Entropy production and time asymmetry in nonequilibrium fluctuations
D. Andrieux and P. Gaspard
arXiv:cond-mat/0703696v2 [cond-mat.stat-mech] 3 May 2007
Center for Nonlinear Phenomena and Complex Systems,
Université Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium
S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan
Laboratoire de Physique, CNRS UMR 5672, Ecole Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cédex 07, France
The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two
out-of-equilibrium systems namely, a Brownian particle in a trap moving at constant speed and an
electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium
fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of
dynamical systems theory. We present experimental results showing that their difference equals the
thermodynamic entropy production in units of Boltzmann’s constant.
PACS numbers: 05.70.Ln; 05.40.-a; 02.50.Ey
Newton’s equations ruling the motion of particles in
matter are known to be time-reversal symmetric. Yet,
macroscopic processes present irreversible behavior in
which entropy is produced according to the second law of
thermodynamics. Recent works suggest that this thermodynamic time asymmetry could be understood in terms
similar as those used for other symmetry breaking phenomena in condensed matter physics. The breaking of
time-reversal symmetry should concern the fluctuations
in systems driven out of equilibrium. These fluctuations
may be described in terms of the probabilities weighting
the different possible trajectories of the systems. Albeit
the time-reversal symmetry of the microscopic Newtonian dynamics says that each trajectory corresponds to
a time-reversed one, it turns out that distinct forward
and backward trajectories may have different probability
weights if the system is out of equilibrium. For example,
the probability for a driven Brownian particle of having
a trajectory from a point A to a point B is different of
having the same reverse trajectory from B to A.
This important observation can be further elaborated
to establish a connection with the entropy production.
We consider the paths or histories z = (z0 , z1 , z2 , ..., zn−1 )
obtained by sampling the trajectories z(t) at regular time
intervals τ . The probability weight of a typical path is
known to decay as
P+ (z0 , z1 , z2 , ..., zn−1 ) ∼ exp(−nτ h)
(1)
as the number n of time intervals increases [1, 2, 3, 4].
The decay rate h is called the entropy per unit time and it
characterizes the temporal disorder, i.e., dynamical randomness, in both deterministic dynamical systems and
stochastic processes [1, 2, 3, 4]. We can compare (1)
with the probability weight of the time-reversed path
zR = (zn−1 , ..., z2 , z1 , z0 ) in the nonequilibrium system
with reversed driving constraints (denoted by the minus
sign):
P− (zn−1 , ..., z2 , z1 , z0 ) ∼ exp(−nτ hR ) .
(2)
It can be shown that, out of equilibrium, the probabilities
of the time-reversed paths decay faster than the probabilities of the paths themselves [5]. We may interpret
this as a breaking of the time-reversal symmetry in the
invariant probability distribution describing the nonequilibrium steady state, the fundamental underlying Newtonian dynamics still being time-reversal symmetric. The
decay rate hR in Eq. (2) is called the time-reversed entropy per unit time and characterizes the dynamical randomness of the time-reversed paths [5, 6]. In the case
of Markovian stochastic processes with discrete fluctuating variables, the difference between both quantities hR
and h gives the entropy production of irreversible thermodynamics [5, 6, 7, 8]. A closely related result has
been obtained for the work dissipated in transient timedependent systems [9]. However, many experimental systems have continuous fluctuating variables and evolve in
nonequilibrium steady states. Therefore, we may wonder
how to measure dynamical randomness in such systems
of experimental interest and whether the time asymmetry of this property can be experimentally detected and
related to the thermodynamic entropy production.
In the present Letter, we provide experimental evidence for the aforementioned time asymmetry in two
nonequilibrium systems, namely, a driven Brownian motion and a fluctuating electric circuit. For this purpose, the decay rates h and hR are considered as socalled (ǫ, τ )-entropies per unit time, which characterize
dynamical randomness in continuous-variable stochastic
processes [4]. These entropies per unit time can be obtained by applying to the present stochastic systems a
method originally proposed for the study of deterministic
dynamical systems [1, 2, 3]. Thanks to this method, the
(ǫ, τ )-entropies per unit time of the paths and the corresponding time-reversed paths can be evaluated from two
long time series measured with sufficient temporal and
spatial resolutions, in two similar runs but one driven
with an opposite nonequilibrium constraint. The experiment thus consists in recording a pair of long time series
2
in each system. The dissipated heat and thermodynamic
entropy production are thus given by the difference between the two (ǫ, τ )-entropies per unit time.
The first system is a Brownian particle dragged by an
optical tweezer, which is composed by a large aperture
microscope objective (×63, 1.3) and by an infrared laser
beam with a wavelength of 980 nm and a power of 20
mW on the focal plane. The trapped polystyrene particle has a diameter of 2 µm and is suspended in a 20%
glycerol-water solution. The particle is trapped at 20 µm
from the bottom plate of the cell which is 200 µm thick.
The detection of the particle position xt is done using
a He-Ne laser and an interferometric technique [10]. In
order to apply a shear to the trapped particle, the cell
is moved with a feedback-controlled piezo which insures
a perfect linearity of displacement. The motion of the
dragged particle is overdamped and can be modeled as
the Langevin equation
α
dxt
= F (xt − ut) + ξ(t) ,
dt
(3)
where α is the viscous friction coefficient, F = −∂x V is
the force exerted by the potential V = kx2 /2 of the laser
trap moving at constant velocity u, and ξ(t) a Gaussian
white noise [11]. The stiffness of the potential is k =
9.62 10−6 kg s−2 . The relaxation time is τR = α/k =
3.05 10−3 s.
The second system is an electric circuit driven out of
equilibrium by a current source which imposes the mean
current I [12]. The current fluctuates in the circuit because of the intrinsic Nyquist thermal noise [11]. The
electric circuit is composed of a capacitance C = 278 pF
in parallel with a resistance R = 9.22 MΩ so that the
time constant of the circuit is τR = RC = 2.56 10−3 s.
This electric circuit and the dragged Brownian particle,
although physically different, are known to be formally
equivalent by the correspondence α ↔ R, k ↔ 1/C and
u ↔ I while the particle position xt corresponds to the
charge qt inside the resistor at time t [11, 12]. The variables xt and qt are acquired at a sampling frequency
1/τ = 8192 Hz.
In both experiments, the temperature is T = 298 K.
In order to fix the ideas, we describe our method in
the case of the dragged Brownian particle. The heat dissipated along a random trajectory during a time interval
t is given by [11, 13]
Z t
dxt′
(4)
F (xt′ − ut′ ) dt′ .
Qt =
′
dt
0
After a long enough time, the system reaches a nonequilibrium steady state, in which the entropy production is
related to the mean value of the dissipated heat according
to
1 dhQt i
αu2
di S
=
=
.
dt
T dt
T
(5)
Our aim is to show that one can extract the heat dissipated along a fluctuating path by comparing the probability of this path, with the one of the time-reversed
path having also reversed the displacement of the potential, i.e., u → −u. We first make the change to the
frame comoving with the minimum of the potential so
that z ≡ x − ut. After initial transients, the system will
reach a steady state characterized by a stationary probability distribution. As we are interested in the probability of a given succession of states corresponding to a
discretization of the signal at small time intervals τ , a
multi-time random variable is defined according to Z =
[Z(t0 ), Z(t0 + τ ), . . . , Z(t0 + nτ − τ )] which corresponds
to the signal during the time period t − t0 = nτ . For
a stationary process their distribution do not depend on
the initial time t0 . From the point of view of probability
theory, the process is defined by the n-time joint probabilities Pσ (z; dz, τ, n) = Pr{z < Z < z+dz; σ} = pσ (z)dz,
where pσ (z) is the probability density for Z to take the
value z = (z0 , z1 , . . . , zn−1 ) at times t0 + iτ for a nonequilibrium driving σ = u/|u| = ±1. Since the process is
Markovian, the joint probabilities can be decomposed
into the products of the Green functions G(zi , zi−1 ; τ )dzi
for i = 1, . . . , n. G(z, z0 ; t) gives the probability density
for the position to be z at time t given that the initial position was z0 [14, 15]. To extract the dissipation occurring
along a single trajectory, one has to look at the ratio of
the probability of the forward path over the probability
of the reversed path having also reversed the displacement of the potential. Indeed, taking the logarithm of
this ratio and the continuous limit τ → 0, n → ∞ with
nτ = t, we find
ln
P+ (z; dz, τ, n)
= βu
P− (zR ; dz, τ, n)
Z
0
t
h
i
F (zt′ ) dt′ −β V (zt )−V (z0 )
(6)
which is exactly the heat Qt in Eq. (4) expressed in
the z variable and multiplied by the inverse temperature
β = (kB T )−1 . We notice that, alone, the first term gives
the work exerted by the trap [13, 16].
Relations similar to Eq. (6) have been obtained for the
distribution of the work done on a time-dependent system [17, 18] and for Boltzmann’s entropy production [19].
We emphasize that Eq. (6) also holds for anharmonic potentials V and that the reversal of u is essential to get
the dissipated heat from the way the path probabilities
P+ and P− differ.
Now, due to the continuous nature in time and in
space of the process, one has to consider (ǫ, τ ) quantities,
i.e. quantities defined on cells of size ǫ and measured at
time intervals τ . Therefore, we introduce the probability
P+ (Zm ; ǫ, τ, n) for the path to remain within a distance
ǫ of some reference path Zm , made of n successive positions of the Brownian particle observed at time intervals
τ for the forward process. The probability is obtained by
searching for the recurrences of M such reference paths
3
20
100
(a)
15
z (nm)
H(ε,τ,n)
50
10
5
0
0
0
−50
0.001
0.002
nτ (s)
0.003
0.004
0.001
0.002
nτ (s)
0.003
0.004
0.001
0.002
nτ (s)
0.003
0.004
14
(b)
12
0.9
1
0.95
H R(ε,τ,n)
10
−100
1.05
time (s)
8
6
4
2
FIG. 1: Time series of typical paths z(t) for the Brownian
particle in the optical trap moving at the velocity u for the
forward process (upper curve) and −u for the reversed process
(lower curve) with u = 4.24 10−6 m/s.
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.2
(c)
1.1
H(ǫ, τ, n) = −
M
1 X
ln P+ (Zm ; ǫ, τ, n)
M m=1
(7)
H R−H
1
or patterns in the time series. Next, we also introduce
the probability P− (ZR
m ; ǫ, τ, n) for a reversed path of the
reversed process to remain within a distance ǫ of the reference path Zm (of the forward process) during n successive
positions. According to a numerical procedure proposed
by Grassberger, Procaccia and others [1, 2] the entropy
per unit time can be estimated by the linear growth of
the mean ‘pattern entropy’ defined as
0.9
0.8
0.7
0.6
0
FIG. 2: (a) Entropy production of the Brownian particle
versus the driving speed u. The solid line is given by Eq.
(5). (b) Entropy production of the RC electric circuit versus the injected current I. The solid line is the Joule law,
di S/dt = RI 2 /T . The dots are the results of Eq. (9).
By similarity, we introduce
H R (ǫ, τ, n) = −
M
1 X
ln P− (ZR
m ; ǫ, τ, n)
M m=1
(8)
for the reversed process. The (ǫ,τ )-entropies per unit
time, h(ǫ, τ ) and hR (ǫ, τ ), are defined by the linear
growth of these mean pattern entropies as a function of
the time nτ [1, 2, 4]. In the nonequilibrium steady state,
the thermodynamic entropy production should thus be
given by the difference between these two quantities:
1 di S
= lim lim [hR (ǫ, τ ) − h(ǫ, τ )] .
ǫ→0 τ →0
kB dt
(9)
It is important to note that the probabilities of the reversed paths are averaged over the paths of the forward
process in order for Eq. (9) to hold. The entropy production is thus expressed as the difference of two usually
very large quantities which increase with the scaling law
ǫ−2 for ǫ, τ going to zero [4, 20]. Nevertheless, their difference remains finite and gives the entropy production in
terms of the time asymmetry of the dynamical randomness characterized by the (ǫ,τ )-entropies per unit time.
In order to test experimentally that entropy production is related to this time asymmetry according to Eq.
(9), we have analyzed for specific values of |u| or |I|, a pair
of time series up to 2 107 points each, one corresponding
to the forward process and the other corresponding to
the reversed process, having first discarded the transient
evolution. Figure 1 depicts examples of paths z(t) for the
Brownian particle in a moving optical trap.
For different values of ǫ between 5.6-11.2 nm [21], the
mean pattern entropy (7) is calculated with the distance
defined by taking the maximum among the deviations
|Z(t) − Zm (t)| with respect to some reference path Zm
for the times t = 0, τ, . . . , (n − 1)τ . The forward entropy
per unit time h(ǫ, τ ) is evaluated from the linear growth
of the mean pattern entropy (7) with the time nτ . The
backward entropy per unit time hR (ǫ, τ ) is obtained similarly from the time-reversed pattern entropy (8). The
difference of the two dynamical entropies is depicted as
in Fig. 2a. The good agreement with the entropy production (5) is the experimental evidence that this latter
is indeed related to the time asymmetry of dynamical
randomness as predicted by Eq. (9).
4
In conclusion, we measured the entropy production by
searching the recurrences of trajectories in the fluctuating dynamics of two nonequilibrium processes. The experiments we performed consisted in the recording of two
long time series. The first one corresponds to a forward
experiment while the other is measured from the same
experimental setup except that the sign of the constraint
driving the system out of equilibrium has been reversed.
From these two time series, we are able to compute two
dynamical entropies, the difference of which gives the entropy production. Moreover, we tested the possibility to
extract the dissipated heat along a single random path.
This shows that the entropy production arises from the
breaking of the time-reversal symmetry in the probability
distribution of the statistical description of the nonequilibrium steady state. Since the decay rates of the multitime probabilities of the forward and reversed paths characterize their dynamical randomness, the present results
show that the thermodynamic entropy production finds
its origin in the time asymmetry of the dynamical randomness.
Acknowledgments. This research is financially supported by the F .N .R .S . Belgium and the “Communauté
française de Belgique” (contract “Actions de Recherche
Concertées” No. 04/09-312).
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diS/dt (k BT/s)
We also tested the possibility to extract the heat (4)
dissipated along a single stochastic path by searching for
the recurrences in the time series according to Eq. (6).
A randomly selected path as well as the corresponding
heat dissipated are plotted in Fig. 3. We find a very
good agreement so that the relation (6) is also verified for
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150
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u (µm/s)
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(b)
200
diS/dt (k BT/s)
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in Fig. 2b that the entropy production obtained from
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with the known Joule law, which is a further confirmation
of Eq. (9).
150
100
50
0
0
0.1
0.2
0.3
I (pA)
FIG. 3: Measure of the heat dissipated by the Brownian particle along the forward and reversed paths of Fig. 1. The trap
velocities are ±u with u = 4.24 10−6 m/s. We are searching for recurrences between the two processes. (a) Inset: A
randomly selected trajectory in the time series. The probabilities of the corresponding forward (filled circles) and the
backward (open circles) paths for ǫ = 8.4 nm. These probabilities present an exponential decrease modulated by the
fluctuations. (b) The dissipated heat given by the logarithm
of the ratio of the forward and backward probabilities according to Eq. (6) for different values of ǫ = k × 0.558 nm with
k = 11, . . . , 20 in the range 6.1-11.2 nm. They are compared
with the value (squares) directly calculated from Eq. (4). For
small values of ǫ, the agreement is quite good for short time
and within experimental errors for larger time.
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[21] We notice that the statistics is not sufficient for smaller
values of ǫ, while the graining is too coarse for larger
values.