It took five years after the numerical and theoretical observation of the fluc- tuation theorem [43] before the first experiments [21] in turbulent flows were performed to check the validity of the FT. After this experiment, things were very quiet for another four years, until a new set of experiments [126] gave the start for an extensive investigation. Today, the experimental verification of the fluctuation theorem is the core business of several research groups. We discuss some of the experiments that have been performed by them.
A. Gallavotti-Cohen fluctuation theorem in turbulent flows
The very first experiment in 1998 from Ciliberto and coworkers [21] involved the study of the heat flux in an open turbulent flow. They considered a wind tunnel where air was injected through a jet nozzle. A thin plaque was placed in the turbulent flow and the (probability density of the) force on this plaque was found to satisfy a fluctuation theorem.
Recently, a similar experiment was repeated by the same group [20], but this time in the more controlled environment of a closed turbulent flow. We discuss this experiment in detail.
The experimental set-up consists of two discs of radius 20 cm, that rotate in opposite directions at a fixed frequency of 40 Hz, see figure 6.1. This device is well known for producing an intense turbulence of the air between the discs in a compact region of space. A square plate, of side 4 cm and thickness 1 mm, is inserted between the two disks, with its center at 6 cm from the disks rotation axis. The plate is mounted on the end of a 10 cm lever, and at the other end a force detection mechanism is placed with an accuracy of 1 mN. The quantity that is monitored is the total force on the square plate, which is obtained by integrating the pressurep(x, t) over the areaS of the object
F(t) = Z
S
dx p(x, t). The pressure itself is a solution of the Poisson equation3
∆p(x, t) =ρ∇ ·u(x, t)· ∇ ·u(x, t),
3This equation can be derived from the steady-state incompressible Navier-Stokes equa-
tions
(~
∇p+ρ ~u·(∇ ·~ u~)−µ∆~u=f ,~ ~
∇ ·~u= 0,
whereµis the viscosity andf~is the external force. Applying the divergence operator∇~ to the first equation yields the Poisson pressure equation.
2R H Force sensor Pressure measuring surface ω ω
Figure 6.1: The experimental set-up of Ciliberto [20]. Two discs with radiusR= 20 cm are separated withH= 20 cm. They are rotated in opposite directions at a fixed frequency
ω= 40 Hz.
where ρis the fluid density and u(x, t) is the velocity field of the fluid. The time fluctuations of the measured force F(t) on the plaque are related to the small scale turbulent fluctuations in the velocity profile of the flow.
The observable of interest is the time-integrated normalized force Yτ = 1 τ Z τ 0 dsF(s) F0 ,
where F0 is the mean value of the force on the object, depending on the
observation timeτ. In the experiment, it is established that the fluctuations of this force are strongly non-Gaussian and that for sufficiently long observation timesτ, the probability density satisfies the fluctuation theorem
ln Prob[Yτ =y] Prob[Yτ=−y]
= (Aτ+B)y, y∈[0,0.5].
They observe that as the observation time τ grows, the prefactor (Aτ +B) in the right hand side of the above equation converges to a Cτ where C is a constant.
B. Dragging particles through a fluid in a harmonic optical trap
The first experiment that measured the work was performed by Wang and coworkers [126]. The experiment involved a latex particle that is captured in a harmonic potential (experimentally realized by an optical trap) and dragging
v
Fluid
Harmonic
potential opt
Figure 6.2: The experimental set-up of Wang [126]. A latex particle that is captured in an optical trap, is dragged through a fluid with a constant velocityvopt.
it through a fluid by moving the potential at a constant velocity, see figure 6.2. The total workWτ that is performed on the particle is mostly dissipated to
the environment in the form of heatQτ, and some of it is stored in the form
of potential energy.
Fluid
Intensity
low high
Figure 6.3: The working principle of the optical trap: the particle has a higher refractive index than the medium so it bends the incoming light rays more than the medium does. More particle rays are bent in regions with higher light intensity (in the figure on the right side). The net change of momentum in the light rays is towards the left. Due to conservation of momentum, the particle is drawn towards the right, i.e., regions of higher intensity.
The particle has a diameter of 6.3 µm, and is placed in a focused, infrared laser beam (λ= 980 nm,Popt= 1 W). Because of a difference between indices
of refraction between particle and fluid, see figure 6.3, the particle is pulled towards the region with the highest light intensity by a force of the order of picoNewtons. The optical trap is harmonic near the focal point so the optical
force on the particle is
Fopt(t) =−k x(t)−voptt, (6.15)
wherex(t) is the position of the particle, measured with respect to the bottom of the harmonic potential well. The trapping strength isk= 0.1 pN /µm, the constant velocity of translation isvopt= 1.6µm / s.
The quantity of interest is the total (dimensionless) work done on the particle Στ= 1 kBT Z τ 0 dt voptFopt(t).
Due to the limited number of sampled trajectories, they did not check an EFT directly, but looked at the integrated fluctuation theorem (6.13). Their ex- periment confirmed this relation for Στ for sampling times up to τ= 2 s and
values of Στ∈[−1,1].
In a second experiment [18] by the same group, the potential was no longer dragged, but rather the strength of the optical trap kin equation (6.15) was instantly switched from a value k0 to a value k1 in the middle of the experi-
ment. A trajectory now consists of the particle moving freely for 10 seconds in a weak trap with trap strengthk0= 1.22 pN /µm, and then for 10 seconds in
a stronger trap4 with k
1 ≈2.80 pN /µm. This experiment allowed to probe
the exact fluctuation theorem (6.12) for Στ, and again it was confirmed for
sampling times up toτ = 200 ms and for values of Στ ∈[−3,3]. For consis-
tency, also the integrated fluctuation theorem (6.13) was checked and it was found to be verified.
Interestingly, they claimed that the quantity Στ they were measuring was the
entropy production (as in the first theoretical results), but further analysis [78, 122] showed that what they were really measuring was the work done on the particle. This was the motivation for van Zon and Cohen [122] to study the model theoretically.
C. Fluctuations in electrical circuits
Shortly after the publication of the extended fluctuation theorem [121, 122], the first experimental studies of fluctuation theorems for the heat [50] came in. Ciliberto and coworkers studied the fluctuations of the energy injected into, and dissipated from an electrical circuit. Their experiment was supported by theoretical calculations from an earlier paper [120].
4We writek
1≈2.80 pN /µm, but in fact the new trap strength was anisotropic with a
strengthkx
1 = 2.70 pN /µm in thex-direction andk
y
1 = 2.90 pN /µm in they-direction.
The particle was only free to move in two dimensions as the experiment took place under a microscope.
R C U I
Figure 6.4: The circuit studied by Ciliberto [50]. Fluctuations in the voltageUare mea- sured because of thermal fluctuations of charge positions. The system is driven away from equilibrium by imposing a constant flux of electrons in the form of the currentI.
They studied a simple electrical RC-circuit, composed of a resistor R and a capacitor C, as depicted in figure 6.4. The resistance has a nominal value R= 9.52 MΩ and the capacitance has a valueC= 280 pF. The time-constant of the system is τ0 = RC = 2.67 ms, which sets the timescale for further
measurements. A constant current ofI= 6·10−13A is injected in the circuit,
so the (average) injected power isU I =IR2
≈1000kBT/s as the experiment
is conducted at room temperature. For the typical experimental time scales of τ ∼10τ0, the total injected energy is of the order of 10 kBT, which explains
the vulnerability of this set-up to thermal fluctuations. These thermal fluc- tuations in the charge positions lead to fluctuations of the measured voltageU. The observables of interest are the total injected energy in the system, i.e., the work done on the system:
Wτ =
Z τ 0
dt Pin(t), Pin(t) =I U(t),
and the total energy dissipated by the resistor due to the Joule effect, i.e., the heat:
Qτ =
Z τ 0
dt Pdiss(t), Pdiss(t) =iR(t)U(t).
The dissipated heat is not immediately equal to the injected energy since the current iR through the resistor is time-dependent (the total current I is
constant however). From Kirchoff’s laws, we get I=iR+C dU dt , so Pin=Pdiss+ 1 2C dU2 dt ,
which allows the calculation of both Pin and Pdiss as a function of time.
For sufficiently long times with respect to τ0, one expects of course that hQτi = hWτi = τ RI2. We write the mean heat production per unit time
as q=hQτi/τ.
The probability distributions for the work and heat are measured in the exper- iment. The main observation is that fluctuations in the work have a Gaussian distribution for any value ofτ ∈[0.45τ0,100τ0], while for the heat, this is only
for values ofτ larger than approximately 5τ0. For small values of τ, the heat
Qτ has an exponential distribution, so in this temporal regime, large fluctua-
tions of the heat are more likely to occur than large fluctuations of the work Wτ.
This experiment verifies the EFT for the work for all timesτ and values of the workWτ∈[0,5hWτi], in the sense that
kBT
τ ln
Prob[Wτ =wτ]
Prob[Wτ =−wτ]
=Cτw,
where the slopeCτ converges to one asτ→ ∞.
The dissipated heat also verifies the extended fluctuation theorem as predicted by references [120, 121, 123]. The quantity
f(q) = kBT τ ln
Prob[Qτ =qτ]
Prob[Qτ=−qτ]
is only linear for values q < q and saturates to the value 2 as q > 3q and τ→ ∞, see equation (6.14).
D. Beyond Gaussian work: driven two-level systems
The first experiments that go beyond Gaussian distributions of the work, were performed by the Stuttgart group [103]. They considered a two-level system, experimentally realized with a single defect in a diamond. This diamond was externally driven by a combination of a green (λ= 514 nm) and a red (λ= 680 nm) laser.
The diamond with a nickel-based defect can be viewed as two individual two- level systems (TLS). The first TLS is excited by the red laser, see figure 6.5(a). This excited stateE1 decays to the ground stateE0 with a transition rateα.
The second TLS is excited to E2 by the green laser and decays back to the
ground stateE0 with rate β. The two transitions αandβ occur in nanosec-
onds and are not resolved in the experiment. However, for sufficiently large intensities of both lasers, the excited state E1 of the first TLS can decay to
the ground state of the second TLS, where it can be excited to E2 again by
the green laser and on its turn decay to the ground state of the first TLS. This cycle can be viewed as a single two-level system as seen in figure 6.5(b), where the excitation by the lasers represents the workW done on the system, and the decay to the ground state as the heatQdissipated to the environment. The system is driven by varying the intensity Ig of the green laser asIg(t) =
I0+Imλ(t), with external protocolλ(t) and modulation intensity Im. In this
experiment, the system was driven harmonically with λ(t) = sin(ωt). After every timestep ∆t, the state of the two-level system ωi ∈ {0,1} is written
red green (a) (b) W Q E0 E1 E0 E2 E1 E2 α β
Figure 6.5: The Jablonski diagram for the two-level system considered by the Stuttgart group [103]. The two individual two-level systems from the diamond with defect can be viewed as a single two-level system at time scales larger than the nanosecond transitionsα
andβ.
n=τ /∆t measured states of the two-level system. The quantity of interest is the dissipated work
Wτdis =
Z τ 0
dtλ(t)˙ ∂U ∂λ,
where the function U(λ) describes roughly how the energy of the system changes under influence of the modulation of the green laser. They observed that the dissipated workWdis
τ displays non-Gaussian fluctuations in the case
of short trajectories and fast modulation. Furthermore, they observed that the dissipated work satisfies an integrated fluctuation theorem. It was noticed that integrated versions of the fluctuation theorem in their experiment are ob- served only for particular protocolsλ(t), which satisfy a symmetry condition λ(t) =λ(τ−t) for allt∈[0, τ].
E. Dragging particles through a fluid in a nonharmonic optical trap
In the more recent paper [9] from the Stuttgart group, the distribution of the work performed on a particle was computed for a non-harmonic potential. A highly charged polystyrene bead with radius R = 2 µm is immersed in an aqueous solution which is put on top of a glass plate. The particle is made visible by reflecting a red laser (λ= 658 nm) on the bottom of the glass plate, see figure 6.6. An infrared optical trap (λ= 1064 nm, Popt= 2 mW) confines
the particle’s motion to one dimension (up or down in the laser light of the optical trap). To drive the particle, an additional green laser (λ= 532 nm, Popt 6 60 mW) functions as a set of optical tweezers. The laser intensity
(or trapping/tweezing strength) is modulated according to a symmetric time- dependent protocol I(t) = I(τ0−t), where τ0 is the total time required to
Fluid Infrared laser restricts movement to x-direction Red laser illuminates particle
Green laser traps particle
x
Figure 6.6: The optical trap used by the Stuttgart experiment [9]. The infrared laser confines the motion along the light rays of the laser, and the green laser’s change in intensity varies the trapping strength of the particle. The particle is illuminated by scattering a red laser of the glass surface that bears the fluid.
The total potential that the particle is subjected to, is now of the form: U(x, t) =A0exp(−κx) +B0x+C0I(t)x.
The first term describes the interaction of the charged particle with the charged glass plate: A0 depends on the surface charge of the glass plate andκ≈0.04
nm−1 is the Debye screening length, which depends on the concentrations of
ions5 in the solution. The second term describes the light pressure from the
infrared optical trap, which depends linearly on the particle’s positionx. The last term considers the time-dependent optical forces induced by the tweezers. The variables of interest are the total work done on the particle
Wτ = Z τ 0 dtI(t)˙ ∂U ∂I = Z τ 0 dt∂U ∂t, and the total heat dissipated by the particle
Qτ =
Z τ 0
dtx(t)˙ ∂U ∂x.
For W and Q, maximal energies of about 15kBT were observed, which sets
the energy scale of the experiment. For a particular time-symmetric protocol I(t), sketched in figure 6.7, with protocol cycle time τ0 = 1 s, the measured
work distribution is strongly non-Gaussian. It satisfies the EFT, ln Prob[Wτ =wτ]
Prob[Wτ =−wτ] =βwτ,
for values of Wτ∈[−4kBT,4kBT] andτ≈100τ0.
5These ions screen the Coulomb potential of the charged glass plate. This gives rise to
t I(t)
ts tp
τ0
Figure 6.7:The time-symmetric protocolI(t) =I(τ0−t) used in the Stuttgart experiment.
One cycle of the protocol consists of two pulses of durationts, separated in time with length
tp. The pulse duration timets= 120 ms and the waiting timetp= 700 ms. The cycle time
for the protocol isτ0= 1000 ms. No explicit formula forI(t) is given in the paper [9].
6.3
Results
We start with the exact fluctuation theorem for the work (6.9). First, we consider the harmonic case with U(x) =x2/2 in equation (6.6), for a general
protocolγt. Then, we give a general condition under which the work satisfies
an EFT, and we give instances under which that condition is satisfied. Counter examples (for which the work does not satisfy an EFT) will show why these conditions are close to optimal. We end with a discussion on the relevance of temporal boundary terms in the large deviations of the heat (6.11). For the proofs, we refer to section 6.5.