, we have included the feedback strength α explicitly as a parameter for ρ(ǫ, α)
in order to indicate that this distribution is valid also for the non-equilibrium steady
S7
© 2014 Macmillan Publishers Limited. All rights reserved.
0
-2
log P(E)
-4
-6
η=0.000
η=0.001
η=0.010
η=0.100
η=1.000
-8
-10
-12
0
1
2
3
4
5
E
Figure S1: Logarithm ln P (E) of the energy distribution shifted by a constant such that
ln P (0) = 0. The symbols are simulation results obtained for η = 0.000, 0.001, 0.010, 0.100
and 1.000 and the solid lines are the corresponding predictions of Equ. (37). The simulations
were carried out in phase for kB T0 = 1, m = 1, k = 1, and Γ0 = 0.01. We used the algorithm
of Sivak, Chodera and Crooks [1] to integrate the Langevin equation of motion for a total of
2 × 109 time steps of length ∆t = 0.006 for each value of η.
state generated by the feedback mechanism. The effect of the feedback is, however,
reduced to a particular term in this potential. As can be easily seen by a change of
variables from ǫ to E = ǫ2 , the distribution of Equ. (36) corresponds to the energy
distribution
1
β0 α 2
ρ(E, α) =
exp −β0 E −
E ,
Zα
4
R
where the normalisation factor Zα = dEρ(E, α) is given by
r !
r
π β0 /α
β0
Zα =
e
erfc
.
αβ0
α
(37)
(38)
Thus, the energy distribution has the form of the Boltzmann-Gibbs distribution for the
“energy” H = E + αE 2 /4, where E is the energy of the system and αE 2 /4 can be
viewed as a feedback energy. Some energy distributions obtained for different feedback
strengths η ranging from 0 to 1 are shown in Fig. S1.
S8
© 2014 Macmillan Publishers Limited. All rights reserved.
1.3
Effective temperature
The average energy is obtained by integration over the energy distribution of Equ. (37),
Z
β0 α 2
−1
hEi = Zα
dE E exp −β0 E −
E .
(39)
4
Evaluation of the integral yields


r
2 
e−β0 /α
β 
 β0
q − 0 
hEi =
,

β0
α √
α
β0
π erfc
α
(40)
Thus, the effective temperature Teff obtained by applying the feedback mechanism is:



−1/αk
T
0
B
2
e
2 
.
(41)
kB Teff = kB T0 √
−
 αkB T0 √π erfc √ 1
αkB T0 
αk T
B 0
For small values of α this expression turns into
kB Teff = kB T0 (1 − αkB T0 ) .
(42)
Hence, for weak feedback, the relative effective temperature Teff /T0 decreases linearly
as function of α with slope kB T0 . The effective temperature Teff is shown in Fig. S2 as
a function of the feedback strength α.
For large values of α, the asymptotic behaviour of the effective temperature is given
by
r
4kB T0 mΩΓ0
.
(43)
πη
√
Hence, at low friction the effective temperature decreases as Γ0 and is inversely pro√
portional to η.
kB Teff ≈
4kB T0
=
πα
s
An alternative but approximate expression for the effective temperature of the oscillator with feedback can be derived from Equ. (53) for the time evolution of the average
energy. In the steady state, the time derivative of the average energy must vanish such
that
−Γ0 hEi −
η
hE 2 i + Γ0 kB T0 = 0,
2mΩ0
(44)
which implies
1
αhE 2 i + hEi − kB T0 = 0.
2
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© 2014 Macmillan Publishers Limited. All rights reserved.
(45)
This equation cannot be simply solved for hEi, because in general the second moment of
the energy, hE 2 i, cannot be expressed in terms of the average energy hEi. In equilibrium,
however, the energy is distributed exponentially and therefore the relation hE 2 i = 2hEi2
holds. Using this relation in the steady state away from equilibrium, where it is valid
only approximately, we obtain an equation for the average energy hEi,
αhEi2 + hEi − kB T0 = 0.
(46)
Solving this quadratic equation yields
hEi =
−1 +
p
1 + 4α/β0
,
2α
which, because of hEi = kB Teff , is equivalent to
√
1 + 4αkB T0 − 1
kB Teff =
.
2α
(47)
(48)
For small values of α, the effective temperature in this approximation turns into
kB Teff = kB T0 (1 − αkB T0 ) ,
(49)
which is identical to the result of Equ. (42) obtained from the exact energy distribution.
For large values of α, on the other hand, the approximate effective temperature of Equ.
(48) becomes
kB Teff =
r
kB T0
=
α
s
kB T0 mΩ0 Γ0
.
η
(50)
In this approximation the effective temperature at low friction (or strong feedback)
has the same scaling with Γ0 and η as the one predicted by Equ. (43). However
this approximate expression underestimates the exact low-friction effective temperature
p
given in Equ. (43) by a factor of π/4 ≈ 1.1284. As can be seen in Fig. S2, Equ. (48)
correctly reproduces the effective temperature for weak feedback, but underestimates
the effective temperature by about 10% for stronger feedback.
It is interesting to note that Equ. (45) implies that
kB Teff = kB T0 −
α 2
hE i,
2
(51)
which is an exact result holding for the non-equilbrium steady state generated by the
feedback. For the dissipation rate, this implies
P̄ = Γ0 (kB T0 − kB Teff ).
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© 2014 Macmillan Publishers Limited. All rights reserved.
(52)
1
1.1
Teff/T
Teff/Tth
0.8
1
0.9
0.8
0.6
0.7
0.6
0.4
0
10
20
α
30
40
0.2
0
0
10
20
α
30
40
Figure S2: Effective temperature Teff as a function the feedback parameter α = η/mΩ0 Γ0 .
The symbols are results of simulations and the solid lines are the effective temperatures
predicted by Equ. (41) (red) and Equ. (48) (blue), respectively. The simulations were
carried out for kB T0 = 1, m = 1, k = 1, and Γ0 = 0.01. We used the algorithm of Sivak,
Chodera and Crooks [1] to integrate the Langevin equation of motion for a total of 108 time
steps of length ∆t = 0.006 for each value of α. Inset: same effective temperatures normalized
by the effective temperature Tth predicted by Equ. (41).
1.4
Relaxation of average energy
Carrying out an average over noise realizations in Equ. (30) and exploiting that hdW i =
0, the time derivative of the ensemble average of the energy is given by
dhEi
η
= −Γ0 hEi −
hE 2 i + Γ0 kB T0 .
dt
2mΩ0
(53)
Note that in general this average is over a non-equilibrium distribution.
Without feedback (A = 0) the average energy evolves according to
dhEi
= −Γ0 hEi + Γ0 kB T0 .
dt
(54)
This differential equation can be easily solved yielding an exponential relaxation of the
average energy with such that the average energy approaches its equilibrium value of
kB T0 exponentially,
hE(t)i = kB T0 + [hE(0)i − kB T0 ]e−Γ0 t ,
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© 2014 Macmillan Publishers Limited. All rights reserved.
(55)
where hE(0)i is the average energy at time 0. In principle, Equ. (53) can be used to
compute the time evolution of the average energy also in the presence of the feedback.
In this case, however, one needs to be able to compute the second moment of the energy
at each instant during the relaxation process.
In the non-equilibrium steady state obtained with feedback on, the energy change
vanishes on the average. Hence, the energy flow from the oscillator to the heat bath
is exactly compensated by the energy extracted from or added to the oscillator by the
feedback mechanism,
Γ0 (hEi − kB T0 ) = −
η
hE 2 i
2mΩ0
(56)
as already expressed in Equ. (44). Here, the left hand side is the average energy flow
(energy per unit time) from the oscillator to the heat bath and the right hand side is
the energy change due to the feedback. Using the parameter α = η /mΩ0 Γ0 , Equ. (56)
turns into
α 2
hE i = kB T0 − hEi.
2
(57)
This equation, relating the first and second moments of the energy distribution, holds
in the non-equilibrium steady state generated by the feedback mechanism.
We also introduce the average dissipation rate
P̄ =
αΓ0 2
hE i,
2
(58)
which is the average energy per unit time transferred from the heat bath to the oscillator. Note that for positive η, the system is colder than the heat bath and, therefore,
on average the energy flows from the heat bath to the system. This is an interesting
difference to most other non-equilibrium steady states, in which energy flows from the
system to the heat bath compensating for the energy dissipated by an external perturbation.
1.5
Phase space distribution
Based on the energy distribution ρ(E, η) of Equ. (37) we derive the phase space distribution ρ(q, p, η) of the non-equilibrium steady state generated by the application of the
feedback. We start by writing the joint probability distribution function ρ(q, E, η) to
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observe the pair (q, E) as
(59)
ρ(q, E, η) = ρ(q|E, η)ρ(E, η),
where ρ(q|E, η) is the conditional probability to observe the position q for a given energy
E at feedback strength η. Assuming that the motion of the oscillator is essentially
undisturbed during an oscillation period as is the case in the low friction limit, the
distribution of positions is given by


ρ(q|E, η) =

π
√
Ω0
2E/m−Ω20 q 2
0
if q 2 ≤
2E
mΩ20
(60)
else
simply because the probability to find the system at q is inversely proportional to the
p
magnitude of the velocity, |p|/m = 2E/m − Ω20 q 2 the system has at q. This condip
tional probability distribution diverges at the turning points q0 = ± 2E/mΩ20 and it
vanishes for |q| > q0 . Multiplying the conditional distribution ρ(q|E, η) with the energy
distribution from Equ. (37) one obtaines the desired joint distribution ρ(q, E, η).
Next, we change variables from (q, E) to (q, p). The respective distributions are
related by
∂(q, E) 1
.
ρ(q, p, η) = ρ(q, E, η) 2
∂(q, p) The Jacobian of the transformation is given by
∂q ∂q
∂(q, E) ∂q ∂p
∂(q, p) = ∂E ∂E
∂q ∂p
|p|
=
.
m
(61)
(62)
In Equ. (61) we have exploited that the distribution ρ(q, p, η) is symmetric in p, and the
factor 1/2 arises because p and −p correspond to the same energy E = kq 2 /2 + p2 /2m.
Thus, the phase space density ρ(q, p, η) becomes
1
|p|
Ω0
ρ[E(q, p), η] p
2
2
2
π 2E/m − Ω0 q m
Ω0
=
ρ[E(q, p), η],
(63)
2π
p
where we have used that |p|/m = 2E/m − Ω20 q 2 . Note that the second case of Equ.
ρ(q, p, η) =
(60) does not need to be taken into account, because for given q and p, the condition
q 2 ≤ 2E/mΩ20 is always obeyed. Using the energy distribution from Equ. (37) we finally
obtain
ρ(q, p, η) =
Ω0
β0 η
exp −β0 E(q, p) −
E(q, p)2 .
2πZ
4mΓ0 Ω0
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© 2014 Macmillan Publishers Limited. All rights reserved.
(64)
Hence, the motion of the oscillator with feedback samples the equilibrium distribution
of a system with energy
H(q, p) = E(q, p) +
η
E(q, p)2 .
4mΓ0 Ω0
(65)
In the phase space distribution of Equ. (64) the term depending on the squared energy
E(q, p)2 causes correlations between q and p that are absent in equilibrium with feedback
off.
The average of the effective energy H(q, p), which contains the energy E(q, p) plus
a "feedback" energy αE(q, p)2 /4 is given by
hHi = hEi +
α 2
hE i.
4
(66)
Using Equ. (57) and noting that hEi = kB Teff we obtain
hHi =
kB T0 + kB Teff
2
(67)
So the average effective energy H(q, p) is the arithmetic average of the energy of two
harmonic oscillators, one at temperature T0 and the other one at temperature Teff .
From the phase space density ρ(q, p, η) one can get the distribution ρ(q, η) of the
positions by integration over the momenta:
Z ∞
ρ(q, p, η) =
dp ρ(q, p, η).
(68)
−∞
Carrying out the integral yields
r
β0 mΩ20 (4 + αmΩ20 q 2 )
ρ(q, η) =
8π 3
h
i
×
exp −
β0 (4+αmΩ20 q 2 )2
32α
erfc
q
β0
α
K1
4
β0 (4 + αmΩ20 q 2 )2
,
32α
(69)
where K1/4 is a generalised Bessel function of the second kind. This expression is
compared with simulations in Fig. S3.
An analogous calculation yields the non-equilibrium momentum distribution ρ(p, η) =
R∞
−∞ dq ρ(q, p, η),
ρ(p, η) =
r
β0 (4m + αp2 )
3 2
h8π m
2 2
)
exp − β0(4m+αp
32αm2
q ×
β0
erfc
α
i
K1
4
β0 (4m + αp2 )2
.
32αm2
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© 2014 Macmillan Publishers Limited. All rights reserved.
(70)
2
η=0.000
η=0.001
η=0.010
η=0.100
η=1.000
0
ln P(q)
-2
-4
-6
-8
-10
-12
-4
-2
0
q
2
4
Figure S3: Logarithm ln P (q) of the distribution of position q for different feedback strengths.
The symbols are simulation results obtained for η = 0.000, 0.001, 0.010, 0.100 and 1.000 and
the solid lines are the corresponding predictions of Equ. (69). The simulations were carried
out for kB T0 = 1, m = 1, k = 1, and Γ0 = 0.01. We used the algorithm of Sivak, Chodera
and Crooks [1] to integrate the Langevin equation of motion for a total of 2 × 109 time steps
of length ∆t = 0.006 for each value of η.
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Fluctuation theorem for ∆S
2
In this section we define the relative entropy ∆S and discuss the fluctuation theorem
holding for this quantity [10, 11, 12, 3].
2.1
Relative entropy change ∆S
We consider a system with energy E(u), including both potential and kinetic energy,
with u specifying the state of the system. The system is in contact with a heat bath at
reciprocal temperature β0 = 1/kB T0 . If no external perturbation acts on the system, u
is distributed according to the equilibrium distribution
ρeq (u) =
where Z(β0 ) =
R
1
e−β0 E(u) ,
Z(β0 )
(71)
due−β0 E(u) is the partition function related to the free energy by
F (β0 ) = −kB T ln Z(β0 ). If left undisturbed, the system evolves according to a dynamics
which is microscopically reversible, i.e., it obeys the detailed balance condition for the
equilibrium distribution,
e−β0 E(u) p(u → v, t) = e−β0 E(v) p(v ∗ → u∗ , t).
(72)
Here, p(u → v, t) is the probability to move from state u at time 0 to state v at time
t and the star denotes a state with inverted momenta. The dynamics generated by the
Langevin equation (with feedback off) obeys this condition.
The system is initially prepared in a steady state with distribution ρss (u, α), for
instance by letting a feedback mechanism act on it. Here, α denotes one or several parameters such as the strength of the feedback mechanism, which determine the steady
state distribution. In general, ρss (u, α) is not known analytically. At time t = 0 the
feedback is switched off and the system relaxes back to equilibrium. During the relaxation process, the system exchanges energy (heat) with the heat bath. Since no work
is done on the system during the relaxation, the total heat Q exchanged with the heat
bath by the system evolving from u0 at time 0 to ut at time t is given by
Q = −[E(ut ) − E(u0 )].
(73)
Note that the heat is defined in such a way that a positive Q corresponds to energy
absorbed by the bath and lost by the system. For a given steady state distribution
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ρss (u, α), we also define the quantity
(74)
φ(u) = − ln ρss (u, α)
as well as its difference between the initial and final state,
(75)
∆φ = φ(ut ) − φ(u0 ).
Based on this definition, we introduce the relative entropy change
(76)
∆S = β0 Q + ∆φ,
which depends only on the state of the system at times 0 and t.
We call ∆S the relative entropy change for the following reason. Defining
S = ln
ρeq (u)
ρss (u)
(77)
we can write
(78)
∆S = S(ut ) − S(u0 ),
such that ∆S is the change in the quantity S accumulated along the trajectory evolving
from u0 to ut . The average of S(u) over the equilibrium distribution is the relative
entropy [2] between the equilibrium distribution ρeq (u) and the steady state distribution
ρss (u),
D(ρeq kρss ) =
Z
du ρeq (u) ln
ρeq (u)
.
ρss (u)
(79)
This quantity, also known as Kullback-Leibler divergence, is a non-symmetric measure
of the difference between two distributions and it vanishes if the two distributions are
identical. Since the ensemble average of S(u) is a relative entropy, we view S(u) as the
relative entropy associated with state u. While strictly speaking the relative entropy is
a property of the entire distributions, entropies assigned to individual configurations or
trajectories have been considered before and shown to be useful [3]. Accordingly, ∆S
can be viewed as the relative entropy change of the trajectory connecting u0 with ut .
In Equ. (76), the first term on the right hand side is the entropy change of the reservoir
at reciprocal temperature β0 and the second term is the entropy change with respect
to the initial steady state distribution. Note that in contrast to the definition of the
generalized work Y of Hatano and Sasa [4, 5], in the definition of ∆S the initial steady
state distribution (rather than the time-evolved distribution) is evaluated both at the
beginning u0 and the end ut of the trajectory.
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It is also interesting to note that the relative entropy change is ∆S is equal to the
logarithmic ratio of the probability to observe a particular trajectory and the probability
to observe the time reversed trajectory [3, 8, 9]. To be more explicit, consider the
probability P [u(t)] of observing a particular trajectory u(t) of length t evolving from
u0 to ut and the probability P [u∗ (t)] of the time-reversed trajectory u∗ (t). In the time
reversed, or conjugate trajectory, the same microscopic states are visited, but in reversed
order and with inverted momenta, ut′ = u∗t−t′ , where the star indicates momentum
inversion [3, 9, 13]. For dynamics that is microscopically reversible (i.e., detailed balance
holds) and assuming that both the forward and the reversed trajectories are started from
the same steady state distribution ρss (u), the ratio of the probabilities to observe a pair
of conjugate trajectories is then given by [3, 9, 14]
P [u(t)]
= eβQ+∆S = e∆S .
P [u∗ (t)]
(80)
Hence, the relative entropy of the distributions of the forward and the reversed trajectories, obtained as average over the logarithmic probability ratio, reads
Z
P [u(t)]
D(P [u(t)]kP [u∗ (t)]) = du(t) P [u(t)] ln
= h∆Si,
P [u∗ (t)]
(81)
where the integration extends over all trajectories u(t). Therefore, the ensemble average
h∆Si of the relative entropy change is the relative entropy of the forward and reversed
ensembles of trajectoris and, as such, provides a measure for the irreversibility (timeasymmetry) of the relaxation process. As a consequence of Equ. (81), in an equilibrium
system the relative entropy change vanishes, h∆Sieq = 0, because in equilibrium a
particular forward trajectory and its time-reversed trajectory have the same probability
to be observed.
For deterministic thermostatted dynamics ∆S equals the dissipation function introduced by Evans and Searles [10, 11, 12]. It is also worth noting that the relative entropy
change and the total entropy change are related by [3]
∆S = ∆Stot − ln
ρss (ut )
,
ρt (ut )
(82)
where ρt (u) is the statistical state of the system at time t. The relation between relative
entropy change and total entropy change is discussed further below.
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2.2
Detailed fluctuation theorem
As shown by Evans and Searles [10, 11, 12] for thermostatted dynamics and by Seifert
for stochastic dynamics [3], the following transient fluctuation theorem holds for timeindependent driving and microscopically reversible dynamics
Pt (−∆S)
= e−∆S .
Pt (∆S)
(83)
Here, Pt (∆S) is the distribution of ∆S observed at an arbitrary time t over many repetitions of the relaxation experiment. In particular, this fluctuation theorem is valid for a
system relaxing to equilibrium from a non-equilibrium steady state as considered here.
Since it is instructive and emphasises the significance of the microscopic reversibility
of the underlying dynamics, we provide a short derivation of the detailed fluctuation
theorem (83) in the following. The derivation is based on two conditions: (1) the initial
steady state distribution and the equilibrium distribution are symmetric with respect to
momentum reversal, (2) the dynamics is microscopically reversible, i.e., detailed balance
holds. Since the energy is quadratic in the momentum p, the first condition is always
obeyed if the distribution is a function of the energy only, as it is the case for parametric
feedback cooling. The latter condition is fulfilled, for instance, for a system evolving
according to a Langevin equation.
The fluctuation theorem follows most easily by considering the probability P [u(t)] of
observing a particular trajectory u(t) and the probability P [u∗ (t)] of the time-reversed
trajectory u∗ (t). As mentioned above, for dynamics that is microscopically reversible,
this ratio is given by P [u(t)]/P [u∗ (t)] = exp(∆S) [3, 9]. The distribution of ∆S at time
t can be expressed in terms of the probability P [u(t)],
Z
Pt (∆S) = du(t) P [u(t)] δ(∆S[u(t)] − ∆S) ,
(84)
where δ(·) is the Dirac δ-function. Transforming integration variables from u(t) to u∗ (t)
and taking advantage of the symmetry of Q and ∆φ with respect to momentum reversal,
one finds
Pt (∆S) =
Z
du∗ (t) P [u∗ (t)]e−∆S[u
∗ (t)]
δ(−∆S[u∗ (t)] − ∆S) = e∆S Pt (−∆S),
(85)
which holds for any time t > 0. Thus, the transient fluctuation theorem of Equ. (83) is a
direct consequence of the microscopic reversibility of the dynamics during the relaxation
process.
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2.3
Integral fluctuation theorem
From the detailed fluctuation relation of Equ. (83) one easily obtains an integral fluctuation theorem by integration over the probability density P (∆S),
Z
Z
he−∆S i = d∆S P (∆S)e−∆S = d∆S P (−∆S) = 1,
(86)
where the last step involves a variable change from ∆S to −∆S.
Applying Jensen’s inequality, i.e., hex i ≥ ehxi , to the integral fluctuation theorem
one obtains
1 = he−∆S i ≥ e−h∆Si ,
(87)
h∆Si ≥ 0.
(88)
which is equivalent to
Thus, the average change in relative entropy is non-positive. Using the definition of ∆S,
the average of the relative entropy change can be written as
h∆Si = β0 hQi + ∆I + D(ρt kρss ),
(89)
where β0 is the reciprocal temperature of the bath. The first term on the right hand
side of the above equation is the change of thermodynamic entropy of the bath,
∆Sbath = β0 hQi.
(90)
The second term, ∆I, is the change in Shannon entropy (or information entropy) I between the initial statistical state characterised by the distribution ρss and the statistical
state with distribution ρt at time time after the relaxation has started,
∆I = I[ρt ] − I[ρss ],
where the Shannon entropy I of a distribution ρ(u) is defined as
Z
I[ρ] = − du ρ(u) ln ρ(u).
(91)
(92)
If one identify the Shannon entropy with the thermodynamic entropy, then ∆I is nothing
else than the entropy change of the system,
∆Ssystem = ∆I.
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© 2014 Macmillan Publishers Limited. All rights reserved.
(93)
Finally, the last term in Equ. (103) is the relative entropy of the distribution at time t
with respect to the steady state distribution at time 0,
Z
ρt (u)
D(ρt kρss ) = du ρt (u) ln
.
ρss (u)
(94)
Putting things together, one obtains
h∆Si = ∆Sbath + ∆Ssystem + D(ρt kρss ) = ∆Stotal + D(ρt kρss ),
(95)
where ∆Stotal = ∆Ssystem + ∆Sbath is the total entropy change of system and bath
together. The inequality that follows from the integral fluctuation theorem implies the
second law-like inequality
∆Stotal + D(ρt kρss ) ≥ 0.
(96)
An integral fluctuation theorem can be derived [3, 6] also for the quantity
R = β0 Q − ln
ρt (ut )
,
ρss (u0 )
(97)
which, in contrast to ∆S, depends also on the time propagated distribution ρt (ut ).
From the fluctuation theorem for R it follows that the average total entropy change is
non-negative, hStot i ≥ 0, providing a microscopic statement of the second law.
2.4
Relaxation from an initial equilibrium state
If the initial steady state ρss (u, α) is an equilibrium distribution e−βE(u) /Z(β) corresponding to the temperature T = 1 /kB β differing from the temperature T0 of the heat
bath, the expressions become particularly simple. In this case,
φ(u) = − ln
e−βE(u)
= βE(u) − βF (β)
Z(β)
(98)
such that
∆φ = β [E(ut ) − E(u0 )] = −βQ.
(99)
Hence, the relative entropy production is given by
∆S = β0 Q − βQ = (β0 − β)Q.
(100)
The fluctuation theorem for ∆S then becomes a fluctuation theorem for the heat Q
exchanged with the reservoir during the relaxation,
Pt (−Q)
= e−(β0 −β)Q ,
Pt (Q)
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(101)
as shown earlier by Jarzynski [7]. The integral fluctuation theorem then turns into
he−(β0 −β)Q i = 1,
(102)
Due to the convexity of the exponential function, this result implies that
(β0 − β)hQi ≥ 0.
(103)
Now, if the system is initially colder than then bath, i.e., β > β0 , then (β0 − β) < 0
and the above inequality implies that hQi ≤ 0, i.e., the system absorbs energy from the
bath. In other words, heat flows from hot to cold as expected from the second law of
thermodynamics.
2.5
Relaxation from a steady state generated by parametric
feedback
If the initial steady state ρss (u, α) is due to parametric feedback cooling, the total
effective “energy” is given by
H(u, α) = E(u) +
α 2
E (u).
4
(104)
Then, ∆φ = β0 ∆H and
∆S = β0
α 2
E (ut ) − E 2 (u0 ) .
4
(105)
In this case, the inequality following from the integral fluctuation theorem implies that
h∆E 2 i ≥ 0.
(106)
Thus, the average of the squared energy does not decrease during the relaxation process. Experimental results obtained for the relaxation from steady states generated by
parametric feedback are presented and discussed in the main paper.
2.6
Distributions of ∆S for t → ∞
We now consider the distribution Pt (∆S) of the quantity ∆S = (β0 α/4)(Et2 − E02 ).
Since ∆S is completely determined by E0 and Et , the distribution of ∆S can be written
as
Pt (∆S) =
ZZ
dE0 dEt P0 (E0 )P (Et |E0 )δ(∆S − ∆S(Et , E0 )),
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© 2014 Macmillan Publishers Limited. All rights reserved.
(107)
where P (Et |E0 ) is the conditional probability that the energy is Et at time t provided
it was E0 at time 0. In general, P (Et |E0 ) is unknown and to determine it one would
have to know the Green’s function of the SDE for the energy. But what can be done
easily is to compute Pt (∆S) for long times, i.e., in the limit t → ∞. In this case the
final energy, Et , is statistically independent from the initial energy, E0 , such that
(108)
P (Et |E0 ) = P∞ (Et ),
where P∞ (E) is the asymptotic distribution of the energy reached in the long time
limit. For the relaxation process after turning off the feedback, P∞ (E) is the equilibrium
distribution of the energy for temperature T0 . In this limit, the distribution is given by
ZZ
P∞ (∆S) =
dE0 dEt P0 (E0 )P∞ (Et )δ(∆S − ∆S(Et , E0 )).
(109)
To solve the integral, we transform variables from E to M ,
M=
αβ0 2
E .
4
(110)
Then, ∆S is given by
(111)
∆S = (Mt − M0 ).
The distributions of E and M are related by
p
dM −1
= P (E(M ))/ αβ0 M .
P (M ) = P (E) dE
The distributions of M with and without feedback are then given by
!
r
p
4β0 √
P0 (M ) = C0 exp −
M − M / αβ0 M
α
and
P∞ (M ) = C∞ exp −
r
4β0 √
M
α
!
p
/ αβ0 M .
Using the new variable M , the long time distribution of ∆S can be written as
ZZ
P∞ (∆S) =
dM0 dMt P0 (M0 )P∞ (Mt )δ[∆S − (Mt − M0 )].
(112)
(113)
(114)
(115)
Integration over M0 yields
P∞ (∆S) =
Z
dM P0 (M − ∆S)P∞ (M ).
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© 2014 Macmillan Publishers Limited. All rights reserved.
(116)
0
10
-5
10
η=0.001
η=0.010
η=0.100
η=1.000
-10
P(∆S)
10
-15
10
-20
10
-25
10
-30
10 -40
-20
0
∆S
20
40
Figure S4: Logarithm ln P (∆S) of the long time distribution ∆S for different feedback
strengths. The distributions were obtained by numerical integration of Equ. (116) using the
the distributions of Eqs. (113) and (114). The feedback strengths were η = 0.001, 0.010, 0.100
and 1.000 and for the other parameters were used kB T0 = 1, m = 1, k = 1, and Γ0 = 0.01.
The corresponding values of α were α = 0.1, 1, 10 and 100. For η = 0, the distribution is a
delta function centered at ∆S = 0. The dashed line indicates the distribution for η → ∞
(see Equ. (117)).
Note that we have defined the distributions of M such that they vanish for negative
M and the integration extends from −∞ to +∞. The integral of the above equation
cannot be calculated analytically, but we can determine the distribution P∞ (∆S) by
numerical integration to arbitrary precision using the distributions of Eqs. (113) and
(114). Some distributions of ∆S obtained in this way for various values of the feedback
strength η are shown in Fig. S4 and are in excellent agreement with the experimental
data presented in the main text..
In the limit of large η, the long time distribution of ∆S becomes:
P∞ (∆S) = Ce∆S/2 K0 (|∆S|/2),
(117)
where K0 is a modified Bessel function of the second kind. This limiting distribution is
shown in Fig. S4 as a dashed line. This distribution manifestly satisfies the fluctuation
theorem.
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© 2014 Macmillan Publishers Limited. All rights reserved.
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